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Номер 8, страница 115, часть 1 - гдз по алгебре 10 класс учебник Пак, Ардакулы
Авторы: Пак О. В., Ардакулы Д., Ескендирова Е. В.
Тип: Учебник
Издательство: Алматыкітап баспасы
Год издания: 2019 - 2026
Часть: 1
ISBN: 978-601-01-3958-9
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Часть 1. Глава 2. Тригонометрические функции. Параграф 4. Преобразования выражений, содержащих обратные тригонометрические функции. Задачи - номер 8, страница 115.
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" => 1119773 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "115" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/2-4zad-8" "field_display_title" => "8" "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 {#1072 #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" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #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_page_start" => "6" "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" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #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" => 4295 "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 {#1042 #items: array:1 [ 0 => App\Models\Term {#1043 #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 {#1044 #items: array:1 [ 0 => App\Models\Term {#1045 #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: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 #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] …17 } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:3 [ 0 => App\Models\Term {#1049 …30} 1 => App\Models\Term {#1050 …30} 2 => App\Models\Term {#1051 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ 0 => App\Models\Term {#1056 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ 0 => App\Models\Term {#1058 …30} ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 #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" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4295 "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 {#1042} "field_class" => Illuminate\Database\Eloquent\Collection {#1044} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046} "field_author" => Illuminate\Database\Eloquent\Collection {#1048} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053} "field_country" => Illuminate\Database\Eloquent\Collection {#1055} "field_city" => Illuminate\Database\Eloquent\Collection {#1057} "field_series" => Illuminate\Database\Eloquent\Collection {#1059} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #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 {#1069 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "6" "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" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069} ] #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 {#1172 #items: array:4 [ 0 => App\Models\Branch {#1182 #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" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1186 #items: array:1 [ 0 => App\Models\Term {#1183 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #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_page_start" => "6" "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" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1215 #items: array:1 [ 0 => App\Models\Book {#1185 #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" => 4295 "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 {#1187 #items: array:1 [ 0 => App\Models\Term {#1184 #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 {#1188 #items: array:1 [ 0 => App\Models\Term {#1189 #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" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #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 {#1190 #items: array:1 [ 0 => App\Models\Term {#1191 #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" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "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" => "almatykitap baspasy" ] #original: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "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" => "almatykitap baspasy" ] #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 {#1192 #items: array:3 [ 0 => App\Models\Term {#1193 #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" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #original: array:12 [ "id" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "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 {#1194 #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" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "field_surname_rp" => null ] #original: array:12 [ "id" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "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 {#1195 #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" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "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 {#1196 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1197 #items: array:1 [ 0 => App\Models\Term {#1198 #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 {#1199 #items: array:1 [ 0 => App\Models\Term {#1200 #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 {#1201 #items: array:1 [ 0 => App\Models\Term {#1202 #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 {#1203 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1204 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1205 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1206 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1207 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1208 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1209 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1210 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1211 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1212 #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" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4295 "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 {#1187} "field_class" => Illuminate\Database\Eloquent\Collection {#1188} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1190} "field_author" => Illuminate\Database\Eloquent\Collection {#1192} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1196} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1197} "field_country" => Illuminate\Database\Eloquent\Collection {#1199} "field_city" => Illuminate\Database\Eloquent\Collection {#1201} "field_series" => Illuminate\Database\Eloquent\Collection {#1203} "field_umk" => Illuminate\Database\Eloquent\Collection {#1204} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1205} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1206} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1207} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1208} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1209} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1210} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1211} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1212} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #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 {#1213 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1186} "field_page_start" => "6" "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" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1215} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1213} ] #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 {#1214 #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" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ 0 => App\Models\Term {#1216 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => 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_page_start" => "71" "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 {#1218 #items: array:1 [ 0 => App\Models\Book {#1185} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1219} "field_page_start" => "71" "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 {#1218} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1217} ] #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\Branch {#1220 #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" => 1119378 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Преобразования выражений, содержащих обратные тригонометрические функции" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:1 [ 0 => App\Models\Term {#1221 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "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_page_start" => "111" "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 {#1223 #items: array:1 [ 0 => App\Models\Book {#1185} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1292 #items: array:1 [ 0 => App\Models\Branch {#1222 #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" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:1 [ 0 => App\Models\Term {#1225 …30} ] #escapeWhenCastingToString: false } "field_page_start" => "71" "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 {#1227 #items: array:1 [ 0 => App\Models\Book {#1228 …30} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1256 #items: array:1 [ 0 => App\Models\Branch {#1257 …30} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119358 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1226} "field_page_start" => "71" "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 {#1227} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1256} ] #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" => 1119378 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Преобразования выражений, содержащих обратные тригонометрические функции" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1224} "field_page_start" => "111" "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 {#1223} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1292} ] #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\Branch {#1290 #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" => 1119379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1291 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "112" "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 {#1293 #items: array:1 [ 0 => App\Models\Book {#1185} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1302 #items: array:1 [ 0 => App\Models\Branch {#1301 #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" => 1119378 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Преобразования выражений, содержащих обратные тригонометрические функции" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1303 #items: array:1 [ 0 => App\Models\Term {#1221} ] #escapeWhenCastingToString: false } "field_page_start" => "111" "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 {#1304 #items: array:1 [ 0 => App\Models\Book {#1185} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1305 #items: array:1 [ 0 => App\Models\Branch {#1222} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119378 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Преобразования выражений, содержащих обратные тригонометрические функции" "field_branch_order" => "4" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1303} "field_page_start" => "111" "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 {#1304} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1305} ] #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" => 1119379 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1291} "field_page_start" => "112" "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 {#1293} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1302} ] #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 {#1104 #items: array:2 [ 0 => App\Models\Element {#1165 #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" => 1300458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155 #items: array:1 [ 0 => App\Models\Edition {#1164 #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" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1162 #items: array:1 [ 0 => App\Models\Term {#1163 …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 {#1161 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1160 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1159 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1162} "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 {#1161} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1160} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1159} "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" => "1119773" "type" => "task" ] "text" => "<p><strong>8. (5)</strong> Упростите следующие выражения:</p><p><strong>a)</strong> $arcsin(cos0.3\pi)$, $arccos(sin0.2\pi)$, $arccos(sin1.8\pi)$; $arcsin(cos4)$;</p><p><strong>б)</strong> $arcctg(tg1.8\pi)$, $arcctg(ctg1.8\pi)$, $arcctg(tg3)$, $arcctg(ctg7)$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1545" "height" => 