gdz.top
Номер 628, страница 182 - гдз по алгебре 9 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2019 - 2026
ISBN: 978-601-317-424-2
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 9 классе
IV. Тригонометрия. 23. Тригонометрические функции и их свойства - номер 628, страница 182.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108363 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-628" "field_display_title" => "628" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 #items: array:1 [ 0 => App\Models\Term {#1042 …30} ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ 0 => App\Models\Term {#1044 …30} ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ 0 => App\Models\Term {#1046 …30} ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1047 #items: array:3 [ 0 => App\Models\Term {#1048 …30} 1 => App\Models\Term {#1049 …30} 2 => App\Models\Term {#1050 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ 0 => App\Models\Term {#1053 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:1 [ 0 => App\Models\Term {#1057 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041} "field_class" => Illuminate\Database\Eloquent\Collection {#1043} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045} "field_author" => Illuminate\Database\Eloquent\Collection {#1047} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052} "field_country" => Illuminate\Database\Eloquent\Collection {#1054} "field_city" => Illuminate\Database\Eloquent\Collection {#1056} "field_series" => Illuminate\Database\Eloquent\Collection {#1058} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:2 [ 0 => App\Models\Branch {#1112 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1113 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:1 [ 0 => App\Models\Book {#1114 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Term {#1115 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Term {#1118 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Term {#1120 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "Kokshetau" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:3 [ 0 => App\Models\Term {#1122 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1123 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1124 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1125 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\Term {#1127 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Term {#1129 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Term {#1131 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1132 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1133 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1134 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1135 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1136 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1137 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1138 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1140 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1141 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1116} "field_class" => Illuminate\Database\Eloquent\Collection {#1117} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1119} "field_author" => Illuminate\Database\Eloquent\Collection {#1121} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1125} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1126} "field_country" => Illuminate\Database\Eloquent\Collection {#1128} "field_city" => Illuminate\Database\Eloquent\Collection {#1130} "field_series" => Illuminate\Database\Eloquent\Collection {#1132} "field_umk" => Illuminate\Database\Eloquent\Collection {#1133} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1134} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1135} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1136} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1137} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1138} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1139} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1140} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1141} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1142 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1113} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1144} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1142} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1143 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "23. Тригонометрические функции и их свойства" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "172" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Book {#1114} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1147 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "23. Тригонометрические функции и их свойства" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1145} "field_page_start" => "172" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1146} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1147} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1101 #items: array:3 [ 0 => App\Models\Element {#1094 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1276983 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Edition {#1093 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ 0 => App\Models\Term {#1092 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1091} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1089} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108363" "type" => "task" ] "text" => "<p><strong>628.