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Номер 830, страница 227 - 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. 30. Упражнения на повторение раздела «Тригонометрия» - номер 830, страница 227.
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" => 1108576 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "227" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-830" "field_display_title" => "830" "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 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ 0 => App\Models\Term {#1044 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] …8 } ] #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 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"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" => 1107672 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "30. 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Упражнения на повторение раздела «Тригонометрия»" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1145} "field_page_start" => "221" "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 {#1087 #items: array:3 [ 0 => App\Models\Element {#1079 #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" => 1277185 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Edition {#1076 #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 {#1074 #items: array:1 [ 0 => App\Models\Term {#1077 …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 {#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" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1074} "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 {#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" => "1108576" "type" => "task" ] "text" => "<p>830.</p><p><strong>1A) Вычислите:</strong> а) $\cos \frac{15\pi}{4}$; б) $\sin \left(-\frac{17\pi}{6}\right)$; в) $\operatorname{tg} 600^\circ$.</p><p><strong>2</strong>A) Упростите выражение:</p><p>а) $(\sin \alpha - \cos \alpha)^2 - 1 + 4\sin 2\alpha$; б) $1 + \operatorname{tg}\left(\frac{3\pi}{2} - \alpha\right) \cdot \operatorname{ctg}(\pi + \alpha)$.</p><p><strong>3A) Дано:</strong> $\cos \alpha = -\frac{4}{5}$, $\pi < \alpha < \frac{3\pi}{2}$.</p><p>Найдите: а) $\cos 2\alpha$; б) $\sin(60^\circ + \alpha)$; в) $\operatorname{tg}(45^\circ - \alpha)$.</p><p><strong>4B) Докажите тождество</strong> $\frac{2 \sin 2\alpha + \sin 4\alpha}{2(\cos \alpha + \cos 3\alpha)} = \operatorname{tg} 2\alpha \cdot \cos \alpha$.</p><p><strong>5C) Найдите значение выражения</strong> $\cos 72^\circ \cdot \sin 54^\circ$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "830-1.jpg" "alt" => null "width" => "1456" "height" => 777 "path" => "/media/algebra_09/soltan-u19/0-1/830-1.webp?ts=1752834829" ] ] ] #original: array:7 [ "id" => 1277185 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "1108576" "type" => "task" ] "text" => "<p>830.</p><p><strong>1A) Вычислите:</strong> а) $\cos \frac{15\pi}{4}$; б) $\sin \left(-\frac{17\pi}{6}\right)$; в) $\operatorname{tg} 600^\circ$.</p><p><strong>2</strong>A) Упростите выражение:</p><p>а) $(\sin \alpha - \cos \alpha)^2 - 1 + 4\sin 2\alpha$; б) $1 + \operatorname{tg}\left(\frac{3\pi}{2} - \alpha\right) \cdot \operatorname{ctg}(\pi + \alpha)$.</p><p><strong>3A) Дано:</strong> $\cos \alpha = -\frac{4}{5}$, $\pi < \alpha < \frac{3\pi}{2}$.</p><p>Найдите: а) $\cos 2\alpha$; б) $\sin(60^\circ + \alpha)$; в) $\operatorname{tg}(45^\circ - \alpha)$.</p><p><strong>4B) Докажите тождество</strong> $\frac{2 \sin 2\alpha + \sin 4\alpha}{2(\cos \alpha + \cos 3\alpha)} = \operatorname{tg} 2\alpha \cdot \cos \alpha$.