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Номер 2, страница 107, 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 3. Обратные тригонометрические функции. 3.4. Арксинус. Задачи - номер 2, страница 107.
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" => 1119752 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "107" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/2-3-4zad-2" "field_display_title" => "2" "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 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ …1] #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 [ …4] ] "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 [ …4] ] "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 {#1105 #items: array:5 [ 0 => App\Models\Branch {#1117 #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 {#1121 #items: array:1 [ 0 => App\Models\Term {#1118 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1150 #items: array:1 [ 0 => App\Models\Book {#1120 #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 {#1122 #items: array:1 [ 0 => App\Models\Term {#1119 #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 {#1123 #items: array:1 [ 0 => App\Models\Term {#1124 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1125 #items: array:1 [ 0 => App\Models\Term {#1126 #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 [ …6] "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 [ …6] "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 {#1127 #items: array:3 [ 0 => App\Models\Term {#1128 #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 {#1129 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1130 #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 {#1131 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1132 #items: array:1 [ 0 => App\Models\Term {#1133 #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 [ …6] "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 [ …6] "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 {#1134 #items: array:1 [ 0 => App\Models\Term {#1135 #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 [ …6] "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 [ …6] "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 {#1136 #items: array:1 [ 0 => App\Models\Term {#1137 #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 [ …6] "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 [ …6] "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 {#1138 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1140 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1141 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1142 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1143 #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 {#1144 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1146 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1147 #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 {#1122} "field_class" => Illuminate\Database\Eloquent\Collection {#1123} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1125} "field_author" => Illuminate\Database\Eloquent\Collection {#1127} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1131} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1132} "field_country" => Illuminate\Database\Eloquent\Collection {#1134} "field_city" => Illuminate\Database\Eloquent\Collection {#1136} "field_series" => Illuminate\Database\Eloquent\Collection {#1138} "field_umk" => Illuminate\Database\Eloquent\Collection {#1139} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1140} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1141} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1142} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1143} "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 {#1144} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1145} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1146} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1147} "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 {#1148 #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 {#1121} "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 {#1150} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1148} ] #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 {#1149 #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 {#1154 #items: array:1 [ 0 => App\Models\Term {#1151 #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 {#1153 #items: array:1 [ 0 => App\Models\Book {#1120} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1152 #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 {#1154} "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 {#1153} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1152} ] #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 {#1155 #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" => 1119373 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Обратные тригонометрические функции" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1159 #items: array:1 [ 0 => App\Models\Term {#1156 #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" => "99" "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 {#1158 #items: array:1 [ 0 => App\Models\Book {#1120} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1167 #items: array:1 [ 0 => App\Models\Branch {#1166 #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 {#1168 #items: array:1 [ …1] #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 {#1169 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ …1] #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 {#1168} "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 {#1169} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1170} ] #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" => 1119373 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Обратные тригонометрические функции" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1159} "field_page_start" => "99" "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 {#1158} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1167} ] #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 {#1160 #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" => 1119376 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.4. Арксинус" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1157 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "106" "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 {#1163 #items: array:1 [ 0 => App\Models\Book {#1120} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1176 #items: array:1 [ 0 => App\Models\Branch {#1175 #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" => 1119373 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Обратные тригонометрические функции" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1177 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_page_start" => "99" "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 {#1178 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119373 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Обратные тригонометрические функции" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1177} "field_page_start" => "99" "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 {#1178} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1179} ] #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" => 1119376 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.4. Арксинус" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1157} "field_page_start" => "106" "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 {#1163} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1176} ] #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\Branch {#1164 #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" => 1119377 "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 {#1161 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "106" "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 {#1172 #items: array:1 [ 0 => App\Models\Book {#1120} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1185 #items: array:1 [ 0 => App\Models\Branch {#1184 #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" => 1119376 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.4. Арксинус" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1186 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "106" "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 {#1187 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1188 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119376 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.4. Арксинус" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1186} "field_page_start" => "106" "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 {#1187} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1188} ] #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" => 1119377 "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 {#1161} "field_page_start" => "106" "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 {#1172} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1185} ] #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 {#1070 #items: array:2 [ 0 => App\Models\Element {#1097 #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" => 1300435 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Edition {#1096 #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 {#1094 #items: array:1 [ …1] #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 {#1093 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 #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 {#1094} "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 {#1093} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092} "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" => "1119752" "type" => "task" ] "text" => "<p><strong>2. (1)</strong> Упростите следующие выражения:</p><p><strong>а)</strong> $\arcsin 0$, $\arcsin 1$, $\arcsin (-1)$, $\arcsin \frac{1}{2}$, $\arcsin \left(-\frac{1}{2}\right)$;</p><p><strong>б)</strong> $\arccos \left( (\sqrt{3}-1)(\sqrt{3}+1)-2 \right)$, $\arccos (\sin^2 \alpha + \cos^2 \alpha)$, $\arccos (-1)^{105}$, $\arccos \left( \frac{1}{\sqrt{3}-1} - \frac{1}{2} \right)$, $\arccos \left( \frac{1}{2}\text{tg}120^{\circ} \right)$;</p><p><strong>В)</strong> $\text{arctg} \left( \cos \frac{53}{2}\pi \right)$, $\text{arctg} \left( \sin \frac{53}{2}\pi \right)$, $\text{arctg} \left( \sin \frac{55}{2}\pi \right)$, $\text{arctg} \left( \frac{4\cos15^{\circ}}{\sqrt{6}+3\sqrt{2}} \right)$, $\text{arctg} \left( \frac{4\sin15^{\circ}}{\sqrt{6}-3\sqrt{2}} \right)$;</p><p><strong>Г)</strong> $\text{arctg} \left( \cos\beta+\cos(180^{\circ}-\beta) \right)$, $\text{arctg} \frac{\sin(180^{\circ}-\gamma)}{\sin\gamma}$ при $\sin\gamma \ne 0$. $\text{arctg} \frac{\cos93^{\circ}}{\sin3^{\circ}}$, $\text{arctg} \sqrt{\frac{3(1-\cos14^{\circ})}{2\sin^2 7^{\circ}}}$; $\text{arctg} (-\sqrt{3})$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1739" "height" => 1144 "path" => "/media/algebra_10/baspasy-u19/0-2-3-4zad/2-1.webp?ts=1753534584" ] ] ] #original: array:7 [ "id" => 1300435 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087} "task" => array:2 [ "refs" => "1119752" "type" => "task" ] "text" => "<p><strong>2. (1)</strong> Упростите следующие выражения:</p><p><strong>а)</strong> $\arcsin 0$, $\arcsin 1$, $\arcsin (-1)$, $\arcsin \frac{1}{2}$, $\arcsin \left(-\frac{1}{2}\right)$;</p><p><strong>б)</strong> $\arccos \left( (\sqrt{3}-1)(\sqrt{3}+1)-2 \right)$, $\arccos (\sin^2 \alpha + \cos^2 \alpha)$, $\arccos (-1)^{105}$, $\arccos \left( \frac{1}{\sqrt{3}-1} - \frac{1}{2} \right)$, $\arccos \left( \frac{1}{2}\text{tg}120^{\circ} \right)$;</p><p><strong>В)</strong> $\text{arctg} \left( \cos \frac{53}{2}\pi \right)$, $\text{arctg} \left( \sin \frac{53}{2}\pi \right)$, $\text{arctg} \left( \sin \frac{55}{2}\pi \right)$, $\text{arctg} \left( \frac{4\cos15^{\circ}}{\sqrt{6}+3\sqrt{2}} \right)$, $\text{arctg} \left( \frac{4\sin15^{\circ}}{\sqrt{6}-3\sqrt{2}} \right)$;</p><p><strong>Г)</strong> $\text{arctg} \left( \cos\beta+\cos(180^{\circ}-\beta) \right)$, $\text{arctg} \frac{\sin(180^{\circ}-\gamma)}{\sin\gamma}$ при $\sin\gamma \ne 0$. $\text{arctg} \frac{\cos93^{\circ}}{\sin3^{\circ}}$, $\text{arctg} \sqrt{\frac{3(1-\cos14^{\circ})}{2\sin^2 7^{\circ}}}$; $\text{arctg} (-\sqrt{3})$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1739" "height" => 1144 "path" => "/media/algebra_10/baspasy-u19/0-2-3-4zad/2-1.webp?ts=1753534584" ] ] ] #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 {#1089 #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" => 1551471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Edition {#1090 #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 {#1086 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084 #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 {#1086} "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 {#1082} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084} "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" => "1119752" "type" => "task" ] "text" => "<p><strong>a)</strong></p><p> $arcsin(0)$. По определению, это угол из отрезка $[-\frac{\pi}{2}, \frac{\pi}{2}]$, синус которого равен 0. Этим углом является 0.<br> Ответ: $0$.</p><p> $arcsin(1)$. Ищем угол из отрезка $[-\frac{\pi}{2}, \frac{\pi}{2}]$, синус которого равен 1. Этим углом является $\frac{\pi}{2}$.<br> Ответ: $\frac{\pi}{2}$.</p><p> $arcsin(-1)$. Ищем угол из отрезка $[-\frac{\pi}{2}, \frac{\pi}{2}]$, синус которого равен -1. Этим углом является $-\frac{\pi}{2}$.<br> Ответ: $-\frac{\pi}{2}$.</p><p> $arcsin(\frac{1}{2})$. Ищем угол из отрезка $[-\frac{\pi}{2}, \frac{\pi}{2}]$, синус которого равен $\frac{1}{2}$. Этим углом является $\frac{\pi}{6}$.<br> Ответ: $\frac{\pi}{6}$.</p><p> $arcsin(-\frac{1}{2})$. Ищем угол из отрезка $[-\frac{\pi}{2}, \frac{\pi}{2}]$, синус которого равен $-\frac{1}{2}$. Этим углом является $-\frac{\pi}{6}$.<br> Ответ: $-\frac{\pi}{6}$.</p><p><strong>б)</strong></p><p> $arccos((\sqrt{3}-1)(\sqrt{3}+1)-2)$. Упростим выражение в скобках, используя формулу разности квадратов: $(\sqrt{3}-1)(\sqrt{3}+1)-2 = ((\sqrt{3})^2-1^2)-2 = (3-1)-2 = 0$. Получаем $arccos(0) = \frac{\pi}{2}$.<br> Ответ: $\frac{\pi}{2}$.</p><p> $arccos(sin^2\alpha+cos^2\alpha)$. Согласно основному тригонометрическому тождеству, $sin^2\alpha+cos^2\alpha = 1$. Получаем $arccos(1) = 0$.<br> Ответ: $0$.</p><p> $arccos((-1)^{105})$. Поскольку 105 — нечетное число, $(-1)^{105} = -1$. Получаем $arccos(-1) = \pi$.<br> Ответ: $\pi$.</p><p> $arccos(\frac{1}{\sqrt{3}-1}-\frac{1}{2})$. Упростим аргумент: $\frac{1}{\sqrt{3}-1}-\frac{1}{2} = \frac{\sqrt{3}+1}{(\sqrt{3}-1)(\sqrt{3}+1)}-\frac{1}{2} = \frac{\sqrt{3}+1}{3-1}-\frac{1}{2} = \frac{\sqrt{3}+1}{2}-\frac{1}{2} = \frac{\sqrt{3}}{2}$. Получаем $arccos(\frac{\sqrt{3}}{2}) = \frac{\pi}{6}$.<br> Ответ: $\frac{\pi}{6}$.</p><p> $arccos(\frac{1}{2}tg120^\circ)$. Упростим аргумент: $tg120^\circ = tg(180^\circ-60^\circ)=-tg60^\circ=-\sqrt{3}$. Тогда $\frac{1}{2}tg120^\circ = \frac{1}{2}(-\sqrt{3}) = -\frac{\sqrt{3}}{2}$. Получаем $arccos(-\frac{\sqrt{3}}{2}) = \pi - arccos(\frac{\sqrt{3}}{2}) = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$.<br> Ответ: $\frac{5\pi}{6}$.</p><p><strong>в)</strong></p><p> $arctg(cos\frac{53}{2}\pi)$. Упростим аргумент: $cos(\frac{53\pi}{2}) = cos(26\pi + \frac{\pi}{2}) = cos(\frac{\pi}{2}) = 0$. Получаем $arctg(0) = 0$.<br> Ответ: $0$.</p><p> $arctg(sin\frac{53}{2}\pi)$. Упростим аргумент: $sin(\frac{53\pi}{2}) = sin(26\pi + \frac{\pi}{2}) = sin(\frac{\pi}{2}) = 1$. Получаем $arctg(1) = \frac{\pi}{4}$.<br> Ответ: $\frac{\pi}{4}$.