</strong> Используя свойства функций $y = \sin x$ и $y = \cos x$, докажите, что:</p><p><strong>a)</strong> $\sin 136^\circ \cdot \sin 137^\circ \cdot \sin 138^\circ > \frac{1}{3};$</p><p><strong>б)</strong> $\cos 55^\circ \cdot \cos 57^\circ \cdot \cos 59^\circ > \frac{1}{9}.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "628-1.jpg" "alt" => null "width" => "1579" "height" => 335 "path" => "/media/algebra_09/soltan-u19/0-1/628-1.webp?ts=1752834644" ] ] ] #original: array:7 [ "id" => 1276983 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086} "task" => array:2 [ "refs" => "1108363" "type" => "task" ] "text" => "<p><strong>628.</strong> Используя свойства функций $y = \sin x$ и $y = \cos x$, докажите, что:</p><p><strong>a)</strong> $\sin 136^\circ \cdot \sin 137^\circ \cdot \sin 138^\circ > \frac{1}{3};$</p><p><strong>б)</strong> $\cos 55^\circ \cdot \cos 57^\circ \cdot \cos 59^\circ > \frac{1}{9}.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "628-1.jpg" "alt" => null "width" => "1579" "height" => 335 "path" => "/media/algebra_09/soltan-u19/0-1/628-1.webp?ts=1752834644" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1088 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1278274 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\Edition {#1085 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ 0 => App\Models\Term {#1084 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1080} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1083} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1081} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108363" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "628-1.jpg" "alt" => null "width" => "1369" "height" => 2308 "path" => "/media/algebra_09/soltan-u19/1-1/628-1.webp?ts=1752831420" ] 1 => array:5 [ "name" => "628-2.jpg" "alt" => null "width" => "745" "height" => 402 "path" => "/media/algebra_09/soltan-u19/1-1/628-2.webp?ts=1752831420" ] ] ] #original: array:6 [ "id" => 1278274 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1076} "task" => array:2 [ "refs" => "1108363" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "628-1.jpg" "alt" => null "width" => "1369" "height" => 2308 "path" => "/media/algebra_09/soltan-u19/1-1/628-1.webp?ts=1752831420" ] 1 => array:5 [ "name" => "628-2.jpg" "alt" => null "width" => "745" "height" => 402 "path" => "/media/algebra_09/soltan-u19/1-1/628-2.webp?ts=1752831420" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1078 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1554229 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Edition {#1079 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1074} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1075} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1072} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108363" "type" => "task" ] "text" => "<p><strong>a)</strong> Требуется доказать неравенство $ \sin 136^\circ \cdot \sin 137^\circ \cdot \sin 138^\circ > \frac{1}{3} $.<br>Воспользуемся свойством функции синус, а именно формулой приведения $ \sin(180^\circ - x) = \sin x $. Углы $136^\circ, 137^\circ, 138^\circ$ находятся во второй четверти, где синус положителен.<br>$ \sin 136^\circ = \sin(180^\circ - 136^\circ) = \sin 44^\circ $<br>$ \sin 137^\circ = \sin(180^\circ - 137^\circ) = \sin 43^\circ $<br>$ \sin 138^\circ = \sin(180^\circ - 138^\circ) = \sin 42^\circ $<br>Таким образом, неравенство эквивалентно следующему:<br>$ \sin 42^\circ \cdot \sin 43^\circ \cdot \sin 44^\circ > \frac{1}{3} $<br>Проанализируем левую часть. Обозначим произведение $ P = \sin 42^\circ \cdot \sin 43^\circ \cdot \sin 44^\circ $.<br>Можно заметить, что углы представляют собой арифметическую прогрессию с центральным членом $ x = 43^\circ $ и разностью $ d = 1^\circ $.<br>Воспользуемся формулой произведения синусов: $ \sin(x-d)\sin(x+d) = \sin^2 x - \sin^2 d $.<br>Тогда $ \sin 42^\circ \cdot \sin 44^\circ = \sin(43^\circ - 1^\circ)\sin(43^\circ + 1^\circ) = \sin^2 43^\circ - \sin^2 1^\circ $.<br>Следовательно, произведение $ P = (\sin^2 43^\circ - \sin^2 1^\circ) \cdot \sin 43^\circ = (\sin 43^\circ)^3 - \sin 43^\circ \cdot \sin^2 1^\circ $.<br>Так как $ \sin 43^\circ > 0 $ и $ \sin^2 1^\circ > 0 $, то $ P < (\sin 43^\circ)^3 $.<br>Теперь сравним $ (\sin 43^\circ)^3 $ с $ \frac{1}{3} $. Это эквивалентно сравнению $ \sin 43^\circ $ с $ \sqrt[3]{\frac{1}{3}} $.<br>Функция $ y = \sin x $ является вогнутой на интервале $ (0^\circ, 180^\circ) $. Это означает, что ее график лежит ниже любой касательной. Проведем касательную к графику в точке $ x_0 = 45^\circ $.<br>Уравнение касательной: $ y = \sin 45^\circ + (\cos 45^\circ)(x - \frac{\pi}{4}) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}(x - \frac{\pi}{4}) $.<br>Для любого $ x $ из интервала, $ \sin x \le \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}(x - \frac{\pi}{4}) $.<br>Представим $ 43^\circ $ в радианах: $ 43^\circ = 43 \frac{\pi}{180} = \frac{\pi}{4} - \frac{2\pi}{180} = \frac{\pi}{4} - \frac{\pi}{90} $.