</p><p><strong>5C) Найдите значение выражения</strong> $\cos 72^\circ \cdot \sin 54^\circ$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "830-1.jpg" "alt" => null "width" => "1456" "height" => 777 "path" => "/media/algebra_09/soltan-u19/0-1/830-1.webp?ts=1752834829" ] ] ] #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 {#1102 #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" => 1278476 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1152 #items: array:1 [ 0 => App\Models\Edition {#1111 #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 {#1108 #items: array:1 [ 0 => App\Models\Term {#1105 …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-" 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Illuminate\Database\Eloquent\Collection {#1106} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1148} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1149} "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" => "1108576" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "830-1.jpg" "alt" => null "width" => "1433" "height" => 2439 "path" => "/media/algebra_09/soltan-u19/1-1/830-1.webp?ts=1752831659" ] 1 => array:5 [ "name" => "830-2.jpg" "alt" => null "width" => "1363" "height" => 2089 "path" => "/media/algebra_09/soltan-u19/1-1/830-2.webp?ts=1752831659" ] ] ] #original: array:6 [ "id" => 1278476 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1152} "task" => array:2 [ "refs" => "1108576" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "830-1.jpg" "alt" => null "width" => "1433" "height" => 2439 "path" => "/media/algebra_09/soltan-u19/1-1/830-1.webp?ts=1752831659" ] 1 => array:5 [ "name" => "830-2.jpg" "alt" => null "width" => "1363" "height" => 2089 "path" => "/media/algebra_09/soltan-u19/1-1/830-2.webp?ts=1752831659" ] ] ] #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 {#1150 #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" => 1554431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1160 #items: array:1 [ 0 => App\Models\Edition {#1151 #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 {#1154 #items: array:1 [ 0 => App\Models\Term {#1153 …30} ] #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 {#1155 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1156 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1157 #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 {#1154} "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 {#1155} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1156} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1157} "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" => "1108576" "type" => "task" ] "text" => "<p><strong>1</strong>А) Вычислите:</p><p><strong>а)</strong> Для вычисления $ \cos\frac{15\pi}{4} $ воспользуемся периодичностью косинуса ($ 2\pi $) и его четностью.<br>Представим угол в виде: $ \frac{15\pi}{4} = \frac{16\pi - \pi}{4} = 4\pi - \frac{\pi}{4} $.<br>Используя периодичность, отбросим $ 4\pi $ (два полных оборота):<br>$ \cos\frac{15\pi}{4} = \cos(4\pi - \frac{\pi}{4}) = \cos(-\frac{\pi}{4}) $.<br>Так как косинус - четная функция ($ \cos(-x) = \cos(x) $), то $ \cos(-\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} $.<br>Ответ: $ \frac{\sqrt{2}}{2} $.</p><p><strong>б)</strong> Для вычисления $ \sin(-\frac{17\pi}{6}) $ воспользуемся нечетностью синуса и его периодичностью.<br>Синус - нечетная функция ($ \sin(-x) = -\sin(x) $), поэтому $ \sin(-\frac{17\pi}{6}) = -\sin(\frac{17\pi}{6}) $.<br>Выделим целую часть оборотов: $ \frac{17\pi}{6} = \frac{12\pi + 5\pi}{6} = 2\pi + \frac{5\pi}{6} $.<br>$ -\sin(\frac{17\pi}{6}) = -\sin(2\pi + \frac{5\pi}{6}) = -\sin(\frac{5\pi}{6}) $.<br>Используем формулу приведения: $ \sin(\frac{5\pi}{6}) = \sin(\pi - \frac{\pi}{6}) = \sin(\frac{\pi}{6}) = \frac{1}{2} $.<br>Следовательно, искомое значение равно $ -\frac{1}{2} $.<br>Ответ: $ -\frac{1}{2} $.