325 "path" => "/media/algebra_10/baspasy-u19/0-2-4zad/8-1.webp?ts=1753534599" ] ] ] #original: array:7 [ "id" => 1300458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155} "task" => array:2 [ "refs" => "1119773" "type" => "task" ] "text" => "<p><strong>8. (5)</strong> Упростите следующие выражения:</p><p><strong>a)</strong> $arcsin(cos0.3\pi)$, $arccos(sin0.2\pi)$, $arccos(sin1.8\pi)$; $arcsin(cos4)$;</p><p><strong>б)</strong> $arcctg(tg1.8\pi)$, $arcctg(ctg1.8\pi)$, $arcctg(tg3)$, $arcctg(ctg7)$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1545" "height" => 325 "path" => "/media/algebra_10/baspasy-u19/0-2-4zad/8-1.webp?ts=1753534599" ] ] ] #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 {#1157 #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" => 1551494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1149 #items: array:1 [ 0 => App\Models\Edition {#1158 #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" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153 #items: array:1 [ 0 => App\Models\Term {#1156 …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 {#1154 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1152 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1148 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153} "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 {#1154} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1152} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1148} "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" => "1119773" "type" => "task" ] "text" => "<p><strong>а)</strong></p>\t<p>\t\tРешение для $arcsin(cos(0.3\pi))$:<br>\t\tИспользуем формулу приведения $cos(\alpha) = sin(\frac{\pi}{2} - \alpha)$.<br>\t\t$arcsin(cos(0.3\pi)) = arcsin(sin(\frac{\pi}{2} - 0.3\pi)) = arcsin(sin(0.5\pi - 0.3\pi)) = arcsin(sin(0.2\pi))$.<br>\t\tОбласть значений функции $arcsin(y)$ — это отрезок $[-\frac{\pi}{2}, \frac{\pi}{2}]$.<br>\t\tТак как $0.2\pi$ принадлежит этому отрезку ($-\frac{\pi}{2} \le 0.2\pi \le \frac{\pi}{2}$), то по определению арксинуса $arcsin(sin(0.2\pi)) = 0.2\pi$.<br>\t\t<b>Ответ:</b> $0.2\pi$.\t</p>\t<p>\t\tРешение для $arccos(sin(0.2\pi))$:<br>\t\tИспользуем формулу приведения $sin(\alpha) = cos(\frac{\pi}{2} - \alpha)$.<br>\t\t$arccos(sin(0.2\pi)) = arccos(cos(\frac{\pi}{2} - 0.2\pi)) = arccos(cos(0.3\pi))$.<br>\t\tОбласть значений функции $arccos(y)$ — это отрезок $[0, \pi]$.<br>\t\tТак как $0.3\pi$ принадлежит этому отрезку ($0 \le 0.3\pi \le \pi$), то по определению арккосинуса $arccos(cos(0.3\pi)) = 0.3\pi$.<br>\t\t<b>Ответ:</b> $0.3\pi$.\t</p>\t<p>\t\tРешение для $arccos(sin(1.8\pi))$:<br>\t\tИспользуем периодичность синуса $sin(1.8\pi) = sin(1.8\pi - 2\pi) = sin(-0.2\pi) = -sin(0.2\pi)$.<br>\t\tПолучаем $arccos(-sin(0.2\pi))$.<br>\t\tИспользуем свойство арккосинуса $arccos(-y) = \pi - arccos(y)$.<br>\t\t$\pi - arccos(sin(0.2\pi))$.<br>\t\tИз предыдущего примера мы знаем, что $arccos(sin(0.2\pi)) = 0.3\pi$.<br>\t\tСледовательно, выражение равно $\pi - 0.3\pi = 0.7\pi$.<br>\t\t<b>Ответ:</b> $0.7\pi$.\t</p>\t<p>\t\tРешение для $arcsin(cos(4))$:<br>\t\tИспользуем формулу приведения $cos(\alpha) = sin(\frac{\pi}{2} - \alpha)$.<br>\t\t$arcsin(cos(4)) = arcsin(sin(\frac{\pi}{2} - 4))$.<br>\t\tПусть $x = \frac{\pi}{2} - 4$. Область значений арксинуса — $[-\frac{\pi}{2}, \frac{\pi}{2}]$.<br>\t\tОценим $x$: $x \approx 1.57 - 4 = -2.43$. Это значение не входит в область значений арксинуса.<br>\t\tНам нужно найти такое число $y$, что $sin(y) = sin(x)$ и $-\frac{\pi}{2} \le y \le \frac{\pi}{2}$.<br>\t\tИзвестно, что $sin(y) = sin(\alpha)$ если $y = \alpha + 2k\pi$ или $y = \pi - \alpha + 2k\pi$ для целого $k$.<br>\t\tПроверим второй случай: $y = \pi - (\frac{\pi}{2} - 4) + 2k\pi = \frac{\pi}{2} + 4 + 2k\pi$.<br>\t\tПри $k=-1$, $y = \frac{\pi}{2} + 4 - 2\pi = 4 - \frac{3\pi}{2}$.<br>\t\tОценим это значение: $4 - \frac{3 \cdot 3.14159}{2} \approx 4 - 4.712 = -0.712$.<br>\t\tТак как $-\frac{\pi}{2} \approx -1.57$, то $-1.57 \le -0.712 \le 1.57$. Значение $4 - \frac{3\pi}{2}$ входит в область значений арксинуса.<br>\t\t<b>Ответ:</b> $4 - \frac{3\pi}{2}$.\t</p>\t<p><strong>б)</strong></p>\t<p>\t\tРешение для $arcctg(tg(1.8\pi))$:<br>\t\tИспользуем формулу приведения $tg(\alpha) = ctg(\frac{\pi}{2} - \alpha)$.