<br>$ \sin 43^\circ \le \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}(\frac{\pi}{4} - \frac{\pi}{90} - \frac{\pi}{4}) = \frac{\sqrt{2}}{2}(1 - \frac{\pi}{90}) $.<br>Используя приближенные значения $ \sqrt{2} \approx 1.4142 $ и $ \pi \approx 3.1416 $, получаем:<br>$ \frac{\pi}{90} \approx \frac{3.1416}{90} \approx 0.0349 $<br>$ \sin 43^\circ \le \frac{1.4142}{2}(1 - 0.0349) = 0.7071 \cdot 0.9651 \approx 0.6824 $.<br>Теперь оценим $ \sqrt[3]{\frac{1}{3}} $: $ \sqrt[3]{\frac{1}{3}} = \frac{1}{\sqrt[3]{3}} $. Так как $ 1.44^3 \approx 2.986 < 3 $, то $ 1.44 < \sqrt[3]{3} $. $ \frac{1}{\sqrt[3]{3}} > \frac{1}{1.45} \approx 0.689 $. Точное значение $ \sqrt[3]{1/3} \approx 0.6933 $.<br>Таким образом, $ \sin 43^\circ \approx 0.6824 < 0.6933 \approx \sqrt[3]{\frac{1}{3}} $.<br>Возведя обе части в куб, получаем $ (\sin 43^\circ)^3 < \frac{1}{3} $.<br>Так как $ P < (\sin 43^\circ)^3 $, то $ P < \frac{1}{3} $.<br>Это означает, что исходное неравенство $ \sin 136^\circ \cdot \sin 137^\circ \cdot \sin 138^\circ > \frac{1}{3} $ неверно. Вероятно, в условии задачи имеется опечатка, и знак неравенства должен быть противоположным.<br>Доказательство неравенства $ \sin 136^\circ \cdot \sin 137^\circ \cdot \sin 138^\circ < \frac{1}{3} $ приведено выше.<br>Ответ: Утверждение в задаче неверно. Было доказано, что $ \sin 136^\circ \cdot \sin 137^\circ \cdot \sin 138^\circ < \frac{1}{3} $.</p><p><strong>б)</strong> Требуется доказать неравенство $ \cos 55^\circ \cdot \cos 57^\circ \cdot \cos 59^\circ > \frac{1}{9} $.<br>Все углы находятся в первой четверти, где косинус положителен. Функция $ y = \cos x $ на интервале $ (0^\circ, 90^\circ) $ является убывающей и вогнутой. Свойство вогнутости означает, что график функции лежит выше любой хорды (секущей), соединяющей две точки графика.<br>Рассмотрим хорду, соединяющую точки графика $ y = \cos x $ при $ x_1 = 45^\circ $ и $ x_2 = 60^\circ $.<br>$ A(45^\circ, \cos 45^\circ) = (45^\circ, \frac{\sqrt{2}}{2}) $<br>$ B(60^\circ, \cos 60^\circ) = (60^\circ, \frac{1}{2}) $<br>Уравнение прямой, проходящей через точки A и B, имеет вид: $ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) $.<br>Для $ x \in (45^\circ, 60^\circ) $ выполняется неравенство $ \cos x > y_{line}(x) $.<br>$ y_{line}(x) = \frac{1}{2} + \frac{\frac{\sqrt{2}}{2} - \frac{1}{2}}{45-60}(x - 60) = \frac{1}{2} + \frac{\sqrt{2}-1}{-30}(x-60) = \frac{1}{2} + \frac{1-\sqrt{2}}{30}(x-60) $.<br>Все углы $ 55^\circ, 57^\circ, 59^\circ $ лежат в интервале $ (45^\circ, 60^\circ) $. Применим неравенство для каждого из них:<br>$ \cos 55^\circ > \frac{1}{2} + \frac{1-\sqrt{2}}{30}(55-60) = \frac{1}{2} - \frac{5(1-\sqrt{2})}{30} = \frac{1}{2} - \frac{1-\sqrt{2}}{6} = \frac{3-1+\sqrt{2}}{6} = \frac{2+\sqrt{2}}{6} $.<br>$ \cos 57^\circ > \frac{1}{2} + \frac{1-\sqrt{2}}{30}(57-60) = \frac{1}{2} - \frac{3(1-\sqrt{2})}{30} = \frac{1}{2} - \frac{1-\sqrt{2}}{10} = \frac{5-1+\sqrt{2}}{10} = \frac{4+\sqrt{2}}{10} $.<br>$ \cos 59^\circ > \frac{1}{2} + \frac{1-\sqrt{2}}{30}(59-60) = \frac{1}{2} - \frac{1-\sqrt{2}}{30} = \frac{15-1+\sqrt{2}}{30} = \frac{14+\sqrt{2}}{30} $.<br>Перемножим полученные неравенства (все части положительны):<br>$ \cos 55^\circ \cdot \cos 57^\circ \cdot \cos 59^\circ > \frac{2+\sqrt{2}}{6} \cdot \frac{4+\sqrt{2}}{10} \cdot \frac{14+\sqrt{2}}{30} $.<br>Оценим правую часть:<br>$ \frac{(2+\sqrt{2})(4+\sqrt{2})(14+\sqrt{2})}{1800} = \frac{(8+2\sqrt{2}+4\sqrt{2}+2)(14+\sqrt{2})}{1800} = \frac{(10+6\sqrt{2})(14+\sqrt{2})}{1800} $<br>$ = \frac{140+10\sqrt{2}+84\sqrt{2}+12}{1800} = \frac{152+94\sqrt{2}}{1800} = \frac{76+47\sqrt{2}}{900} $.<br>Теперь докажем, что $ \frac{76+47\sqrt{2}}{900} > \frac{1}{9} $.<br>$ \frac{76+47\sqrt{2}}{900} > \frac{100}{900} $<br>$ 76+47\sqrt{2} > 100 $<br>$ 47\sqrt{2} > 24 $<br>Возведем обе части в квадрат (они положительны):<br>$ (47\sqrt{2})^2 > 24^2 $<br>$ 47^2 \cdot 2 > 576 $<br>$ 2209 \cdot 2 > 576 $<br>$ 4418 > 576 $<br>Это неравенство очевидно истинно. Следовательно, исходное неравенство также истинно.<br>Ответ: Неравенство доказано.</p>" ] #original: array:6 [ "id" => 1554229 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "1108363" "type" => "task" ] "text" => "<p><strong>a)</strong> Требуется доказать неравенство $ \sin 136^\circ \cdot \sin 137^\circ \cdot \sin 138^\circ > \frac{1}{3} $.<br>Воспользуемся свойством функции синус, а именно формулой приведения $ \sin(180^\circ - x) = \sin x $. Углы $136^\circ, 137^\circ, 138^\circ$ находятся во второй четверти, где синус положителен.<br>$ \sin 136^\circ = \sin(180^\circ - 136^\circ) = \sin 44^\circ $<br>$ \sin 137^\circ = \sin(180^\circ - 137^\circ) = \sin 43^\circ $<br>$ \sin 138^\circ = \sin(180^\circ - 138^\circ) = \sin 42^\circ $<br>Таким образом, неравенство эквивалентно следующему:<br>$ \sin 42^\circ \cdot \sin 43^\circ \cdot \sin 44^\circ > \frac{1}{3} $<br>Проанализируем левую часть. Обозначим произведение $ P = \sin 42^\circ \cdot \sin 43^\circ \cdot \sin 44^\circ $.<br>Можно заметить, что углы представляют собой арифметическую прогрессию с центральным членом $ x = 43^\circ $ и разностью $ d = 1^\circ $.<br>Воспользуемся формулой произведения синусов: $ \sin(x-d)\sin(x+d) = \sin^2 x - \sin^2 d $.<br>Тогда $ \sin 42^\circ \cdot \sin 44^\circ = \sin(43^\circ - 1^\circ)\sin(43^\circ + 1^\circ) = \sin^2 43^\circ - \sin^2 1^\circ $.<br>Следовательно, произведение $ P = (\sin^2 43^\circ - \sin^2 1^\circ) \cdot \sin 43^\circ = (\sin 43^\circ)^3 - \sin 43^\circ \cdot \sin^2 1^\circ $.