</p><p><strong>в)</strong> Для вычисления $ \text{tg } 600^\circ $ воспользуемся периодичностью тангенса ($ 180^\circ $).<br>Представим угол в виде: $ 600^\circ = 3 \cdot 180^\circ + 60^\circ $.<br>$ \text{tg } 600^\circ = \text{tg}(3 \cdot 180^\circ + 60^\circ) = \text{tg } 60^\circ = \sqrt{3} $.<br>Ответ: $ \sqrt{3} $.</p><p><strong>2</strong>А) Упростите выражение:</p><p><strong>а)</strong> Раскроем скобки и используем тригонометрические тождества:<br>$ (\sin\alpha - \cos\alpha)^2 - 1 + 4\sin2\alpha = (\sin^2\alpha - 2\sin\alpha\cos\alpha + \cos^2\alpha) - 1 + 4\sin2\alpha $.<br>Используя основное тригонометрическое тождество $ \sin^2\alpha + \cos^2\alpha = 1 $ и формулу двойного угла $ \sin2\alpha = 2\sin\alpha\cos\alpha $, получаем:<br>$ (1 - \sin2\alpha) - 1 + 4\sin2\alpha = 1 - \sin2\alpha - 1 + 4\sin2\alpha = 3\sin2\alpha $.<br>Ответ: $ 3\sin2\alpha $.</p><p><strong>б)</strong> Используем формулы приведения:<br>$ \text{tg}(\frac{3\pi}{2} - \alpha) = \text{ctg}\alpha $ (угол в III четверти, тангенс положителен, функция меняется на кофункцию).<br>$ \text{ctg}(\pi + \alpha) = \text{ctg}\alpha $ (угол в III четверти, котангенс положителен, функция не меняется).<br>Подставляем в исходное выражение:<br>$ 1 + \text{tg}(\frac{3\pi}{2} - \alpha) \cdot \text{ctg}(\pi + \alpha) = 1 + \text{ctg}\alpha \cdot \text{ctg}\alpha = 1 + \text{ctg}^2\alpha $.<br>По основному тригонометрическому тождеству $ 1 + \text{ctg}^2\alpha = \frac{1}{\sin^2\alpha} $.<br>Ответ: $ \frac{1}{\sin^2\alpha} $.</p><p><strong>3</strong>А) Дано: $ \cos\alpha = -\frac{4}{5} $, $ \pi < \alpha < \frac{3\pi}{2} $.</p><p>Сначала найдем $ \sin\alpha $. Из основного тригонометрического тождества $ \sin^2\alpha + \cos^2\alpha = 1 $:<br>$ \sin^2\alpha = 1 - \cos^2\alpha = 1 - (-\frac{4}{5})^2 = 1 - \frac{16}{25} = \frac{9}{25} $.<br>Так как угол $ \alpha $ находится в III четверти ($ \pi < \alpha < \frac{3\pi}{2} $), синус отрицателен: $ \sin\alpha = -\sqrt{\frac{9}{25}} = -\frac{3}{5} $.</p><p><strong>а)</strong> Найдем $ \cos2\alpha $ по формуле двойного угла:<br>$ \cos2\alpha = 2\cos^2\alpha - 1 = 2(-\frac{4}{5})^2 - 1 = 2(\frac{16}{25}) - 1 = \frac{32}{25} - \frac{25}{25} = \frac{7}{25} $.<br>Ответ: $ \frac{7}{25} $.</p><p><strong>б)</strong> Найдем $ \sin(60^\circ + \alpha) $ по формуле синуса суммы:<br>$ \sin(60^\circ + \alpha) = \sin60^\circ\cos\alpha + \cos60^\circ\sin\alpha $.<br>Подставляем известные значения $ \sin60^\circ = \frac{\sqrt{3}}{2} $, $ \cos60^\circ = \frac{1}{2} $, $ \cos\alpha = -\frac{4}{5} $ и $ \sin\alpha = -\frac{3}{5} $:<br>$ \sin(60^\circ + \alpha) = \frac{\sqrt{3}}{2} \cdot (-\frac{4}{5}) + \frac{1}{2} \cdot (-\frac{3}{5}) = -\frac{4\sqrt{3}}{10} - \frac{3}{10} = -\frac{3+4\sqrt{3}}{10} $.<br>Ответ: $ -\frac{3+4\sqrt{3}}{10} $.</p><p><strong>в)</strong> Найдем $ \text{tg}(45^\circ - \alpha) $ по формуле тангенса разности. Сначала найдем $ \text{tg}\alpha $:<br>$ \text{tg}\alpha = \frac{\sin\alpha}{\cos\alpha} = \frac{-3/5}{-4/5} = \frac{3}{4} $.<br>Теперь используем формулу $ \text{tg}(45^\circ - \alpha) = \frac{\text{tg}45^\circ - \text{tg}\alpha}{1 + \text{tg}45^\circ\text{tg}\alpha} $.<br>Подставляем $ \text{tg}45^\circ = 1 $ и $ \text{tg}\alpha = \frac{3}{4} $:<br>$ \text{tg}(45^\circ - \alpha) = \frac{1 - \frac{3}{4}}{1 + 1 \cdot \frac{3}{4}} = \frac{\frac{4-3}{4}}{\frac{4+3}{4}} = \frac{1/4}{7/4} = \frac{1}{7} $.<br>Ответ: $ \frac{1}{7} $.</p><p><strong>4</strong>B) Докажите тождество</p><p>Требуется доказать тождество $ \frac{2 \sin 2\alpha + \sin 4\alpha}{2(\cos \alpha + \cos 3\alpha)} = \text{tg } 2\alpha \cdot \cos \alpha $.<br>Преобразуем левую часть (ЛЧ) тождества.<br>Преобразуем числитель, используя формулу синуса двойного угла $ \sin 4\alpha = 2 \sin 2\alpha \cos 2\alpha $:<br>$ 2 \sin 2\alpha + \sin 4\alpha = 2 \sin 2\alpha + 2 \sin 2\alpha \cos 2\alpha = 2 \sin 2\alpha (1 + \cos 2\alpha) $.