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Номер 681, страница 195 - 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. 25. Формулы приведения - номер 681, страница 195.
App\Models\Task {#1024 // 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" => 1108418 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-681" "field_display_title" => "681" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #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 {#333 #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 {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #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 {#1032 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1035 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1050 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 #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 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 #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 {#1032} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037} "field_author" => Illuminate\Database\Eloquent\Collection {#1039} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044} "field_country" => Illuminate\Database\Eloquent\Collection {#1046} "field_city" => Illuminate\Database\Eloquent\Collection {#1048} "field_series" => Illuminate\Database\Eloquent\Collection {#1050} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055} "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 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059} "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 {#1060 #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 {#333} "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 {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #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 {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #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" => 1107667 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "25. Формулы приведения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "189" "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 {#1064 #items: array:1 [ 0 => App\Models\Book {#1065 #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 {#1066 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1066} "field_class" => Illuminate\Database\Eloquent\Collection {#1068} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070} "field_author" => Illuminate\Database\Eloquent\Collection {#1072} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077} "field_country" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1093 #items: array:1 [ 0 => App\Models\Branch {#1094 #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 {#1095 …2} "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 {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 …2} ] #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 {#1095 …2} "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 {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 …2} ] #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" => 1107667 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "25. Формулы приведения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "189" "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 {#1064} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093} ] #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 {#1098 #items: array:3 [ 0 => App\Models\Element {#1099 #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" => 1277036 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1101 #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 {#1102 …2} "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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …2} "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 {#1102 …2} "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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …2} "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" => "1108418" "type" => "task" ] "text" => "<p><strong>681.</strong> Докажите, что верно неравенство:</p><p><strong>а)</strong> $ \sin^2(0,5\pi + x) - \operatorname{tg}^2(0,5\pi + x) \le 0 $ при всех допустимых значениях x;</p><p><strong>б)</strong> $ \operatorname{tg}(1,5\pi - x) + \operatorname{ctg}(1,5\pi - x) > \sqrt{\pi} $ при $ x \in (0; \frac{\pi}{2}) $.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "681-1.jpg" "alt" => null "width" => "1543" "height" => 299 "path" => "/media/algebra_09/soltan-u19/0-1/681-1.webp?ts=1752834689" ] ] ] #original: array:7 [ "id" => 1277036 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1108418" "type" => "task" ] "text" => "<p><strong>681.</strong> Докажите, что верно неравенство:</p><p><strong>а)</strong> $ \sin^2(0,5\pi + x) - \operatorname{tg}^2(0,5\pi + x) \le 0 $ при всех допустимых значениях x;</p><p><strong>б)</strong> $ \operatorname{tg}(1,5\pi - x) + \operatorname{ctg}(1,5\pi - x) > \sqrt{\pi} $ при $ x \in (0; \frac{\pi}{2}) $.