<br>\t\t$arcctg(tg(1.8\pi)) = arcctg(ctg(\frac{\pi}{2} - 1.8\pi)) = arcctg(ctg(-1.3\pi))$.<br>\t\tОбласть значений арккотангенса — интервал $(0, \pi)$. Значение $-1.3\pi$ не входит в этот интервал.<br>\t\tИспользуем периодичность котангенса (период $T=\pi$): $ctg(-1.3\pi) = ctg(-1.3\pi + 2\pi) = ctg(0.7\pi)$.<br>\t\tТак как $0 < 0.7\pi < \pi$, то $arcctg(ctg(0.7\pi)) = 0.7\pi$.<br>\t\t<b>Ответ:</b> $0.7\pi$.\t</p>\t<p>\t\tРешение для $arctg(ctg(1.8\pi))$:<br>\t\tИспользуем формулу приведения $ctg(\alpha) = tg(\frac{\pi}{2} - \alpha)$.<br>\t\t$arctg(ctg(1.8\pi)) = arctg(tg(\frac{\pi}{2} - 1.8\pi)) = arctg(tg(-1.3\pi))$.<br>\t\tОбласть значений арктангенса — интервал $(-\frac{\pi}{2}, \frac{\pi}{2})$. Значение $-1.3\pi$ не входит в этот интервал.<br>\t\tИспользуем периодичность тангенса (период $T=\pi$): $tg(-1.3\pi) = tg(-1.3\pi + \pi) = tg(-0.3\pi)$.<br>\t\tТак как $-\frac{\pi}{2} < -0.3\pi < \frac{\pi}{2}$, то $arctg(tg(-0.3\pi)) = -0.3\pi$.<br>\t\t<b>Ответ:</b> $-0.3\pi$.\t</p>\t<p>\t\tРешение для $arcctg(tg(3))$:<br>\t\t$arcctg(tg(3)) = arcctg(ctg(\frac{\pi}{2} - 3))$.<br>\t\tОбласть значений арккотангенса — $(0, \pi)$.<br>\t\tАргумент котангенса $\frac{\pi}{2} - 3 \approx 1.57 - 3 = -1.43$ не принадлежит этому интервалу.<br>\t\tИспользуем периодичность котангенса: $ctg(\frac{\pi}{2} - 3) = ctg(\frac{\pi}{2} - 3 + \pi) = ctg(\frac{3\pi}{2} - 3)$.<br>\t\tНовый аргумент $\frac{3\pi}{2} - 3 \approx 4.71 - 3 = 1.71$.<br>\t\tТак как $0 < 1.71 < \pi \approx 3.14$, то значение $ \frac{3\pi}{2} - 3 $ находится в области значений арккотангенса.<br>\t\t<b>Ответ:</b> $\frac{3\pi}{2} - 3$.\t</p>\t<p>\t\tРешение для $arctg(ctg(7))$:<br>\t\t$arctg(ctg(7)) = arctg(tg(\frac{\pi}{2} - 7))$.<br>\t\tОбласть значений арктангенса — $(-\frac{\pi}{2}, \frac{\pi}{2})$.<br>\t\tАргумент тангенса $\frac{\pi}{2} - 7 \approx 1.57 - 7 = -5.43$ не принадлежит этому интервалу.<br>\t\tИспользуем периодичность тангенса: $tg(\alpha) = tg(\alpha + k\pi)$.<br>\t\tПри $k=1$: $\frac{\pi}{2} - 7 + \pi = \frac{3\pi}{2} - 7 \approx 4.71 - 7 = -2.29$, что не входит в $(-\frac{\pi}{2}, \frac{\pi}{2})$.<br>\t\tПри $k=2$: $\frac{\pi}{2} - 7 + 2\pi = \frac{5\pi}{2} - 7 \approx 7.85 - 7 = 0.85$, что входит в $(-\frac{\pi}{2}, \frac{\pi}{2}) \approx (-1.57, 1.57)$.<br>\t\tСледовательно, итоговое значение равно $\frac{5\pi}{2} - 7$.<br>\t\t<b>Ответ:</b> $\frac{5\pi}{2} - 7$.\t</p>" ] #original: array:6 [ "id" => 1551494 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1149} "task" => array:2 [ "refs" => "1119773" "type" => "task" ] "text" => "<p><strong>а)</strong></p>\t<p>\t\tРешение для $arcsin(cos(0.3\pi))$:<br>\t\tИспользуем формулу приведения $cos(\alpha) = sin(\frac{\pi}{2} - \alpha)$.<br>\t\t$arcsin(cos(0.3\pi)) = arcsin(sin(\frac{\pi}{2} - 0.3\pi)) = arcsin(sin(0.5\pi - 0.3\pi)) = arcsin(sin(0.2\pi))$.<br>\t\tОбласть значений функции $arcsin(y)$ — это отрезок $[-\frac{\pi}{2}, \frac{\pi}{2}]$.<br>\t\tТак как $0.2\pi$ принадлежит этому отрезку ($-\frac{\pi}{2} \le 0.2\pi \le \frac{\pi}{2}$), то по определению арксинуса $arcsin(sin(0.2\pi)) = 0.2\pi$.<br>\t\t<b>Ответ:</b> $0.2\pi$.\t</p>\t<p>\t\tРешение для $arccos(sin(0.2\pi))$:<br>\t\tИспользуем формулу приведения $sin(\alpha) = cos(\frac{\pi}{2} - \alpha)$.<br>\t\t$arccos(sin(0.2\pi)) = arccos(cos(\frac{\pi}{2} - 0.2\pi)) = arccos(cos(0.3\pi))$.<br>\t\tОбласть значений функции $arccos(y)$ — это отрезок $[0, \pi]$.<br>\t\tТак как $0.3\pi$ принадлежит этому отрезку ($0 \le 0.3\pi \le \pi$), то по определению арккосинуса $arccos(cos(0.3\pi)) = 0.3\pi$.<br>\t\t<b>Ответ:</b> $0.3\pi$.\t</p>\t<p>\t\tРешение для $arccos(sin(1.8\pi))$:<br>\t\tИспользуем периодичность синуса $sin(1.8\pi) = sin(1.8\pi - 2\pi) = sin(-0.2\pi) = -sin(0.2\pi)$.<br>\t\tПолучаем $arccos(-sin(0.2\pi))$.<br>\t\tИспользуем свойство арккосинуса $arccos(-y) = \pi - arccos(y)$.<br>\t\t$\pi - arccos(sin(0.2\pi))$.<br>\t\tИз предыдущего примера мы знаем, что $arccos(sin(0.2\pi)) = 0.3\pi$.<br>\t\tСледовательно, выражение равно $\pi - 0.3\pi = 0.7\pi$.<br>\t\t<b>Ответ:</b> $0.7\pi$.\t</p>\t<p>\t\tРешение для $arcsin(cos(4))$:<br>\t\tИспользуем формулу приведения $cos(\alpha) = sin(\frac{\pi}{2} - \alpha)$.<br>\t\t$arcsin(cos(4)) = arcsin(sin(\frac{\pi}{2} - 4))$.<br>\t\tПусть $x = \frac{\pi}{2} - 4$. Область значений арксинуса — $[-\frac{\pi}{2}, \frac{\pi}{2}]$.