<br>Преобразуем знаменатель, используя формулу суммы косинусов $ \cos x + \cos y = 2\cos\frac{x+y}{2}\cos\frac{x-y}{2} $:<br>$ 2(\cos \alpha + \cos 3\alpha) = 2 \cdot (2\cos\frac{\alpha+3\alpha}{2}\cos\frac{3\alpha-\alpha}{2}) = 4\cos 2\alpha \cos\alpha $.<br>Запишем преобразованную дробь:<br>ЛЧ $ = \frac{2 \sin 2\alpha (1 + \cos 2\alpha)}{4\cos 2\alpha \cos\alpha} $.<br>Используем формулу косинуса двойного угла для понижения степени $ 1 + \cos 2\alpha = 2\cos^2\alpha $:<br>ЛЧ $ = \frac{2 \sin 2\alpha \cdot (2\cos^2\alpha)}{4\cos 2\alpha \cos\alpha} = \frac{4 \sin 2\alpha \cos^2\alpha}{4\cos 2\alpha \cos\alpha} $.<br>Сокращаем общие множители $ 4 $ и $ \cos\alpha $:<br>ЛЧ $ = \frac{\sin 2\alpha \cos\alpha}{\cos 2\alpha} = \frac{\sin 2\alpha}{\cos 2\alpha} \cdot \cos\alpha = \text{tg } 2\alpha \cdot \cos\alpha $.<br>Левая часть равна правой, тождество доказано.</p><p><strong>5</strong>С) Найдите значение выражения</p><p>Требуется найти значение $ \cos 72^\circ \cdot \sin 54^\circ $.<br>Используем формулу приведения: $ \sin 54^\circ = \sin(90^\circ - 36^\circ) = \cos 36^\circ $.<br>Выражение принимает вид: $ \cos 72^\circ \cdot \cos 36^\circ $.<br>Домножим и разделим выражение на $ 2\sin 36^\circ $ (это возможно, так как $ \sin 36^\circ \neq 0 $):<br>$ \frac{2\sin 36^\circ \cos 36^\circ \cdot \cos 72^\circ}{2\sin 36^\circ} $.<br>Используем формулу синуса двойного угла $ \sin 2x = 2\sin x \cos x $ для числителя:<br>$ \frac{\sin(2 \cdot 36^\circ) \cdot \cos 72^\circ}{2\sin 36^\circ} = \frac{\sin 72^\circ \cdot \cos 72^\circ}{2\sin 36^\circ} $.<br>Снова применяем ту же формулу для числителя:<br>$ \frac{\frac{1}{2} \cdot (2\sin 72^\circ \cos 72^\circ)}{2\sin 36^\circ} = \frac{\frac{1}{2} \sin(2 \cdot 72^\circ)}{2\sin 36^\circ} = \frac{\frac{1}{2} \sin 144^\circ}{2\sin 36^\circ} $.<br>Используем формулу приведения: $ \sin 144^\circ = \sin(180^\circ - 36^\circ) = \sin 36^\circ $.<br>Подставляем и сокращаем:<br>$ \frac{\frac{1}{2} \sin 36^\circ}{2\sin 36^\circ} = \frac{1/2}{2} = \frac{1}{4} $.<br>Ответ: $ \frac{1}{4} $.</p>" ] #original: array:6 [ "id" => 1554431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1160} "task" => array:2 [ "refs" => "1108576" "type" => "task" ] "text" => "<p><strong>1</strong>А) Вычислите:</p><p><strong>а)</strong> Для вычисления $ \cos\frac{15\pi}{4} $ воспользуемся периодичностью косинуса ($ 2\pi $) и его четностью.<br>Представим угол в виде: $ \frac{15\pi}{4} = \frac{16\pi - \pi}{4} = 4\pi - \frac{\pi}{4} $.<br>Используя периодичность, отбросим $ 4\pi $ (два полных оборота):<br>$ \cos\frac{15\pi}{4} = \cos(4\pi - \frac{\pi}{4}) = \cos(-\frac{\pi}{4}) $.<br>Так как косинус - четная функция ($ \cos(-x) = \cos(x) $), то $ \cos(-\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} $.<br>Ответ: $ \frac{\sqrt{2}}{2} $.</p><p><strong>б)</strong> Для вычисления $ \sin(-\frac{17\pi}{6}) $ воспользуемся нечетностью синуса и его периодичностью.<br>Синус - нечетная функция ($ \sin(-x) = -\sin(x) $), поэтому $ \sin(-\frac{17\pi}{6}) = -\sin(\frac{17\pi}{6}) $.<br>Выделим целую часть оборотов: $ \frac{17\pi}{6} = \frac{12\pi + 5\pi}{6} = 2\pi + \frac{5\pi}{6} $.<br>$ -\sin(\frac{17\pi}{6}) = -\sin(2\pi + \frac{5\pi}{6}) = -\sin(\frac{5\pi}{6}) $.<br>Используем формулу приведения: $ \sin(\frac{5\pi}{6}) = \sin(\pi - \frac{\pi}{6}) = \sin(\frac{\pi}{6}) = \frac{1}{2} $.<br>Следовательно, искомое значение равно $ -\frac{1}{2} $.<br>Ответ: $ -\frac{1}{2} $.</p><p><strong>в)</strong> Для вычисления $ \text{tg } 600^\circ $ воспользуемся периодичностью тангенса ($ 180^\circ $).<br>Представим угол в виде: $ 600^\circ = 3 \cdot 180^\circ + 60^\circ $.