</p><p> $arctg(sin\frac{55}{2}\pi)$. Упростим аргумент: $sin(\frac{55\pi}{2}) = sin(28\pi - \frac{\pi}{2}) = sin(-\frac{\pi}{2}) = -1$. Получаем $arctg(-1) = -\frac{\pi}{4}$.<br> Ответ: $-\frac{\pi}{4}$.</p><p> $arctg(\frac{4cos15^\circ}{\sqrt{6}+3\sqrt{2}})$. Упростим аргумент. $cos15^\circ = \frac{\sqrt{6}+\sqrt{2}}{4}$. Знаменатель: $\sqrt{6}+3\sqrt{2} = \sqrt{2}(\sqrt{3}+3)$. Аргумент: $\frac{4(\frac{\sqrt{6}+\sqrt{2}}{4})}{\sqrt{2}(\sqrt{3}+3)} = \frac{\sqrt{2}(\sqrt{3}+1)}{\sqrt{2}(\sqrt{3}+3)} = \frac{\sqrt{3}+1}{3+\sqrt{3}} = \frac{\sqrt{3}+1}{\sqrt{3}(\sqrt{3}+1)} = \frac{1}{\sqrt{3}}$. Получаем $arctg(\frac{1}{\sqrt{3}}) = \frac{\pi}{6}$.<br> Ответ: $\frac{\pi}{6}$.</p><p> $arctg(\frac{4sin15^\circ}{\sqrt{6}-3\sqrt{2}})$. Упростим аргумент. $sin15^\circ = \frac{\sqrt{6}-\sqrt{2}}{4}$. Знаменатель: $\sqrt{6}-3\sqrt{2} = \sqrt{2}(\sqrt{3}-3)$. Аргумент: $\frac{4(\frac{\sqrt{6}-\sqrt{2}}{4})}{\sqrt{2}(\sqrt{3}-3)} = \frac{\sqrt{2}(\sqrt{3}-1)}{\sqrt{2}(\sqrt{3}-3)} = \frac{\sqrt{3}-1}{\sqrt{3}-3} = \frac{\sqrt{3}-1}{-(\sqrt{3}-1)\sqrt{3}} = -\frac{1}{\sqrt{3}}$. Получаем $arctg(-\frac{1}{\sqrt{3}}) = -\frac{\pi}{6}$.<br> Ответ: $-\frac{\pi}{6}$.</p><p><strong>г)</strong></p><p> $arcctg(cos\beta+cos(180^\circ-\beta))$. По формуле приведения $cos(180^\circ-\beta)=-cos\beta$. Тогда аргумент равен $cos\beta-cos\beta = 0$. Получаем $arcctg(0) = \frac{\pi}{2}$.<br> Ответ: $\frac{\pi}{2}$.</p><p> $arcctg(\frac{sin(180^\circ-\gamma)}{sin\gamma})$ при $sin\gamma \neq 0$. По формуле приведения $sin(180^\circ-\gamma)=sin\gamma$. Тогда аргумент равен $\frac{sin\gamma}{sin\gamma} = 1$. Получаем $arcctg(1) = \frac{\pi}{4}$.<br> Ответ: $\frac{\pi}{4}$.</p><p> $arcctg(\frac{cos93^\circ}{sin3^\circ})$. По формуле приведения $cos93^\circ = cos(90^\circ+3^\circ) = -sin3^\circ$. Тогда аргумент равен $\frac{-sin3^\circ}{sin3^\circ} = -1$. Получаем $arcctg(-1) = \pi - arcctg(1) = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$.<br> Ответ: $\frac{3\pi}{4}$.</p><p> $arcctg\sqrt{\frac{3(1-cos14^\circ)}{2sin^27^\circ}}$. По формуле двойного угла $1-cos14^\circ = 2sin^27^\circ$. Тогда выражение под корнем равно $\frac{3(2sin^27^\circ)}{2sin^27^\circ} = 3$. Получаем $arcctg(\sqrt{3}) = \frac{\pi}{6}$.<br> Ответ: $\frac{\pi}{6}$.</p><p> $arcctg(-\sqrt{3})$. Используем свойство арккотангенса: $arcctg(-x) = \pi - arcctg(x)$. Получаем $arcctg(-\sqrt{3}) = \pi - arcctg(\sqrt{3}) = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$.<br> Ответ: $\frac{5\pi}{6}$.</p>" ] #original: array:6 [ "id" => 1551471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081} "task" => array:2 [ "refs" => "1119752" "type" => "task" ] "text" => "<p><strong>a)</strong></p><p> $arcsin(0)$. По определению, это угол из отрезка $[-\frac{\pi}{2}, \frac{\pi}{2}]$, синус которого равен 0. Этим углом является 0.<br> Ответ: $0$.</p><p> $arcsin(1)$. Ищем угол из отрезка $[-\frac{\pi}{2}, \frac{\pi}{2}]$, синус которого равен 1. Этим углом является $\frac{\pi}{2}$.<br> Ответ: $\frac{\pi}{2}$.</p><p> $arcsin(-1)$. Ищем угол из отрезка $[-\frac{\pi}{2}, \frac{\pi}{2}]$, синус которого равен -1. Этим углом является $-\frac{\pi}{2}$.<br> Ответ: $-\frac{\pi}{2}$.</p><p> $arcsin(\frac{1}{2})$. Ищем угол из отрезка $[-\frac{\pi}{2}, \frac{\pi}{2}]$, синус которого равен $\frac{1}{2}$. Этим углом является $\frac{\pi}{6}$.<br> Ответ: $\frac{\pi}{6}$.</p><p> $arcsin(-\frac{1}{2})$. Ищем угол из отрезка $[-\frac{\pi}{2}, \frac{\pi}{2}]$, синус которого равен $-\frac{1}{2}$. Этим углом является $-\frac{\pi}{6}$.<br> Ответ: $-\frac{\pi}{6}$.</p><p><strong>б)</strong></p><p> $arccos((\sqrt{3}-1)(\sqrt{3}+1)-2)$. Упростим выражение в скобках, используя формулу разности квадратов: $(\sqrt{3}-1)(\sqrt{3}+1)-2 = ((\sqrt{3})^2-1^2)-2 = (3-1)-2 = 0$. Получаем $arccos(0) = \frac{\pi}{2}$.<br> Ответ: $\frac{\pi}{2}$.</p><p> $arccos(sin^2\alpha+cos^2\alpha)$. Согласно основному тригонометрическому тождеству, $sin^2\alpha+cos^2\alpha = 1$. Получаем $arccos(1) = 0$.