<br>\t\tОценим $x$: $x \approx 1.57 - 4 = -2.43$. Это значение не входит в область значений арксинуса.<br>\t\tНам нужно найти такое число $y$, что $sin(y) = sin(x)$ и $-\frac{\pi}{2} \le y \le \frac{\pi}{2}$.<br>\t\tИзвестно, что $sin(y) = sin(\alpha)$ если $y = \alpha + 2k\pi$ или $y = \pi - \alpha + 2k\pi$ для целого $k$.<br>\t\tПроверим второй случай: $y = \pi - (\frac{\pi}{2} - 4) + 2k\pi = \frac{\pi}{2} + 4 + 2k\pi$.<br>\t\tПри $k=-1$, $y = \frac{\pi}{2} + 4 - 2\pi = 4 - \frac{3\pi}{2}$.<br>\t\tОценим это значение: $4 - \frac{3 \cdot 3.14159}{2} \approx 4 - 4.712 = -0.712$.<br>\t\tТак как $-\frac{\pi}{2} \approx -1.57$, то $-1.57 \le -0.712 \le 1.57$. Значение $4 - \frac{3\pi}{2}$ входит в область значений арксинуса.<br>\t\t<b>Ответ:</b> $4 - \frac{3\pi}{2}$.\t</p>\t<p><strong>б)</strong></p>\t<p>\t\tРешение для $arcctg(tg(1.8\pi))$:<br>\t\tИспользуем формулу приведения $tg(\alpha) = ctg(\frac{\pi}{2} - \alpha)$.<br>\t\t$arcctg(tg(1.8\pi)) = arcctg(ctg(\frac{\pi}{2} - 1.8\pi)) = arcctg(ctg(-1.3\pi))$.<br>\t\tОбласть значений арккотангенса — интервал $(0, \pi)$. Значение $-1.3\pi$ не входит в этот интервал.<br>\t\tИспользуем периодичность котангенса (период $T=\pi$): $ctg(-1.3\pi) = ctg(-1.3\pi + 2\pi) = ctg(0.7\pi)$.<br>\t\tТак как $0 < 0.7\pi < \pi$, то $arcctg(ctg(0.7\pi)) = 0.7\pi$.<br>\t\t<b>Ответ:</b> $0.7\pi$.\t</p>\t<p>\t\tРешение для $arctg(ctg(1.8\pi))$:<br>\t\tИспользуем формулу приведения $ctg(\alpha) = tg(\frac{\pi}{2} - \alpha)$.<br>\t\t$arctg(ctg(1.8\pi)) = arctg(tg(\frac{\pi}{2} - 1.8\pi)) = arctg(tg(-1.3\pi))$.<br>\t\tОбласть значений арктангенса — интервал $(-\frac{\pi}{2}, \frac{\pi}{2})$. Значение $-1.3\pi$ не входит в этот интервал.<br>\t\tИспользуем периодичность тангенса (период $T=\pi$): $tg(-1.3\pi) = tg(-1.3\pi + \pi) = tg(-0.3\pi)$.<br>\t\tТак как $-\frac{\pi}{2} < -0.3\pi < \frac{\pi}{2}$, то $arctg(tg(-0.3\pi)) = -0.3\pi$.<br>\t\t<b>Ответ:</b> $-0.3\pi$.\t</p>\t<p>\t\tРешение для $arcctg(tg(3))$:<br>\t\t$arcctg(tg(3)) = arcctg(ctg(\frac{\pi}{2} - 3))$.<br>\t\tОбласть значений арккотангенса — $(0, \pi)$.<br>\t\tАргумент котангенса $\frac{\pi}{2} - 3 \approx 1.57 - 3 = -1.43$ не принадлежит этому интервалу.<br>\t\tИспользуем периодичность котангенса: $ctg(\frac{\pi}{2} - 3) = ctg(\frac{\pi}{2} - 3 + \pi) = ctg(\frac{3\pi}{2} - 3)$.<br>\t\tНовый аргумент $\frac{3\pi}{2} - 3 \approx 4.71 - 3 = 1.71$.<br>\t\tТак как $0 < 1.71 < \pi \approx 3.14$, то значение $ \frac{3\pi}{2} - 3 $ находится в области значений арккотангенса.<br>\t\t<b>Ответ:</b> $\frac{3\pi}{2} - 3$.\t</p>\t<p>\t\tРешение для $arctg(ctg(7))$:<br>\t\t$arctg(ctg(7)) = arctg(tg(\frac{\pi}{2} - 7))$.<br>\t\tОбласть значений арктангенса — $(-\frac{\pi}{2}, \frac{\pi}{2})$.<br>\t\tАргумент тангенса $\frac{\pi}{2} - 7 \approx 1.57 - 7 = -5.43$ не принадлежит этому интервалу.<br>\t\tИспользуем периодичность тангенса: $tg(\alpha) = tg(\alpha + k\pi)$.<br>\t\tПри $k=1$: $\frac{\pi}{2} - 7 + \pi = \frac{3\pi}{2} - 7 \approx 4.71 - 7 = -2.29$, что не входит в $(-\frac{\pi}{2}, \frac{\pi}{2})$.<br>\t\tПри $k=2$: $\frac{\pi}{2} - 7 + 2\pi = \frac{5\pi}{2} - 7 \approx 7.85 - 7 = 0.85$, что входит в $(-\frac{\pi}{2}, \frac{\pi}{2}) \approx (-1.57, 1.57)$.<br>\t\tСледовательно, итоговое значение равно $\frac{5\pi}{2} - 7$.<br>\t\t<b>Ответ:</b> $\frac{5\pi}{2} - 7$.\t</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" => "1119774" "type" => "task" ] "previous" => array:2 [ "refs" => "1119772" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1168 #items: array:1 [ 0 => App\Models\Book {#1185} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1169 #items: array:1 [ 0 => App\Models\BookPage {#1141 #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" => 1099268 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "115" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/page-2-115" "field_display_title" => "115" "field_folder" => "2" "field_image_name" => "115" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\Branch 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№8 (с. 115)
Условие. №8 (с. 115)
скриншот условия
8. (5) Упростите следующие выражения:
a) $arcsin(cos0.3\pi)$, $arccos(sin0.2\pi)$, $arccos(sin1.8\pi)$; $arcsin(cos4)$;
б) $arcctg(tg1.8\pi)$, $arcctg(ctg1.8\pi)$, $arcctg(tg3)$, $arcctg(ctg7)$.