<br>$ \text{tg } 600^\circ = \text{tg}(3 \cdot 180^\circ + 60^\circ) = \text{tg } 60^\circ = \sqrt{3} $.<br>Ответ: $ \sqrt{3} $.</p><p><strong>2</strong>А) Упростите выражение:</p><p><strong>а)</strong> Раскроем скобки и используем тригонометрические тождества:<br>$ (\sin\alpha - \cos\alpha)^2 - 1 + 4\sin2\alpha = (\sin^2\alpha - 2\sin\alpha\cos\alpha + \cos^2\alpha) - 1 + 4\sin2\alpha $.<br>Используя основное тригонометрическое тождество $ \sin^2\alpha + \cos^2\alpha = 1 $ и формулу двойного угла $ \sin2\alpha = 2\sin\alpha\cos\alpha $, получаем:<br>$ (1 - \sin2\alpha) - 1 + 4\sin2\alpha = 1 - \sin2\alpha - 1 + 4\sin2\alpha = 3\sin2\alpha $.<br>Ответ: $ 3\sin2\alpha $.</p><p><strong>б)</strong> Используем формулы приведения:<br>$ \text{tg}(\frac{3\pi}{2} - \alpha) = \text{ctg}\alpha $ (угол в III четверти, тангенс положителен, функция меняется на кофункцию).<br>$ \text{ctg}(\pi + \alpha) = \text{ctg}\alpha $ (угол в III четверти, котангенс положителен, функция не меняется).<br>Подставляем в исходное выражение:<br>$ 1 + \text{tg}(\frac{3\pi}{2} - \alpha) \cdot \text{ctg}(\pi + \alpha) = 1 + \text{ctg}\alpha \cdot \text{ctg}\alpha = 1 + \text{ctg}^2\alpha $.<br>По основному тригонометрическому тождеству $ 1 + \text{ctg}^2\alpha = \frac{1}{\sin^2\alpha} $.<br>Ответ: $ \frac{1}{\sin^2\alpha} $.</p><p><strong>3</strong>А) Дано: $ \cos\alpha = -\frac{4}{5} $, $ \pi < \alpha < \frac{3\pi}{2} $.</p><p>Сначала найдем $ \sin\alpha $. Из основного тригонометрического тождества $ \sin^2\alpha + \cos^2\alpha = 1 $:<br>$ \sin^2\alpha = 1 - \cos^2\alpha = 1 - (-\frac{4}{5})^2 = 1 - \frac{16}{25} = \frac{9}{25} $.<br>Так как угол $ \alpha $ находится в III четверти ($ \pi < \alpha < \frac{3\pi}{2} $), синус отрицателен: $ \sin\alpha = -\sqrt{\frac{9}{25}} = -\frac{3}{5} $.</p><p><strong>а)</strong> Найдем $ \cos2\alpha $ по формуле двойного угла:<br>$ \cos2\alpha = 2\cos^2\alpha - 1 = 2(-\frac{4}{5})^2 - 1 = 2(\frac{16}{25}) - 1 = \frac{32}{25} - \frac{25}{25} = \frac{7}{25} $.<br>Ответ: $ \frac{7}{25} $.</p><p><strong>б)</strong> Найдем $ \sin(60^\circ + \alpha) $ по формуле синуса суммы:<br>$ \sin(60^\circ + \alpha) = \sin60^\circ\cos\alpha + \cos60^\circ\sin\alpha $.<br>Подставляем известные значения $ \sin60^\circ = \frac{\sqrt{3}}{2} $, $ \cos60^\circ = \frac{1}{2} $, $ \cos\alpha = -\frac{4}{5} $ и $ \sin\alpha = -\frac{3}{5} $:<br>$ \sin(60^\circ + \alpha) = \frac{\sqrt{3}}{2} \cdot (-\frac{4}{5}) + \frac{1}{2} \cdot (-\frac{3}{5}) = -\frac{4\sqrt{3}}{10} - \frac{3}{10} = -\frac{3+4\sqrt{3}}{10} $.<br>Ответ: $ -\frac{3+4\sqrt{3}}{10} $.</p><p><strong>в)</strong> Найдем $ \text{tg}(45^\circ - \alpha) $ по формуле тангенса разности. Сначала найдем $ \text{tg}\alpha $:<br>$ \text{tg}\alpha = \frac{\sin\alpha}{\cos\alpha} = \frac{-3/5}{-4/5} = \frac{3}{4} $.<br>Теперь используем формулу $ \text{tg}(45^\circ - \alpha) = \frac{\text{tg}45^\circ - \text{tg}\alpha}{1 + \text{tg}45^\circ\text{tg}\alpha} $.<br>Подставляем $ \text{tg}45^\circ = 1 $ и $ \text{tg}\alpha = \frac{3}{4} $:<br>$ \text{tg}(45^\circ - \alpha) = \frac{1 - \frac{3}{4}}{1 + 1 \cdot \frac{3}{4}} = \frac{\frac{4-3}{4}}{\frac{4+3}{4}} = \frac{1/4}{7/4} = \frac{1}{7} $.<br>Ответ: $ \frac{1}{7} $.</p><p><strong>4</strong>B) Докажите тождество</p><p>Требуется доказать тождество $ \frac{2 \sin 2\alpha + \sin 4\alpha}{2(\cos \alpha + \cos 3\alpha)} = \text{tg } 2\alpha \cdot \cos \alpha $.<br>Преобразуем левую часть (ЛЧ) тождества.<br>Преобразуем числитель, используя формулу синуса двойного угла $ \sin 4\alpha = 2 \sin 2\alpha \cos 2\alpha $:<br>$ 2 \sin 2\alpha + \sin 4\alpha = 2 \sin 2\alpha + 2 \sin 2\alpha \cos 2\alpha = 2 \sin 2\alpha (1 + \cos 2\alpha) $.