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "681-1.jpg" "alt" => null "width" => "1543" "height" => 299 "path" => "/media/algebra_09/soltan-u19/0-1/681-1.webp?ts=1752834689" ] ] ] #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 {#1107 #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" => 1278327 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1109 #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 {#1110 …2} "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 {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …2} "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 {#1110 …2} "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 {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …2} "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" => "1108418" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "681-1.jpg" "alt" => null "width" => "946" "height" => 575 "path" => "/media/algebra_09/soltan-u19/1-1/681-1.webp?ts=1752831477" ] 1 => array:5 [ "name" => "681-2.jpg" "alt" => null "width" => "1364" "height" => 2040 "path" => "/media/algebra_09/soltan-u19/1-1/681-2.webp?ts=1752831477" ] ] ] #original: array:6 [ "id" => 1278327 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1108418" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "681-1.jpg" "alt" => null "width" => "946" "height" => 575 "path" => "/media/algebra_09/soltan-u19/1-1/681-1.webp?ts=1752831477" ] 1 => array:5 [ "name" => "681-2.jpg" "alt" => null "width" => "1364" "height" => 2040 "path" => "/media/algebra_09/soltan-u19/1-1/681-2.webp?ts=1752831477" ] ] ] #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 {#1115 #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" => 1554282 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1117 #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 {#1118 …2} "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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "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 {#1118 …2} "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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "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" => "1108418" "type" => "task" ] "text" => "<p><strong>а)</strong> Требуется доказать неравенство $ \sin^2(0,5\pi + x) - \tg^2(0,5\pi + x) \le 0 $ при всех допустимых значениях $ x $.</p><p>Сначала преобразуем левую часть неравенства, используя формулы приведения. Аргумент функций равен $ 0,5\pi + x = \frac{\pi}{2} + x $.</p><p>Для синуса: $ \sin(\frac{\pi}{2} + x) = \cos(x) $.</p><p>Для тангенса: $ \tg(\frac{\pi}{2} + x) = -\ctg(x) $.</p><p>Подставив эти выражения в левую часть неравенства, получим:</p><p>$ \sin^2(\frac{\pi}{2} + x) - \tg^2(\frac{\pi}{2} + x) = (\cos(x))^2 - (-\ctg(x))^2 = \cos^2(x) - \ctg^2(x) $.</p><p>Область допустимых значений (ОДЗ) исходного выражения определяется условием существования $ \tg(\frac{\pi}{2} + x) $. Тангенс не определен, когда его аргумент равен $ \frac{\pi}{2} + \pi k $, где $ k $ - целое число. Таким образом, $ \frac{\pi}{2} + x \neq \frac{\pi}{2} + \pi k $, что приводит к условию $ x \neq \pi k $, $ k \in \mathbb{Z} $. При этих значениях $ x $ также выполняется условие $ \sin(x) \neq 0 $, необходимое для существования котангенса.</p><p>Теперь преобразуем полученное выражение, используя определение котангенса $ \ctg(x) = \frac{\cos(x)}{\sin(x)} $:</p><p>$ \cos^2(x) - \ctg^2(x) = \cos^2(x) - \frac{\cos^2(x)}{\sin^2(x)} $.</p><p>Вынесем $ \cos^2(x) $ за скобки:</p><p>$ \cos^2(x) \left(1 - \frac{1}{\sin^2(x)}\right) = \cos^2(x) \left(\frac{\sin^2(x) - 1}{\sin^2(x)}\right) $.</p><p>Из основного тригонометрического тождества $ \sin^2(x) + \cos^2(x) = 1 $ следует, что $ \sin^2(x) - 1 = -\cos^2(x) $. Подставим это в наше выражение:</p><p>$ \cos^2(x) \left(\frac{-\cos^2(x)}{\sin^2(x)}\right) = -\frac{\cos^4(x)}{\sin^2(x)} $.</p><p>Проанализируем знак полученного выражения. Поскольку $ \cos^4(x) = (\cos^2(x))^2 \ge 0 $ для любого $ x $, и $ \sin^2(x) > 0 $ на области допустимых значений, то дробь $ \frac{\cos^4(x)}{\sin^2(x)} $ всегда неотрицательна. Следовательно, выражение $ -\frac{\cos^4(x)}{\sin^2(x)} $ всегда меньше или равно нулю.</p><p>Таким образом, мы доказали, что $ \sin^2(0,5\pi + x) - \tg^2(0,5\pi + x) \le 0 $ для всех допустимых $ x $.</p><p><strong>Ответ:</strong> Неравенство доказано.</p><p><strong>б)</strong> Требуется доказать неравенство $ \tg(1,5\pi - x) + \ctg(1,5\pi - x) > \sqrt{\pi} $ при $ x \in (0; \frac{\pi}{2}) $.</p><p>Преобразуем левую часть неравенства с помощью формул приведения. Аргумент функций равен $ 1,5\pi - x = \frac{3\pi}{2} - x $.</p><p>Для тангенса: $ \tg(\frac{3\pi}{2} - x) = \ctg(x) $.</p><p>Для котангенса: $ \ctg(\frac{3\pi}{2} - x) = \tg(x) $.</p><p>Левая часть неравенства принимает вид:</p><p>$ \ctg(x) + \tg(x) $.</p><p>Приведем выражение к общему знаменателю:</p><p>$ \frac{\cos(x)}{\sin(x)} + \frac{\sin(x)}{\cos(x)} = \frac{\cos^2(x) + \sin^2(x)}{\sin(x)\cos(x)} $.