<br>Так как $ \sin 43^\circ > 0 $ и $ \sin^2 1^\circ > 0 $, то $ P < (\sin 43^\circ)^3 $.<br>Теперь сравним $ (\sin 43^\circ)^3 $ с $ \frac{1}{3} $. Это эквивалентно сравнению $ \sin 43^\circ $ с $ \sqrt[3]{\frac{1}{3}} $.<br>Функция $ y = \sin x $ является вогнутой на интервале $ (0^\circ, 180^\circ) $. Это означает, что ее график лежит ниже любой касательной. Проведем касательную к графику в точке $ x_0 = 45^\circ $.<br>Уравнение касательной: $ y = \sin 45^\circ + (\cos 45^\circ)(x - \frac{\pi}{4}) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}(x - \frac{\pi}{4}) $.<br>Для любого $ x $ из интервала, $ \sin x \le \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}(x - \frac{\pi}{4}) $.<br>Представим $ 43^\circ $ в радианах: $ 43^\circ = 43 \frac{\pi}{180} = \frac{\pi}{4} - \frac{2\pi}{180} = \frac{\pi}{4} - \frac{\pi}{90} $.<br>$ \sin 43^\circ \le \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}(\frac{\pi}{4} - \frac{\pi}{90} - \frac{\pi}{4}) = \frac{\sqrt{2}}{2}(1 - \frac{\pi}{90}) $.<br>Используя приближенные значения $ \sqrt{2} \approx 1.4142 $ и $ \pi \approx 3.1416 $, получаем:<br>$ \frac{\pi}{90} \approx \frac{3.1416}{90} \approx 0.0349 $<br>$ \sin 43^\circ \le \frac{1.4142}{2}(1 - 0.0349) = 0.7071 \cdot 0.9651 \approx 0.6824 $.<br>Теперь оценим $ \sqrt[3]{\frac{1}{3}} $: $ \sqrt[3]{\frac{1}{3}} = \frac{1}{\sqrt[3]{3}} $. Так как $ 1.44^3 \approx 2.986 < 3 $, то $ 1.44 < \sqrt[3]{3} $. $ \frac{1}{\sqrt[3]{3}} > \frac{1}{1.45} \approx 0.689 $. Точное значение $ \sqrt[3]{1/3} \approx 0.6933 $.<br>Таким образом, $ \sin 43^\circ \approx 0.6824 < 0.6933 \approx \sqrt[3]{\frac{1}{3}} $.<br>Возведя обе части в куб, получаем $ (\sin 43^\circ)^3 < \frac{1}{3} $.<br>Так как $ P < (\sin 43^\circ)^3 $, то $ P < \frac{1}{3} $.<br>Это означает, что исходное неравенство $ \sin 136^\circ \cdot \sin 137^\circ \cdot \sin 138^\circ > \frac{1}{3} $ неверно. Вероятно, в условии задачи имеется опечатка, и знак неравенства должен быть противоположным.<br>Доказательство неравенства $ \sin 136^\circ \cdot \sin 137^\circ \cdot \sin 138^\circ < \frac{1}{3} $ приведено выше.<br>Ответ: Утверждение в задаче неверно. Было доказано, что $ \sin 136^\circ \cdot \sin 137^\circ \cdot \sin 138^\circ < \frac{1}{3} $.</p><p><strong>б)</strong> Требуется доказать неравенство $ \cos 55^\circ \cdot \cos 57^\circ \cdot \cos 59^\circ > \frac{1}{9} $.<br>Все углы находятся в первой четверти, где косинус положителен. Функция $ y = \cos x $ на интервале $ (0^\circ, 90^\circ) $ является убывающей и вогнутой. Свойство вогнутости означает, что график функции лежит выше любой хорды (секущей), соединяющей две точки графика.<br>Рассмотрим хорду, соединяющую точки графика $ y = \cos x $ при $ x_1 = 45^\circ $ и $ x_2 = 60^\circ $.<br>$ A(45^\circ, \cos 45^\circ) = (45^\circ, \frac{\sqrt{2}}{2}) $<br>$ B(60^\circ, \cos 60^\circ) = (60^\circ, \frac{1}{2}) $<br>Уравнение прямой, проходящей через точки A и B, имеет вид: $ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) $.<br>Для $ x \in (45^\circ, 60^\circ) $ выполняется неравенство $ \cos x > y_{line}(x) $.<br>$ y_{line}(x) = \frac{1}{2} + \frac{\frac{\sqrt{2}}{2} - \frac{1}{2}}{45-60}(x - 60) = \frac{1}{2} + \frac{\sqrt{2}-1}{-30}(x-60) = \frac{1}{2} + \frac{1-\sqrt{2}}{30}(x-60) $.<br>Все углы $ 55^\circ, 57^\circ, 59^\circ $ лежат в интервале $ (45^\circ, 60^\circ) $. Применим неравенство для каждого из них:<br>$ \cos 55^\circ > \frac{1}{2} + \frac{1-\sqrt{2}}{30}(55-60) = \frac{1}{2} - \frac{5(1-\sqrt{2})}{30} = \frac{1}{2} - \frac{1-\sqrt{2}}{6} = \frac{3-1+\sqrt{2}}{6} = \frac{2+\sqrt{2}}{6} $.<br>$ \cos 57^\circ > \frac{1}{2} + \frac{1-\sqrt{2}}{30}(57-60) = \frac{1}{2} - \frac{3(1-\sqrt{2})}{30} = \frac{1}{2} - \frac{1-\sqrt{2}}{10} = \frac{5-1+\sqrt{2}}{10} = \frac{4+\sqrt{2}}{10} $.<br>$ \cos 59^\circ > \frac{1}{2} + \frac{1-\sqrt{2}}{30}(59-60) = \frac{1}{2} - \frac{1-\sqrt{2}}{30} = \frac{15-1+\sqrt{2}}{30} = \frac{14+\sqrt{2}}{30} $.<br>Перемножим полученные неравенства (все части положительны):<br>$ \cos 55^\circ \cdot \cos 57^\circ \cdot \cos 59^\circ > \frac{2+\sqrt{2}}{6} \cdot \frac{4+\sqrt{2}}{10} \cdot \frac{14+\sqrt{2}}{30} $.<br>Оценим правую часть:<br>$ \frac{(2+\sqrt{2})(4+\sqrt{2})(14+\sqrt{2})}{1800} = \frac{(8+2\sqrt{2}+4\sqrt{2}+2)(14+\sqrt{2})}{1800} = \frac{(10+6\sqrt{2})(14+\sqrt{2})}{1800} $<br>$ = \frac{140+10\sqrt{2}+84\sqrt{2}+12}{1800} = \frac{152+94\sqrt{2}}{1800} = \frac{76+47\sqrt{2}}{900} $.<br>Теперь докажем, что $ \frac{76+47\sqrt{2}}{900} > \frac{1}{9} $.<br>$ \frac{76+47\sqrt{2}}{900} > \frac{100}{900} $<br>$ 76+47\sqrt{2} > 100 $<br>$ 47\sqrt{2} > 24 $<br>Возведем обе части в квадрат (они положительны):<br>$ (47\sqrt{2})^2 > 24^2 $<br>$ 47^2 \cdot 2 > 576 $<br>$ 2209 \cdot 2 > 576 $<br>$ 4418 > 576 $<br>Это неравенство очевидно истинно. Следовательно, исходное неравенство также истинно.<br>Ответ: Неравенство доказано.