<br>Преобразуем знаменатель, используя формулу суммы косинусов $ \cos x + \cos y = 2\cos\frac{x+y}{2}\cos\frac{x-y}{2} $:<br>$ 2(\cos \alpha + \cos 3\alpha) = 2 \cdot (2\cos\frac{\alpha+3\alpha}{2}\cos\frac{3\alpha-\alpha}{2}) = 4\cos 2\alpha \cos\alpha $.<br>Запишем преобразованную дробь:<br>ЛЧ $ = \frac{2 \sin 2\alpha (1 + \cos 2\alpha)}{4\cos 2\alpha \cos\alpha} $.<br>Используем формулу косинуса двойного угла для понижения степени $ 1 + \cos 2\alpha = 2\cos^2\alpha $:<br>ЛЧ $ = \frac{2 \sin 2\alpha \cdot (2\cos^2\alpha)}{4\cos 2\alpha \cos\alpha} = \frac{4 \sin 2\alpha \cos^2\alpha}{4\cos 2\alpha \cos\alpha} $.<br>Сокращаем общие множители $ 4 $ и $ \cos\alpha $:<br>ЛЧ $ = \frac{\sin 2\alpha \cos\alpha}{\cos 2\alpha} = \frac{\sin 2\alpha}{\cos 2\alpha} \cdot \cos\alpha = \text{tg } 2\alpha \cdot \cos\alpha $.<br>Левая часть равна правой, тождество доказано.</p><p><strong>5</strong>С) Найдите значение выражения</p><p>Требуется найти значение $ \cos 72^\circ \cdot \sin 54^\circ $.<br>Используем формулу приведения: $ \sin 54^\circ = \sin(90^\circ - 36^\circ) = \cos 36^\circ $.<br>Выражение принимает вид: $ \cos 72^\circ \cdot \cos 36^\circ $.<br>Домножим и разделим выражение на $ 2\sin 36^\circ $ (это возможно, так как $ \sin 36^\circ \neq 0 $):<br>$ \frac{2\sin 36^\circ \cos 36^\circ \cdot \cos 72^\circ}{2\sin 36^\circ} $.<br>Используем формулу синуса двойного угла $ \sin 2x = 2\sin x \cos x $ для числителя:<br>$ \frac{\sin(2 \cdot 36^\circ) \cdot \cos 72^\circ}{2\sin 36^\circ} = \frac{\sin 72^\circ \cdot \cos 72^\circ}{2\sin 36^\circ} $.<br>Снова применяем ту же формулу для числителя:<br>$ \frac{\frac{1}{2} \cdot (2\sin 72^\circ \cos 72^\circ)}{2\sin 36^\circ} = \frac{\frac{1}{2} \sin(2 \cdot 72^\circ)}{2\sin 36^\circ} = \frac{\frac{1}{2} \sin 144^\circ}{2\sin 36^\circ} $.<br>Используем формулу приведения: $ \sin 144^\circ = \sin(180^\circ - 36^\circ) = \sin 36^\circ $.<br>Подставляем и сокращаем:<br>$ \frac{\frac{1}{2} \sin 36^\circ}{2\sin 36^\circ} = \frac{1/2}{2} = \frac{1}{4} $.<br>Ответ: $ \frac{1}{4} $.</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" => "1108577" "type" => "task" ] "previous" => array:2 [ "refs" => "1108575" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Book {#1114} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1094 #items: array:1 [ 0 => App\Models\BookPage {#1092 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false 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№830 (с. 227)
Условие. №830 (с. 227)
скриншот условия
830.
1A) Вычислите: а) $\cos \frac{15\pi}{4}$; б) $\sin \left(-\frac{17\pi}{6}\right)$; в) $\operatorname{tg} 600^\circ$.
2A) Упростите выражение:
а) $(\sin \alpha - \cos \alpha)^2 - 1 + 4\sin 2\alpha$; б) $1 + \operatorname{tg}\left(\frac{3\pi}{2} - \alpha\right) \cdot \operatorname{ctg}(\pi + \alpha)$.
3A) Дано: $\cos \alpha = -\frac{4}{5}$, $\pi < \alpha < \frac{3\pi}{2}$.
Найдите: а) $\cos 2\alpha$; б) $\sin(60^\circ + \alpha)$; в) $\operatorname{tg}(45^\circ - \alpha)$.
4B) Докажите тождество $\frac{2 \sin 2\alpha + \sin 4\alpha}{2(\cos \alpha + \cos 3\alpha)} = \operatorname{tg} 2\alpha \cdot \cos \alpha$.
5C) Найдите значение выражения $\cos 72^\circ \cdot \sin 54^\circ$.
Решение. №830 (с. 227)
Решение 2 (rus). №830 (с. 227)
1А) Вычислите:
а) Для вычисления $ \cos\frac{15\pi}{4} $ воспользуемся периодичностью косинуса ($ 2\pi $) и его четностью.
Представим угол в виде: $ \frac{15\pi}{4} = \frac{16\pi - \pi}{4} = 4\pi - \frac{\pi}{4} $.
Используя периодичность, отбросим $ 4\pi $ (два полных оборота):
$ \cos\frac{15\pi}{4} = \cos(4\pi - \frac{\pi}{4}) = \cos(-\frac{\pi}{4}) $.
Так как косинус - четная функция ($ \cos(-x) = \cos(x) $), то $ \cos(-\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} $.
Ответ: $ \frac{\sqrt{2}}{2} $.
б) Для вычисления $ \sin(-\frac{17\pi}{6}) $ воспользуемся нечетностью синуса и его периодичностью.