</p><p>Используя основное тригонометрическое тождество $ \sin^2(x) + \cos^2(x) = 1 $ и формулу синуса двойного угла $ \sin(2x) = 2\sin(x)\cos(x) $, получим:</p><p>$ \frac{1}{\sin(x)\cos(x)} = \frac{2}{2\sin(x)\cos(x)} = \frac{2}{\sin(2x)} $.</p><p>Теперь нужно исследовать значения этого выражения при $ x \in (0; \frac{\pi}{2}) $.</p><p>Если $ x $ изменяется в интервале $ (0; \frac{\pi}{2}) $, то аргумент $ 2x $ изменяется в интервале $ (0; \pi) $.</p><p>В интервале $ (0; \pi) $ синус принимает положительные значения, причем $ 0 < \sin(2x) \le 1 $. Максимальное значение $ \sin(2x) = 1 $ достигается при $ 2x = \frac{\pi}{2} $, то есть $ x = \frac{\pi}{4} $.</p><p>Наименьшее значение выражения $ \frac{2}{\sin(2x)} $ достигается тогда, когда знаменатель $ \sin(2x) $ максимален. Таким образом, минимальное значение левой части неравенства равно $ \frac{2}{1} = 2 $.</p><p>Итак, для всех $ x \in (0; \frac{\pi}{2}) $ выполняется неравенство $ \tg(1,5\pi - x) + \ctg(1,5\pi - x) \ge 2 $.</p><p>Осталось сравнить $ 2 $ и $ \sqrt{\pi} $. Мы знаем, что число $ \pi \approx 3,14159... $, следовательно $ \pi < 4 $. Так как функция квадратного корня является возрастающей для положительных чисел, из $ \pi < 4 $ следует, что $ \sqrt{\pi} < \sqrt{4} $, то есть $ \sqrt{\pi} < 2 $.</p><p>Мы установили, что $ \tg(1,5\pi - x) + \ctg(1,5\pi - x) \ge 2 $ и $ 2 > \sqrt{\pi} $. Отсюда следует, что $ \tg(1,5\pi - x) + \ctg(1,5\pi - x) > \sqrt{\pi} $ для всех $ x \in (0; \frac{\pi}{2}) $.</p><p><strong>Ответ:</strong> Неравенство доказано.</p>" ] #original: array:6 [ "id" => 1554282 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1108418" "type" => "task" ] "text" => "<p><strong>а)</strong> Требуется доказать неравенство $ \sin^2(0,5\pi + x) - \tg^2(0,5\pi + x) \le 0 $ при всех допустимых значениях $ x $.</p><p>Сначала преобразуем левую часть неравенства, используя формулы приведения. Аргумент функций равен $ 0,5\pi + x = \frac{\pi}{2} + x $.</p><p>Для синуса: $ \sin(\frac{\pi}{2} + x) = \cos(x) $.</p><p>Для тангенса: $ \tg(\frac{\pi}{2} + x) = -\ctg(x) $.</p><p>Подставив эти выражения в левую часть неравенства, получим:</p><p>$ \sin^2(\frac{\pi}{2} + x) - \tg^2(\frac{\pi}{2} + x) = (\cos(x))^2 - (-\ctg(x))^2 = \cos^2(x) - \ctg^2(x) $.</p><p>Область допустимых значений (ОДЗ) исходного выражения определяется условием существования $ \tg(\frac{\pi}{2} + x) $. Тангенс не определен, когда его аргумент равен $ \frac{\pi}{2} + \pi k $, где $ k $ - целое число. Таким образом, $ \frac{\pi}{2} + x \neq \frac{\pi}{2} + \pi k $, что приводит к условию $ x \neq \pi k $, $ k \in \mathbb{Z} $. При этих значениях $ x $ также выполняется условие $ \sin(x) \neq 0 $, необходимое для существования котангенса.</p><p>Теперь преобразуем полученное выражение, используя определение котангенса $ \ctg(x) = \frac{\cos(x)}{\sin(x)} $:</p><p>$ \cos^2(x) - \ctg^2(x) = \cos^2(x) - \frac{\cos^2(x)}{\sin^2(x)} $.</p><p>Вынесем $ \cos^2(x) $ за скобки:</p><p>$ \cos^2(x) \left(1 - \frac{1}{\sin^2(x)}\right) = \cos^2(x) \left(\frac{\sin^2(x) - 1}{\sin^2(x)}\right) $.</p><p>Из основного тригонометрического тождества $ \sin^2(x) + \cos^2(x) = 1 $ следует, что $ \sin^2(x) - 1 = -\cos^2(x) $. Подставим это в наше выражение:</p><p>$ \cos^2(x) \left(\frac{-\cos^2(x)}{\sin^2(x)}\right) = -\frac{\cos^4(x)}{\sin^2(x)} $.</p><p>Проанализируем знак полученного выражения. Поскольку $ \cos^4(x) = (\cos^2(x))^2 \ge 0 $ для любого $ x $, и $ \sin^2(x) > 0 $ на области допустимых значений, то дробь $ \frac{\cos^4(x)}{\sin^2(x)} $ всегда неотрицательна. Следовательно, выражение $ -\frac{\cos^4(x)}{\sin^2(x)} $ всегда меньше или равно нулю.</p><p>Таким образом, мы доказали, что $ \sin^2(0,5\pi + x) - \tg^2(0,5\pi + x) \le 0 $ для всех допустимых $ x $.</p><p><strong>Ответ:</strong> Неравенство доказано.</p><p><strong>б)</strong> Требуется доказать неравенство $ \tg(1,5\pi - x) + \ctg(1,5\pi - x) > \sqrt{\pi} $ при $ x \in (0; \frac{\pi}{2}) $.</p><p>Преобразуем левую часть неравенства с помощью формул приведения. Аргумент функций равен $ 1,5\pi - x = \frac{3\pi}{2} - x $.</p><p>Для тангенса: $ \tg(\frac{3\pi}{2} - x) = \ctg(x) $.</p><p>Для котангенса: $ \ctg(\frac{3\pi}{2} - x) = \tg(x) $.</p><p>Левая часть неравенства принимает вид:</p><p>$ \ctg(x) + \tg(x) $.</p><p>Приведем выражение к общему знаменателю:</p><p>$ \frac{\cos(x)}{\sin(x)} + \frac{\sin(x)}{\cos(x)} = \frac{\cos^2(x) + \sin^2(x)}{\sin(x)\cos(x)} $.</p><p>Используя основное тригонометрическое тождество $ \sin^2(x) + \cos^2(x) = 1 $ и формулу синуса двойного угла $ \sin(2x) = 2\sin(x)\cos(x) $, получим:</p><p>$ \frac{1}{\sin(x)\cos(x)} = \frac{2}{2\sin(x)\cos(x)} = \frac{2}{\sin(2x)} $.</p><p>Теперь нужно исследовать значения этого выражения при $ x \in (0; \frac{\pi}{2}) $.</p><p>Если $ x $ изменяется в интервале $ (0; \frac{\pi}{2}) $, то аргумент $ 2x $ изменяется в интервале $ (0; \pi) $.