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108364" "type" => "task" ] "previous" => array:2 [ "refs" => "1108362" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Book {#1114} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1099 #items: array:1 [ 0 => App\Models\BookPage {#1070 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1097924 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-182" "field_display_title" => "182" "field_folder" => "folder1" "field_image_name" => "182" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ 0 => App\Models\Book {#1114} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1111 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1154 #items: array:1 [ 0 => App\Models\Element {#1153 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1261075 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1097924" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1261075 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155} "book_page" => array:2 [ "refs" => "1097924" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1097925" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097923" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1251 #items: array:6 [ 0 => App\Models\Task {#1263 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108361 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-626" "field_display_title" => "626" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1264 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1265 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1266 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1267 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1270 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108362" "type" => "task" ] "previous" => array:2 [ "refs" => "1108360" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1276 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097924" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108361 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-626" "field_display_title" => "626" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1264} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1265} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1266} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1267} "content" => Illuminate\Database\Eloquent\Collection {#1270} "next" => array:2 [ "refs" => "1108362" "type" => "task" ] "previous" => array:2 [ "refs" => "1108360" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1276} "page" => array:2 [ "refs" => "1097924" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1269 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108362 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-627" "field_display_title" => "627" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1268 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1275 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1277 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1273 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1272 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108363" "type" => "task" ] "previous" => array:2 [ "refs" => "1108361" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1290 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097924" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108362 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-627" "field_display_title" => "627" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1268} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1275} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1277} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1273} "content" => Illuminate\Database\Eloquent\Collection {#1272} "next" => array:2 [ "refs" => "1108363" "type" => "task" ] "previous" => array:2 [ "refs" => "1108361" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1290} "page" => array:2 [ "refs" => "1097924" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1274 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108363 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-628" "field_display_title" => "628" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1271 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1289 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1291 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1287 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1285 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108364" "type" => "task" ] "previous" => array:2 [ "refs" => "1108362" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1288 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097924" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108363 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-628" "field_display_title" => "628" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1271} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1289} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1291} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1287} "content" => Illuminate\Database\Eloquent\Collection {#1285} "next" => array:2 [ "refs" => "1108364" "type" => "task" ] "previous" => array:2 [ "refs" => "1108362" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1288} "page" => array:2 [ "refs" => "1097924" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1286 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-629" "field_display_title" => "629" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1299 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1300 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1301 