Решение 2 (rus). №8 (с. 115)
а)
Решение для $arcsin(cos(0.3\pi))$:
Используем формулу приведения $cos(\alpha) = sin(\frac{\pi}{2} - \alpha)$.
$arcsin(cos(0.3\pi)) = arcsin(sin(\frac{\pi}{2} - 0.3\pi)) = arcsin(sin(0.5\pi - 0.3\pi)) = arcsin(sin(0.2\pi))$.
Область значений функции $arcsin(y)$ — это отрезок $[-\frac{\pi}{2}, \frac{\pi}{2}]$.
Так как $0.2\pi$ принадлежит этому отрезку ($-\frac{\pi}{2} \le 0.2\pi \le \frac{\pi}{2}$), то по определению арксинуса $arcsin(sin(0.2\pi)) = 0.2\pi$.
Ответ: $0.2\pi$.
Решение для $arccos(sin(0.2\pi))$:
Используем формулу приведения $sin(\alpha) = cos(\frac{\pi}{2} - \alpha)$.
$arccos(sin(0.2\pi)) = arccos(cos(\frac{\pi}{2} - 0.2\pi)) = arccos(cos(0.3\pi))$.
Область значений функции $arccos(y)$ — это отрезок $[0, \pi]$.
Так как $0.3\pi$ принадлежит этому отрезку ($0 \le 0.3\pi \le \pi$), то по определению арккосинуса $arccos(cos(0.3\pi)) = 0.3\pi$.
Ответ: $0.3\pi$.
Решение для $arccos(sin(1.8\pi))$:
Используем периодичность синуса $sin(1.8\pi) = sin(1.8\pi - 2\pi) = sin(-0.2\pi) = -sin(0.2\pi)$.
Получаем $arccos(-sin(0.2\pi))$.
Используем свойство арккосинуса $arccos(-y) = \pi - arccos(y)$.
$\pi - arccos(sin(0.2\pi))$.
Из предыдущего примера мы знаем, что $arccos(sin(0.2\pi)) = 0.3\pi$.
Следовательно, выражение равно $\pi - 0.3\pi = 0.7\pi$.
Ответ: $0.7\pi$.
Решение для $arcsin(cos(4))$:
Используем формулу приведения $cos(\alpha) = sin(\frac{\pi}{2} - \alpha)$.
$arcsin(cos(4)) = arcsin(sin(\frac{\pi}{2} - 4))$.
Пусть $x = \frac{\pi}{2} - 4$. Область значений арксинуса — $[-\frac{\pi}{2}, \frac{\pi}{2}]$.
Оценим $x$: $x \approx 1.57 - 4 = -2.43$. Это значение не входит в область значений арксинуса.
Нам нужно найти такое число $y$, что $sin(y) = sin(x)$ и $-\frac{\pi}{2} \le y \le \frac{\pi}{2}$.
Известно, что $sin(y) = sin(\alpha)$ если $y = \alpha + 2k\pi$ или $y = \pi - \alpha + 2k\pi$ для целого $k$.
Проверим второй случай: $y = \pi - (\frac{\pi}{2} - 4) + 2k\pi = \frac{\pi}{2} + 4 + 2k\pi$.