</p><p>В интервале $ (0; \pi) $ синус принимает положительные значения, причем $ 0 < \sin(2x) \le 1 $. Максимальное значение $ \sin(2x) = 1 $ достигается при $ 2x = \frac{\pi}{2} $, то есть $ x = \frac{\pi}{4} $.</p><p>Наименьшее значение выражения $ \frac{2}{\sin(2x)} $ достигается тогда, когда знаменатель $ \sin(2x) $ максимален. Таким образом, минимальное значение левой части неравенства равно $ \frac{2}{1} = 2 $.</p><p>Итак, для всех $ x \in (0; \frac{\pi}{2}) $ выполняется неравенство $ \tg(1,5\pi - x) + \ctg(1,5\pi - x) \ge 2 $.</p><p>Осталось сравнить $ 2 $ и $ \sqrt{\pi} $. Мы знаем, что число $ \pi \approx 3,14159... $, следовательно $ \pi < 4 $. Так как функция квадратного корня является возрастающей для положительных чисел, из $ \pi < 4 $ следует, что $ \sqrt{\pi} < \sqrt{4} $, то есть $ \sqrt{\pi} < 2 $.</p><p>Мы установили, что $ \tg(1,5\pi - x) + \ctg(1,5\pi - x) \ge 2 $ и $ 2 > \sqrt{\pi} $. Отсюда следует, что $ \tg(1,5\pi - x) + \ctg(1,5\pi - x) > \sqrt{\pi} $ для всех $ x \in (0; \frac{\pi}{2}) $.</p><p><strong>Ответ:</strong> 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" => "1108419" "type" => "task" ] "previous" => array:2 [ "refs" => "1108417" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Book {#1124 #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" => 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Illuminate\Database\Eloquent\Collection {#1142} "field_umk" => Illuminate\Database\Eloquent\Collection {#1143} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1144} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1145} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1147} "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 {#1148} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151} "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 } "page" => Illuminate\Database\Eloquent\Collection {#1392 #items: array:1 [ 0 => App\Models\BookPage {#1155 #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" => 1097937 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-195" "field_display_title" => "195" "field_folder" => "folder1" "field_image_name" => "195" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1159 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Book {#1160 #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 {#1161 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1183 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1161 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1183 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 "edition_groups" => Illuminate\Database\Eloquent\Collection {#1188 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1189 #items: array:1 [ 0 => App\Models\Element {#1190 #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" => 1261088 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261088 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #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" => "1097938" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097936" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:6 [ 0 => App\Models\Task {#1199 #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" => 1108413 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-676" "field_display_title" => "676" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200 …2} "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 {#1202 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236 …2} "content" => Illuminate\Database\Eloquent\Collection {#1305 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1324 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108413 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-676" "field_display_title" => "676" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200 …2} "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 {#1202 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236 …2} "content" => Illuminate\Database\Eloquent\Collection {#1305 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1324 …2} "page" => array:2 [ …2] ] #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 {#1325 #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" => 1108414 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-677" "field_display_title" => "677" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1326 …2} "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 {#1327 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1328 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1329 …2} "content" => Illuminate\Database\Eloquent\Collection {#1330 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1337 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108414 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-677" "field_display_title" => "677" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1326 …2} "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 {#1327 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1328 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1329 …2} "content" => Illuminate\Database\Eloquent\Collection {#1330 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1337 …2} "page" => array:2 [ …2] ] #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 {#1338 #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" => 1108415 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-678" "field_display_title" => "678" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1339 …2} "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 {#1340 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1341 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1342 …2} "content" => Illuminate\Database\Eloquent\Collection {#1343 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1350 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108415 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-678" "field_display_title" => "678" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1339 …2} "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 {#1340 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1341 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1342 …2} "content" => Illuminate\Database\Eloquent\Collection {#1343 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1350 …2} "page" => array:2 [ …2] ] #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 {#1351 #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" => 1108416 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-679" "field_display_title" => "679" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1352 …2} "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 {#1353 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1354 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1355 …2} "content" => Illuminate\Database\Eloquent\Collection {#1356 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1363 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108416 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-679" "field_display_title" => "679" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1352 …2} "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 {#1353 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1354 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1355 …2} "content" => Illuminate\Database\Eloquent\Collection {#1356 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1363 …2} "page" => array:2 [ …2] ] #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 {#1364 #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" => 1108417 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-680" "field_display_title" => "680" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1365 …2} "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 {#1366 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1367 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1368 …2} "content" => Illuminate\Database\Eloquent\Collection {#1369 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1376 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108417 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-680" "field_display_title" => "680" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1365 …2} "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 {#1366 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1367 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1368 …2} "content" => Illuminate\Database\Eloquent\Collection {#1369 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1376 …2} "page" => array:2 [ …2] ] #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 {#1377 #connection: "mysql" 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№681 (с. 195)
Условие. №681 (с. 195)
скриншот условия
681. Докажите, что верно неравенство:
а) $ \sin^2(0,5\pi + x) - \operatorname{tg}^2(0,5\pi + x) \le 0 $ при всех допустимых значениях x;
б) $ \operatorname{tg}(1,5\pi - x) + \operatorname{ctg}(1,5\pi - x) > \sqrt{\pi} $ при $ x \in (0; \frac{\pi}{2}) $.
Решение. №681 (с. 195)
Решение 2 (rus). №681 (с. 195)
а) Требуется доказать неравенство $ \sin^2(0,5\pi + x) - \tg^2(0,5\pi + x) \le 0 $ при всех допустимых значениях $ x $.
Сначала преобразуем левую часть неравенства, используя формулы приведения. Аргумент функций равен $ 0,5\pi + x = \frac{\pi}{2} + x $.
Для синуса: $ \sin(\frac{\pi}{2} + x) = \cos(x) $.
Для тангенса: $ \tg(\frac{\pi}{2} + x) = -\ctg(x) $.
Подставив эти выражения в левую часть неравенства, получим:
$ \sin^2(\frac{\pi}{2} + x) - \tg^2(\frac{\pi}{2} + x) = (\cos(x))^2 - (-\ctg(x))^2 = \cos^2(x) - \ctg^2(x) $.
Область допустимых значений (ОДЗ) исходного выражения определяется условием существования $ \tg(\frac{\pi}{2} + x) $. Тангенс не определен, когда его аргумент равен $ \frac{\pi}{2} + \pi k $, где $ k $ - целое число. Таким образом, $ \frac{\pi}{2} + x \neq \frac{\pi}{2} + \pi k $, что приводит к условию $ x \neq \pi k $, $ k \in \mathbb{Z} $. При этих значениях $ x $ также выполняется условие $ \sin(x) \neq 0 $, необходимое для существования котангенса.