<br> Ответ: $0$.</p><p> $arccos((-1)^{105})$. Поскольку 105 — нечетное число, $(-1)^{105} = -1$. Получаем $arccos(-1) = \pi$.<br> Ответ: $\pi$.</p><p> $arccos(\frac{1}{\sqrt{3}-1}-\frac{1}{2})$. Упростим аргумент: $\frac{1}{\sqrt{3}-1}-\frac{1}{2} = \frac{\sqrt{3}+1}{(\sqrt{3}-1)(\sqrt{3}+1)}-\frac{1}{2} = \frac{\sqrt{3}+1}{3-1}-\frac{1}{2} = \frac{\sqrt{3}+1}{2}-\frac{1}{2} = \frac{\sqrt{3}}{2}$. Получаем $arccos(\frac{\sqrt{3}}{2}) = \frac{\pi}{6}$.<br> Ответ: $\frac{\pi}{6}$.</p><p> $arccos(\frac{1}{2}tg120^\circ)$. Упростим аргумент: $tg120^\circ = tg(180^\circ-60^\circ)=-tg60^\circ=-\sqrt{3}$. Тогда $\frac{1}{2}tg120^\circ = \frac{1}{2}(-\sqrt{3}) = -\frac{\sqrt{3}}{2}$. Получаем $arccos(-\frac{\sqrt{3}}{2}) = \pi - arccos(\frac{\sqrt{3}}{2}) = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$.<br> Ответ: $\frac{5\pi}{6}$.</p><p><strong>в)</strong></p><p> $arctg(cos\frac{53}{2}\pi)$. Упростим аргумент: $cos(\frac{53\pi}{2}) = cos(26\pi + \frac{\pi}{2}) = cos(\frac{\pi}{2}) = 0$. Получаем $arctg(0) = 0$.<br> Ответ: $0$.</p><p> $arctg(sin\frac{53}{2}\pi)$. Упростим аргумент: $sin(\frac{53\pi}{2}) = sin(26\pi + \frac{\pi}{2}) = sin(\frac{\pi}{2}) = 1$. Получаем $arctg(1) = \frac{\pi}{4}$.<br> Ответ: $\frac{\pi}{4}$.</p><p> $arctg(sin\frac{55}{2}\pi)$. Упростим аргумент: $sin(\frac{55\pi}{2}) = sin(28\pi - \frac{\pi}{2}) = sin(-\frac{\pi}{2}) = -1$. Получаем $arctg(-1) = -\frac{\pi}{4}$.<br> Ответ: $-\frac{\pi}{4}$.</p><p> $arctg(\frac{4cos15^\circ}{\sqrt{6}+3\sqrt{2}})$. Упростим аргумент. $cos15^\circ = \frac{\sqrt{6}+\sqrt{2}}{4}$. Знаменатель: $\sqrt{6}+3\sqrt{2} = \sqrt{2}(\sqrt{3}+3)$. Аргумент: $\frac{4(\frac{\sqrt{6}+\sqrt{2}}{4})}{\sqrt{2}(\sqrt{3}+3)} = \frac{\sqrt{2}(\sqrt{3}+1)}{\sqrt{2}(\sqrt{3}+3)} = \frac{\sqrt{3}+1}{3+\sqrt{3}} = \frac{\sqrt{3}+1}{\sqrt{3}(\sqrt{3}+1)} = \frac{1}{\sqrt{3}}$. Получаем $arctg(\frac{1}{\sqrt{3}}) = \frac{\pi}{6}$.<br> Ответ: $\frac{\pi}{6}$.</p><p> $arctg(\frac{4sin15^\circ}{\sqrt{6}-3\sqrt{2}})$. Упростим аргумент. $sin15^\circ = \frac{\sqrt{6}-\sqrt{2}}{4}$. Знаменатель: $\sqrt{6}-3\sqrt{2} = \sqrt{2}(\sqrt{3}-3)$. Аргумент: $\frac{4(\frac{\sqrt{6}-\sqrt{2}}{4})}{\sqrt{2}(\sqrt{3}-3)} = \frac{\sqrt{2}(\sqrt{3}-1)}{\sqrt{2}(\sqrt{3}-3)} = \frac{\sqrt{3}-1}{\sqrt{3}-3} = \frac{\sqrt{3}-1}{-(\sqrt{3}-1)\sqrt{3}} = -\frac{1}{\sqrt{3}}$. Получаем $arctg(-\frac{1}{\sqrt{3}}) = -\frac{\pi}{6}$.<br> Ответ: $-\frac{\pi}{6}$.</p><p><strong>г)</strong></p><p> $arcctg(cos\beta+cos(180^\circ-\beta))$. По формуле приведения $cos(180^\circ-\beta)=-cos\beta$. Тогда аргумент равен $cos\beta-cos\beta = 0$. Получаем $arcctg(0) = \frac{\pi}{2}$.<br> Ответ: $\frac{\pi}{2}$.</p><p> $arcctg(\frac{sin(180^\circ-\gamma)}{sin\gamma})$ при $sin\gamma \neq 0$. По формуле приведения $sin(180^\circ-\gamma)=sin\gamma$. Тогда аргумент равен $\frac{sin\gamma}{sin\gamma} = 1$. Получаем $arcctg(1) = \frac{\pi}{4}$.<br> Ответ: $\frac{\pi}{4}$.</p><p> $arcctg(\frac{cos93^\circ}{sin3^\circ})$. По формуле приведения $cos93^\circ = cos(90^\circ+3^\circ) = -sin3^\circ$. Тогда аргумент равен $\frac{-sin3^\circ}{sin3^\circ} = -1$. Получаем $arcctg(-1) = \pi - arcctg(1) = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$.<br> Ответ: $\frac{3\pi}{4}$.</p><p> $arcctg\sqrt{\frac{3(1-cos14^\circ)}{2sin^27^\circ}}$. По формуле двойного угла $1-cos14^\circ = 2sin^27^\circ$. Тогда выражение под корнем равно $\frac{3(2sin^27^\circ)}{2sin^27^\circ} = 3$. Получаем $arcctg(\sqrt{3}) = \frac{\pi}{6}$.<br> Ответ: $\frac{\pi}{6}$.</p><p> $arcctg(-\sqrt{3})$. Используем свойство арккотангенса: $arcctg(-x) = \pi - arcctg(x)$. Получаем $arcctg(-\sqrt{3}) = \pi - arcctg(\sqrt{3}) = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$.<br> Ответ: $\frac{5\pi}{6}$.