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1302 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1305 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108365" "type" => "task" ] "previous" => array:2 [ "refs" => "1108363" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1311 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097924" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-629" "field_display_title" => "629" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1299} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1300} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1301} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1302} "content" => Illuminate\Database\Eloquent\Collection {#1305} "next" => array:2 [ "refs" => "1108365" "type" => "task" ] "previous" => array:2 [ "refs" => "1108363" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1311} "page" => array:2 [ "refs" => "1097924" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1304 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108365 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-630" "field_display_title" => "630" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1303 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1310 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1312 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1308 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1307 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108366" "type" => "task" ] "previous" => array:2 [ "refs" => "1108364" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1325 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097924" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108365 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-630" "field_display_title" => "630" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1303} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1310} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1312} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1308} "content" => Illuminate\Database\Eloquent\Collection {#1307} "next" => array:2 [ "refs" => "1108366" "type" => "task" ] "previous" => array:2 [ "refs" => "1108364" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1325} "page" => array:2 [ "refs" => "1097924" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1309 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108366 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-631" "field_display_title" => "631" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1306 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1324 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1326 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1322 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1321 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108367" "type" => "task" ] "previous" => array:2 [ "refs" => "1108365" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1339 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097924" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108366 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-631" "field_display_title" => "631" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1306} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1324} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1326} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1322} "content" => Illuminate\Database\Eloquent\Collection {#1321} "next" => array:2 [ "refs" => "1108367" "type" => "task" ] "previous" => array:2 [ "refs" => "1108365" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1339} "page" => array:2 [ "refs" => "1097924" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1097924 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-182" "field_display_title" => "182" "field_folder" => "folder1" "field_image_name" => "182" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1073} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1102} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1111} "content" => Illuminate\Database\Eloquent\Collection {#1154} "next" => array:2 [ "refs" => "1097925" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097923" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1251} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1108363 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "182" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-628" "field_display_title" => "628" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1104} "content" => Illuminate\Database\Eloquent\Collection {#1101} "next" => array:2 [ "refs" => "1108364" "type" => "task" ] "previous" => array:2 [ "refs" => "1108362" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1097} "page" => array:2 [ "refs" => "1097924" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№628 (с. 182)
Условие. №628 (с. 182)
скриншот условия
628. Используя свойства функций $y = \sin x$ и $y = \cos x$, докажите, что:
a) $\sin 136^\circ \cdot \sin 137^\circ \cdot \sin 138^\circ > \frac{1}{3};$
б) $\cos 55^\circ \cdot \cos 57^\circ \cdot \cos 59^\circ > \frac{1}{9}.$
Решение. №628 (с. 182)
Решение 2 (rus). №628 (с. 182)
a) Требуется доказать неравенство $ \sin 136^\circ \cdot \sin 137^\circ \cdot \sin 138^\circ > \frac{1}{3} $.