Теперь преобразуем полученное выражение, используя определение котангенса $ \ctg(x) = \frac{\cos(x)}{\sin(x)} $:
$ \cos^2(x) - \ctg^2(x) = \cos^2(x) - \frac{\cos^2(x)}{\sin^2(x)} $.
Вынесем $ \cos^2(x) $ за скобки:
$ \cos^2(x) \left(1 - \frac{1}{\sin^2(x)}\right) = \cos^2(x) \left(\frac{\sin^2(x) - 1}{\sin^2(x)}\right) $.
Из основного тригонометрического тождества $ \sin^2(x) + \cos^2(x) = 1 $ следует, что $ \sin^2(x) - 1 = -\cos^2(x) $. Подставим это в наше выражение:
$ \cos^2(x) \left(\frac{-\cos^2(x)}{\sin^2(x)}\right) = -\frac{\cos^4(x)}{\sin^2(x)} $.
Проанализируем знак полученного выражения. Поскольку $ \cos^4(x) = (\cos^2(x))^2 \ge 0 $ для любого $ x $, и $ \sin^2(x) > 0 $ на области допустимых значений, то дробь $ \frac{\cos^4(x)}{\sin^2(x)} $ всегда неотрицательна. Следовательно, выражение $ -\frac{\cos^4(x)}{\sin^2(x)} $ всегда меньше или равно нулю.
Таким образом, мы доказали, что $ \sin^2(0,5\pi + x) - \tg^2(0,5\pi + x) \le 0 $ для всех допустимых $ x $.
Ответ: Неравенство доказано.
б) Требуется доказать неравенство $ \tg(1,5\pi - x) + \ctg(1,5\pi - x) > \sqrt{\pi} $ при $ x \in (0; \frac{\pi}{2}) $.
Преобразуем левую часть неравенства с помощью формул приведения. Аргумент функций равен $ 1,5\pi - x = \frac{3\pi}{2} - x $.
Для тангенса: $ \tg(\frac{3\pi}{2} - x) = \ctg(x) $.
Для котангенса: $ \ctg(\frac{3\pi}{2} - x) = \tg(x) $.
Левая часть неравенства принимает вид:
$ \ctg(x) + \tg(x) $.
Приведем выражение к общему знаменателю:
$ \frac{\cos(x)}{\sin(x)} + \frac{\sin(x)}{\cos(x)} = \frac{\cos^2(x) + \sin^2(x)}{\sin(x)\cos(x)} $.
Используя основное тригонометрическое тождество $ \sin^2(x) + \cos^2(x) = 1 $ и формулу синуса двойного угла $ \sin(2x) = 2\sin(x)\cos(x) $, получим:
$ \frac{1}{\sin(x)\cos(x)} = \frac{2}{2\sin(x)\cos(x)} = \frac{2}{\sin(2x)} $.
Теперь нужно исследовать значения этого выражения при $ x \in (0; \frac{\pi}{2}) $.
Если $ x $ изменяется в интервале $ (0; \frac{\pi}{2}) $, то аргумент $ 2x $ изменяется в интервале $ (0; \pi) $.
В интервале $ (0; \pi) $ синус принимает положительные значения, причем $ 0 < \sin(2x) \le 1 $. Максимальное значение $ \sin(2x) = 1 $ достигается при $ 2x = \frac{\pi}{2} $, то есть $ x = \frac{\pi}{4} $.
Наименьшее значение выражения $ \frac{2}{\sin(2x)} $ достигается тогда, когда знаменатель $ \sin(2x) $ максимален. Таким образом, минимальное значение левой части неравенства равно $ \frac{2}{1} = 2 $.
Итак, для всех $ x \in (0; \frac{\pi}{2}) $ выполняется неравенство $ \tg(1,5\pi - x) + \ctg(1,5\pi - x) \ge 2 $.
Осталось сравнить $ 2 $ и $ \sqrt{\pi} $. Мы знаем, что число $ \pi \approx 3,14159... $, следовательно $ \pi < 4 $. Так как функция квадратного корня является возрастающей для положительных чисел, из $ \pi < 4 $ следует, что $ \sqrt{\pi} < \sqrt{4} $, то есть $ \sqrt{\pi} < 2 $.
Мы установили, что $ \tg(1,5\pi - x) + \ctg(1,5\pi - x) \ge 2 $ и $ 2 > \sqrt{\pi} $. Отсюда следует, что $ \tg(1,5\pi - x) + \ctg(1,5\pi - x) > \sqrt{\pi} $ для всех $ x \in (0; \frac{\pi}{2}) $.
Ответ: Неравенство доказано.
Другие задания:
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