Синус - нечетная функция ($ \sin(-x) = -\sin(x) $), поэтому $ \sin(-\frac{17\pi}{6}) = -\sin(\frac{17\pi}{6}) $.
Выделим целую часть оборотов: $ \frac{17\pi}{6} = \frac{12\pi + 5\pi}{6} = 2\pi + \frac{5\pi}{6} $.
$ -\sin(\frac{17\pi}{6}) = -\sin(2\pi + \frac{5\pi}{6}) = -\sin(\frac{5\pi}{6}) $.
Используем формулу приведения: $ \sin(\frac{5\pi}{6}) = \sin(\pi - \frac{\pi}{6}) = \sin(\frac{\pi}{6}) = \frac{1}{2} $.
Следовательно, искомое значение равно $ -\frac{1}{2} $.
Ответ: $ -\frac{1}{2} $.
в) Для вычисления $ \text{tg } 600^\circ $ воспользуемся периодичностью тангенса ($ 180^\circ $).
Представим угол в виде: $ 600^\circ = 3 \cdot 180^\circ + 60^\circ $.
$ \text{tg } 600^\circ = \text{tg}(3 \cdot 180^\circ + 60^\circ) = \text{tg } 60^\circ = \sqrt{3} $.
Ответ: $ \sqrt{3} $.
2А) Упростите выражение:
а) Раскроем скобки и используем тригонометрические тождества:
$ (\sin\alpha - \cos\alpha)^2 - 1 + 4\sin2\alpha = (\sin^2\alpha - 2\sin\alpha\cos\alpha + \cos^2\alpha) - 1 + 4\sin2\alpha $.
Используя основное тригонометрическое тождество $ \sin^2\alpha + \cos^2\alpha = 1 $ и формулу двойного угла $ \sin2\alpha = 2\sin\alpha\cos\alpha $, получаем:
$ (1 - \sin2\alpha) - 1 + 4\sin2\alpha = 1 - \sin2\alpha - 1 + 4\sin2\alpha = 3\sin2\alpha $.
Ответ: $ 3\sin2\alpha $.
б) Используем формулы приведения:
$ \text{tg}(\frac{3\pi}{2} - \alpha) = \text{ctg}\alpha $ (угол в III четверти, тангенс положителен, функция меняется на кофункцию).
$ \text{ctg}(\pi + \alpha) = \text{ctg}\alpha $ (угол в III четверти, котангенс положителен, функция не меняется).
Подставляем в исходное выражение:
$ 1 + \text{tg}(\frac{3\pi}{2} - \alpha) \cdot \text{ctg}(\pi + \alpha) = 1 + \text{ctg}\alpha \cdot \text{ctg}\alpha = 1 + \text{ctg}^2\alpha $.
По основному тригонометрическому тождеству $ 1 + \text{ctg}^2\alpha = \frac{1}{\sin^2\alpha} $.
Ответ: $ \frac{1}{\sin^2\alpha} $.
3А) Дано: $ \cos\alpha = -\frac{4}{5} $, $ \pi < \alpha < \frac{3\pi}{2} $.
Сначала найдем $ \sin\alpha $. Из основного тригонометрического тождества $ \sin^2\alpha + \cos^2\alpha = 1 $:
$ \sin^2\alpha = 1 - \cos^2\alpha = 1 - (-\frac{4}{5})^2 = 1 - \frac{16}{25} = \frac{9}{25} $.
Так как угол $ \alpha $ находится в III четверти ($ \pi < \alpha < \frac{3\pi}{2} $), синус отрицателен: $ \sin\alpha = -\sqrt{\frac{9}{25}} = -\frac{3}{5} $.
а) Найдем $ \cos2\alpha $ по формуле двойного угла:
$ \cos2\alpha = 2\cos^2\alpha - 1 = 2(-\frac{4}{5})^2 - 1 = 2(\frac{16}{25}) - 1 = \frac{32}{25} - \frac{25}{25} = \frac{7}{25} $.
Ответ: $ \frac{7}{25} $.
б) Найдем $ \sin(60^\circ + \alpha) $ по формуле синуса суммы:
$ \sin(60^\circ + \alpha) = \sin60^\circ\cos\alpha + \cos60^\circ\sin\alpha $.
Подставляем известные значения $ \sin60^\circ = \frac{\sqrt{3}}{2} $, $ \cos60^\circ = \frac{1}{2} $, $ \cos\alpha = -\frac{4}{5} $ и $ \sin\alpha = -\frac{3}{5} $:
$ \sin(60^\circ + \alpha) = \frac{\sqrt{3}}{2} \cdot (-\frac{4}{5}) + \frac{1}{2} \cdot (-\frac{3}{5}) = -\frac{4\sqrt{3}}{10} - \frac{3}{10} = -\frac{3+4\sqrt{3}}{10} $.
Ответ: $ -\frac{3+4\sqrt{3}}{10} $.