Воспользуемся свойством функции синус, а именно формулой приведения $ \sin(180^\circ - x) = \sin x $. Углы $136^\circ, 137^\circ, 138^\circ$ находятся во второй четверти, где синус положителен.
$ \sin 136^\circ = \sin(180^\circ - 136^\circ) = \sin 44^\circ $
$ \sin 137^\circ = \sin(180^\circ - 137^\circ) = \sin 43^\circ $
$ \sin 138^\circ = \sin(180^\circ - 138^\circ) = \sin 42^\circ $
Таким образом, неравенство эквивалентно следующему:
$ \sin 42^\circ \cdot \sin 43^\circ \cdot \sin 44^\circ > \frac{1}{3} $
Проанализируем левую часть. Обозначим произведение $ P = \sin 42^\circ \cdot \sin 43^\circ \cdot \sin 44^\circ $.
Можно заметить, что углы представляют собой арифметическую прогрессию с центральным членом $ x = 43^\circ $ и разностью $ d = 1^\circ $.
Воспользуемся формулой произведения синусов: $ \sin(x-d)\sin(x+d) = \sin^2 x - \sin^2 d $.
Тогда $ \sin 42^\circ \cdot \sin 44^\circ = \sin(43^\circ - 1^\circ)\sin(43^\circ + 1^\circ) = \sin^2 43^\circ - \sin^2 1^\circ $.
Следовательно, произведение $ P = (\sin^2 43^\circ - \sin^2 1^\circ) \cdot \sin 43^\circ = (\sin 43^\circ)^3 - \sin 43^\circ \cdot \sin^2 1^\circ $.
Так как $ \sin 43^\circ > 0 $ и $ \sin^2 1^\circ > 0 $, то $ P < (\sin 43^\circ)^3 $.
Теперь сравним $ (\sin 43^\circ)^3 $ с $ \frac{1}{3} $. Это эквивалентно сравнению $ \sin 43^\circ $ с $ \sqrt[3]{\frac{1}{3}} $.
Функция $ y = \sin x $ является вогнутой на интервале $ (0^\circ, 180^\circ) $. Это означает, что ее график лежит ниже любой касательной. Проведем касательную к графику в точке $ x_0 = 45^\circ $.
Уравнение касательной: $ y = \sin 45^\circ + (\cos 45^\circ)(x - \frac{\pi}{4}) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}(x - \frac{\pi}{4}) $.
Для любого $ x $ из интервала, $ \sin x \le \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}(x - \frac{\pi}{4}) $.
Представим $ 43^\circ $ в радианах: $ 43^\circ = 43 \frac{\pi}{180} = \frac{\pi}{4} - \frac{2\pi}{180} = \frac{\pi}{4} - \frac{\pi}{90} $.
$ \sin 43^\circ \le \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}(\frac{\pi}{4} - \frac{\pi}{90} - \frac{\pi}{4}) = \frac{\sqrt{2}}{2}(1 - \frac{\pi}{90}) $.
Используя приближенные значения $ \sqrt{2} \approx 1.4142 $ и $ \pi \approx 3.1416 $, получаем:
$ \frac{\pi}{90} \approx \frac{3.1416}{90} \approx 0.0349 $
$ \sin 43^\circ \le \frac{1.4142}{2}(1 - 0.0349) = 0.7071 \cdot 0.9651 \approx 0.6824 $.
Теперь оценим $ \sqrt[3]{\frac{1}{3}} $: $ \sqrt[3]{\frac{1}{3}} = \frac{1}{\sqrt[3]{3}} $. Так как $ 1.44^3 \approx 2.986 < 3 $, то $ 1.44 < \sqrt[3]{3} $. $ \frac{1}{\sqrt[3]{3}} > \frac{1}{1.45} \approx 0.689 $. Точное значение $ \sqrt[3]{1/3} \approx 0.6933 $.
Таким образом, $ \sin 43^\circ \approx 0.6824 < 0.6933 \approx \sqrt[3]{\frac{1}{3}} $.
Возведя обе части в куб, получаем $ (\sin 43^\circ)^3 < \frac{1}{3} $.
Так как $ P < (\sin 43^\circ)^3 $, то $ P < \frac{1}{3} $.
Это означает, что исходное неравенство $ \sin 136^\circ \cdot \sin 137^\circ \cdot \sin 138^\circ > \frac{1}{3} $ неверно. Вероятно, в условии задачи имеется опечатка, и знак неравенства должен быть противоположным.
Доказательство неравенства $ \sin 136^\circ \cdot \sin 137^\circ \cdot \sin 138^\circ < \frac{1}{3} $ приведено выше.
Ответ: Утверждение в задаче неверно. Было доказано, что $ \sin 136^\circ \cdot \sin 137^\circ \cdot \sin 138^\circ < \frac{1}{3} $.