</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" => "1119753" "type" => "task" ] "previous" => array:2 [ "refs" => "1119751" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Book {#1120} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1075 #items: array:1 [ 0 => App\Models\BookPage {#1101 #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" => 1099260 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "107" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/page-2-107" "field_display_title" => "107" "field_folder" => "2" "field_image_name" => "107" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:1 [ 0 => App\Models\Branch {#1083 #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" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_page_start" => "5" "field_branch_display" => "0" 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"field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1199 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1210 #items: array:1 [ 0 => App\Models\Element {#1209 #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" => 1262685 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1211 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1262685 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1211 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] 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[] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№2 (с. 107)
Условие. №2 (с. 107)
скриншот условия
2. (1) Упростите следующие выражения:
а) $\arcsin 0$, $\arcsin 1$, $\arcsin (-1)$, $\arcsin \frac{1}{2}$, $\arcsin \left(-\frac{1}{2}\right)$;
б) $\arccos \left( (\sqrt{3}-1)(\sqrt{3}+1)-2 \right)$, $\arccos (\sin^2 \alpha + \cos^2 \alpha)$, $\arccos (-1)^{105}$, $\arccos \left( \frac{1}{\sqrt{3}-1} - \frac{1}{2} \right)$, $\arccos \left( \frac{1}{2}\text{tg}120^{\circ} \right)$;
В) $\text{arctg} \left( \cos \frac{53}{2}\pi \right)$, $\text{arctg} \left( \sin \frac{53}{2}\pi \right)$, $\text{arctg} \left( \sin \frac{55}{2}\pi \right)$, $\text{arctg} \left( \frac{4\cos15^{\circ}}{\sqrt{6}+3\sqrt{2}} \right)$, $\text{arctg} \left( \frac{4\sin15^{\circ}}{\sqrt{6}-3\sqrt{2}} \right)$;
Г) $\text{arctg} \left( \cos\beta+\cos(180^{\circ}-\beta) \right)$, $\text{arctg} \frac{\sin(180^{\circ}-\gamma)}{\sin\gamma}$ при $\sin\gamma \ne 0$. $\text{arctg} \frac{\cos93^{\circ}}{\sin3^{\circ}}$, $\text{arctg} \sqrt{\frac{3(1-\cos14^{\circ})}{2\sin^2 7^{\circ}}}$; $\text{arctg} (-\sqrt{3})$.
Решение 2 (rus). №2 (с. 107)
a)
$arcsin(0)$. По определению, это угол из отрезка $[-\frac{\pi}{2}, \frac{\pi}{2}]$, синус которого равен 0. Этим углом является 0.
Ответ: $0$.
$arcsin(1)$. Ищем угол из отрезка $[-\frac{\pi}{2}, \frac{\pi}{2}]$, синус которого равен 1. Этим углом является $\frac{\pi}{2}$.
Ответ: $\frac{\pi}{2}$.
$arcsin(-1)$. Ищем угол из отрезка $[-\frac{\pi}{2}, \frac{\pi}{2}]$, синус которого равен -1. Этим углом является $-\frac{\pi}{2}$.
Ответ: $-\frac{\pi}{2}$.
$arcsin(\frac{1}{2})$. Ищем угол из отрезка $[-\frac{\pi}{2}, \frac{\pi}{2}]$, синус которого равен $\frac{1}{2}$. Этим углом является $\frac{\pi}{6}$.
Ответ: $\frac{\pi}{6}$.
$arcsin(-\frac{1}{2})$. Ищем угол из отрезка $[-\frac{\pi}{2}, \frac{\pi}{2}]$, синус которого равен $-\frac{1}{2}$. Этим углом является $-\frac{\pi}{6}$.
Ответ: $-\frac{\pi}{6}$.
б)
$arccos((\sqrt{3}-1)(\sqrt{3}+1)-2)$. Упростим выражение в скобках, используя формулу разности квадратов: $(\sqrt{3}-1)(\sqrt{3}+1)-2 = ((\sqrt{3})^2-1^2)-2 = (3-1)-2 = 0$. Получаем $arccos(0) = \frac{\pi}{2}$.
Ответ: $\frac{\pi}{2}$.
$arccos(sin^2\alpha+cos^2\alpha)$. Согласно основному тригонометрическому тождеству, $sin^2\alpha+cos^2\alpha = 1$. Получаем $arccos(1) = 0$.
Ответ: $0$.
$arccos((-1)^{105})$. Поскольку 105 — нечетное число, $(-1)^{105} = -1$. Получаем $arccos(-1) = \pi$.
Ответ: $\pi$.
$arccos(\frac{1}{\sqrt{3}-1}-\frac{1}{2})$. Упростим аргумент: $\frac{1}{\sqrt{3}-1}-\frac{1}{2} = \frac{\sqrt{3}+1}{(\sqrt{3}-1)(\sqrt{3}+1)}-\frac{1}{2} = \frac{\sqrt{3}+1}{3-1}-\frac{1}{2} = \frac{\sqrt{3}+1}{2}-\frac{1}{2} = \frac{\sqrt{3}}{2}$. Получаем $arccos(\frac{\sqrt{3}}{2}) = \frac{\pi}{6}$.