в) Найдем $ \text{tg}(45^\circ - \alpha) $ по формуле тангенса разности. Сначала найдем $ \text{tg}\alpha $:
$ \text{tg}\alpha = \frac{\sin\alpha}{\cos\alpha} = \frac{-3/5}{-4/5} = \frac{3}{4} $.
Теперь используем формулу $ \text{tg}(45^\circ - \alpha) = \frac{\text{tg}45^\circ - \text{tg}\alpha}{1 + \text{tg}45^\circ\text{tg}\alpha} $.
Подставляем $ \text{tg}45^\circ = 1 $ и $ \text{tg}\alpha = \frac{3}{4} $:
$ \text{tg}(45^\circ - \alpha) = \frac{1 - \frac{3}{4}}{1 + 1 \cdot \frac{3}{4}} = \frac{\frac{4-3}{4}}{\frac{4+3}{4}} = \frac{1/4}{7/4} = \frac{1}{7} $.
Ответ: $ \frac{1}{7} $.
4B) Докажите тождество
Требуется доказать тождество $ \frac{2 \sin 2\alpha + \sin 4\alpha}{2(\cos \alpha + \cos 3\alpha)} = \text{tg } 2\alpha \cdot \cos \alpha $.
Преобразуем левую часть (ЛЧ) тождества.
Преобразуем числитель, используя формулу синуса двойного угла $ \sin 4\alpha = 2 \sin 2\alpha \cos 2\alpha $:
$ 2 \sin 2\alpha + \sin 4\alpha = 2 \sin 2\alpha + 2 \sin 2\alpha \cos 2\alpha = 2 \sin 2\alpha (1 + \cos 2\alpha) $.
Преобразуем знаменатель, используя формулу суммы косинусов $ \cos x + \cos y = 2\cos\frac{x+y}{2}\cos\frac{x-y}{2} $:
$ 2(\cos \alpha + \cos 3\alpha) = 2 \cdot (2\cos\frac{\alpha+3\alpha}{2}\cos\frac{3\alpha-\alpha}{2}) = 4\cos 2\alpha \cos\alpha $.
Запишем преобразованную дробь:
ЛЧ $ = \frac{2 \sin 2\alpha (1 + \cos 2\alpha)}{4\cos 2\alpha \cos\alpha} $.
Используем формулу косинуса двойного угла для понижения степени $ 1 + \cos 2\alpha = 2\cos^2\alpha $:
ЛЧ $ = \frac{2 \sin 2\alpha \cdot (2\cos^2\alpha)}{4\cos 2\alpha \cos\alpha} = \frac{4 \sin 2\alpha \cos^2\alpha}{4\cos 2\alpha \cos\alpha} $.
Сокращаем общие множители $ 4 $ и $ \cos\alpha $:
ЛЧ $ = \frac{\sin 2\alpha \cos\alpha}{\cos 2\alpha} = \frac{\sin 2\alpha}{\cos 2\alpha} \cdot \cos\alpha = \text{tg } 2\alpha \cdot \cos\alpha $.
Левая часть равна правой, тождество доказано.
5С) Найдите значение выражения
Требуется найти значение $ \cos 72^\circ \cdot \sin 54^\circ $.
Используем формулу приведения: $ \sin 54^\circ = \sin(90^\circ - 36^\circ) = \cos 36^\circ $.
Выражение принимает вид: $ \cos 72^\circ \cdot \cos 36^\circ $.
Домножим и разделим выражение на $ 2\sin 36^\circ $ (это возможно, так как $ \sin 36^\circ \neq 0 $):
$ \frac{2\sin 36^\circ \cos 36^\circ \cdot \cos 72^\circ}{2\sin 36^\circ} $.
Используем формулу синуса двойного угла $ \sin 2x = 2\sin x \cos x $ для числителя:
$ \frac{\sin(2 \cdot 36^\circ) \cdot \cos 72^\circ}{2\sin 36^\circ} = \frac{\sin 72^\circ \cdot \cos 72^\circ}{2\sin 36^\circ} $.
Снова применяем ту же формулу для числителя:
$ \frac{\frac{1}{2} \cdot (2\sin 72^\circ \cos 72^\circ)}{2\sin 36^\circ} = \frac{\frac{1}{2} \sin(2 \cdot 72^\circ)}{2\sin 36^\circ} = \frac{\frac{1}{2} \sin 144^\circ}{2\sin 36^\circ} $.
Используем формулу приведения: $ \sin 144^\circ = \sin(180^\circ - 36^\circ) = \sin 36^\circ $.
Подставляем и сокращаем:
$ \frac{\frac{1}{2} \sin 36^\circ}{2\sin 36^\circ} = \frac{1/2}{2} = \frac{1}{4} $.
Ответ: $ \frac{1}{4} $.
Другие задания:
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ПрисоединитьсяМы подготовили для вас ответ c подробным объяснением домашего задания по алгебре за 9 класс, для упражнения номер 830 расположенного на странице 227 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 №830 (с. 227), авторов: Солтан (Генадий Николаевич), Солтан (Алла Евгеньевна), Жумадилова (Аманбала Жумадиловна), учебного пособия издательства Кокшетау.