При $k=-1$, $y = \frac{\pi}{2} + 4 - 2\pi = 4 - \frac{3\pi}{2}$.
Оценим это значение: $4 - \frac{3 \cdot 3.14159}{2} \approx 4 - 4.712 = -0.712$.
Так как $-\frac{\pi}{2} \approx -1.57$, то $-1.57 \le -0.712 \le 1.57$. Значение $4 - \frac{3\pi}{2}$ входит в область значений арксинуса.
Ответ: $4 - \frac{3\pi}{2}$.
б)
Решение для $arcctg(tg(1.8\pi))$:
Используем формулу приведения $tg(\alpha) = ctg(\frac{\pi}{2} - \alpha)$.
$arcctg(tg(1.8\pi)) = arcctg(ctg(\frac{\pi}{2} - 1.8\pi)) = arcctg(ctg(-1.3\pi))$.
Область значений арккотангенса — интервал $(0, \pi)$. Значение $-1.3\pi$ не входит в этот интервал.
Используем периодичность котангенса (период $T=\pi$): $ctg(-1.3\pi) = ctg(-1.3\pi + 2\pi) = ctg(0.7\pi)$.
Так как $0 < 0.7\pi < \pi$, то $arcctg(ctg(0.7\pi)) = 0.7\pi$.
Ответ: $0.7\pi$.
Решение для $arctg(ctg(1.8\pi))$:
Используем формулу приведения $ctg(\alpha) = tg(\frac{\pi}{2} - \alpha)$.
$arctg(ctg(1.8\pi)) = arctg(tg(\frac{\pi}{2} - 1.8\pi)) = arctg(tg(-1.3\pi))$.
Область значений арктангенса — интервал $(-\frac{\pi}{2}, \frac{\pi}{2})$. Значение $-1.3\pi$ не входит в этот интервал.
Используем периодичность тангенса (период $T=\pi$): $tg(-1.3\pi) = tg(-1.3\pi + \pi) = tg(-0.3\pi)$.
Так как $-\frac{\pi}{2} < -0.3\pi < \frac{\pi}{2}$, то $arctg(tg(-0.3\pi)) = -0.3\pi$.
Ответ: $-0.3\pi$.
Решение для $arcctg(tg(3))$:
$arcctg(tg(3)) = arcctg(ctg(\frac{\pi}{2} - 3))$.
Область значений арккотангенса — $(0, \pi)$.
Аргумент котангенса $\frac{\pi}{2} - 3 \approx 1.57 - 3 = -1.43$ не принадлежит этому интервалу.
Используем периодичность котангенса: $ctg(\frac{\pi}{2} - 3) = ctg(\frac{\pi}{2} - 3 + \pi) = ctg(\frac{3\pi}{2} - 3)$.
Новый аргумент $\frac{3\pi}{2} - 3 \approx 4.71 - 3 = 1.71$.
Так как $0 < 1.71 < \pi \approx 3.14$, то значение $ \frac{3\pi}{2} - 3 $ находится в области значений арккотангенса.
Ответ: $\frac{3\pi}{2} - 3$.
Решение для $arctg(ctg(7))$:
$arctg(ctg(7)) = arctg(tg(\frac{\pi}{2} - 7))$.
Область значений арктангенса — $(-\frac{\pi}{2}, \frac{\pi}{2})$.
Аргумент тангенса $\frac{\pi}{2} - 7 \approx 1.57 - 7 = -5.43$ не принадлежит этому интервалу.
Используем периодичность тангенса: $tg(\alpha) = tg(\alpha + k\pi)$.
При $k=1$: $\frac{\pi}{2} - 7 + \pi = \frac{3\pi}{2} - 7 \approx 4.71 - 7 = -2.29$, что не входит в $(-\frac{\pi}{2}, \frac{\pi}{2})$.
При $k=2$: $\frac{\pi}{2} - 7 + 2\pi = \frac{5\pi}{2} - 7 \approx 7.85 - 7 = 0.85$, что входит в $(-\frac{\pi}{2}, \frac{\pi}{2}) \approx (-1.57, 1.57)$.
Следовательно, итоговое значение равно $\frac{5\pi}{2} - 7$.
Ответ: $\frac{5\pi}{2} - 7$.
Другие задания:
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