б) Требуется доказать неравенство $ \cos 55^\circ \cdot \cos 57^\circ \cdot \cos 59^\circ > \frac{1}{9} $.
Все углы находятся в первой четверти, где косинус положителен. Функция $ y = \cos x $ на интервале $ (0^\circ, 90^\circ) $ является убывающей и вогнутой. Свойство вогнутости означает, что график функции лежит выше любой хорды (секущей), соединяющей две точки графика.
Рассмотрим хорду, соединяющую точки графика $ y = \cos x $ при $ x_1 = 45^\circ $ и $ x_2 = 60^\circ $.
$ A(45^\circ, \cos 45^\circ) = (45^\circ, \frac{\sqrt{2}}{2}) $
$ B(60^\circ, \cos 60^\circ) = (60^\circ, \frac{1}{2}) $
Уравнение прямой, проходящей через точки A и B, имеет вид: $ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) $.
Для $ x \in (45^\circ, 60^\circ) $ выполняется неравенство $ \cos x > y_{line}(x) $.
$ y_{line}(x) = \frac{1}{2} + \frac{\frac{\sqrt{2}}{2} - \frac{1}{2}}{45-60}(x - 60) = \frac{1}{2} + \frac{\sqrt{2}-1}{-30}(x-60) = \frac{1}{2} + \frac{1-\sqrt{2}}{30}(x-60) $.
Все углы $ 55^\circ, 57^\circ, 59^\circ $ лежат в интервале $ (45^\circ, 60^\circ) $. Применим неравенство для каждого из них:
$ \cos 55^\circ > \frac{1}{2} + \frac{1-\sqrt{2}}{30}(55-60) = \frac{1}{2} - \frac{5(1-\sqrt{2})}{30} = \frac{1}{2} - \frac{1-\sqrt{2}}{6} = \frac{3-1+\sqrt{2}}{6} = \frac{2+\sqrt{2}}{6} $.
$ \cos 57^\circ > \frac{1}{2} + \frac{1-\sqrt{2}}{30}(57-60) = \frac{1}{2} - \frac{3(1-\sqrt{2})}{30} = \frac{1}{2} - \frac{1-\sqrt{2}}{10} = \frac{5-1+\sqrt{2}}{10} = \frac{4+\sqrt{2}}{10} $.
$ \cos 59^\circ > \frac{1}{2} + \frac{1-\sqrt{2}}{30}(59-60) = \frac{1}{2} - \frac{1-\sqrt{2}}{30} = \frac{15-1+\sqrt{2}}{30} = \frac{14+\sqrt{2}}{30} $.
Перемножим полученные неравенства (все части положительны):
$ \cos 55^\circ \cdot \cos 57^\circ \cdot \cos 59^\circ > \frac{2+\sqrt{2}}{6} \cdot \frac{4+\sqrt{2}}{10} \cdot \frac{14+\sqrt{2}}{30} $.
Оценим правую часть:
$ \frac{(2+\sqrt{2})(4+\sqrt{2})(14+\sqrt{2})}{1800} = \frac{(8+2\sqrt{2}+4\sqrt{2}+2)(14+\sqrt{2})}{1800} = \frac{(10+6\sqrt{2})(14+\sqrt{2})}{1800} $
$ = \frac{140+10\sqrt{2}+84\sqrt{2}+12}{1800} = \frac{152+94\sqrt{2}}{1800} = \frac{76+47\sqrt{2}}{900} $.
Теперь докажем, что $ \frac{76+47\sqrt{2}}{900} > \frac{1}{9} $.
$ \frac{76+47\sqrt{2}}{900} > \frac{100}{900} $
$ 76+47\sqrt{2} > 100 $
$ 47\sqrt{2} > 24 $
Возведем обе части в квадрат (они положительны):
$ (47\sqrt{2})^2 > 24^2 $
$ 47^2 \cdot 2 > 576 $
$ 2209 \cdot 2 > 576 $
$ 4418 > 576 $
Это неравенство очевидно истинно. Следовательно, исходное неравенство также истинно.
Ответ: Неравенство доказано.
Другие задания:
Помогло решение? Оставьте отзыв в комментариях ниже.
Присоединяйтесь к Телеграм-группе @top_gdz
ПрисоединитьсяМы подготовили для вас ответ c подробным объяснением домашего задания по алгебре за 9 класс, для упражнения номер 628 расположенного на странице 182 к учебнику 2019 года издания для учащихся школ и гимназий.
Теперь на нашем сайте ГДЗ.ТОП вы всегда легко и бесплатно найдёте условие с правильным ответом на вопрос «Как решить ДЗ» и «Как сделать» задание по алгебре к упражнению №628 (с. 182), авторов: Солтан (Генадий Николаевич), Солтан (Алла Евгеньевна), Жумадилова (Аманбала Жумадиловна), учебного пособия издательства Кокшетау.