Ответ: $\frac{\pi}{6}$.
$arccos(\frac{1}{2}tg120^\circ)$. Упростим аргумент: $tg120^\circ = tg(180^\circ-60^\circ)=-tg60^\circ=-\sqrt{3}$. Тогда $\frac{1}{2}tg120^\circ = \frac{1}{2}(-\sqrt{3}) = -\frac{\sqrt{3}}{2}$. Получаем $arccos(-\frac{\sqrt{3}}{2}) = \pi - arccos(\frac{\sqrt{3}}{2}) = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$.
Ответ: $\frac{5\pi}{6}$.
в)
$arctg(cos\frac{53}{2}\pi)$. Упростим аргумент: $cos(\frac{53\pi}{2}) = cos(26\pi + \frac{\pi}{2}) = cos(\frac{\pi}{2}) = 0$. Получаем $arctg(0) = 0$.
Ответ: $0$.
$arctg(sin\frac{53}{2}\pi)$. Упростим аргумент: $sin(\frac{53\pi}{2}) = sin(26\pi + \frac{\pi}{2}) = sin(\frac{\pi}{2}) = 1$. Получаем $arctg(1) = \frac{\pi}{4}$.
Ответ: $\frac{\pi}{4}$.
$arctg(sin\frac{55}{2}\pi)$. Упростим аргумент: $sin(\frac{55\pi}{2}) = sin(28\pi - \frac{\pi}{2}) = sin(-\frac{\pi}{2}) = -1$. Получаем $arctg(-1) = -\frac{\pi}{4}$.
Ответ: $-\frac{\pi}{4}$.
$arctg(\frac{4cos15^\circ}{\sqrt{6}+3\sqrt{2}})$. Упростим аргумент. $cos15^\circ = \frac{\sqrt{6}+\sqrt{2}}{4}$. Знаменатель: $\sqrt{6}+3\sqrt{2} = \sqrt{2}(\sqrt{3}+3)$. Аргумент: $\frac{4(\frac{\sqrt{6}+\sqrt{2}}{4})}{\sqrt{2}(\sqrt{3}+3)} = \frac{\sqrt{2}(\sqrt{3}+1)}{\sqrt{2}(\sqrt{3}+3)} = \frac{\sqrt{3}+1}{3+\sqrt{3}} = \frac{\sqrt{3}+1}{\sqrt{3}(\sqrt{3}+1)} = \frac{1}{\sqrt{3}}$. Получаем $arctg(\frac{1}{\sqrt{3}}) = \frac{\pi}{6}$.
Ответ: $\frac{\pi}{6}$.
$arctg(\frac{4sin15^\circ}{\sqrt{6}-3\sqrt{2}})$. Упростим аргумент. $sin15^\circ = \frac{\sqrt{6}-\sqrt{2}}{4}$. Знаменатель: $\sqrt{6}-3\sqrt{2} = \sqrt{2}(\sqrt{3}-3)$. Аргумент: $\frac{4(\frac{\sqrt{6}-\sqrt{2}}{4})}{\sqrt{2}(\sqrt{3}-3)} = \frac{\sqrt{2}(\sqrt{3}-1)}{\sqrt{2}(\sqrt{3}-3)} = \frac{\sqrt{3}-1}{\sqrt{3}-3} = \frac{\sqrt{3}-1}{-(\sqrt{3}-1)\sqrt{3}} = -\frac{1}{\sqrt{3}}$. Получаем $arctg(-\frac{1}{\sqrt{3}}) = -\frac{\pi}{6}$.
Ответ: $-\frac{\pi}{6}$.
г)
$arcctg(cos\beta+cos(180^\circ-\beta))$. По формуле приведения $cos(180^\circ-\beta)=-cos\beta$. Тогда аргумент равен $cos\beta-cos\beta = 0$. Получаем $arcctg(0) = \frac{\pi}{2}$.
Ответ: $\frac{\pi}{2}$.
$arcctg(\frac{sin(180^\circ-\gamma)}{sin\gamma})$ при $sin\gamma \neq 0$. По формуле приведения $sin(180^\circ-\gamma)=sin\gamma$. Тогда аргумент равен $\frac{sin\gamma}{sin\gamma} = 1$. Получаем $arcctg(1) = \frac{\pi}{4}$.
Ответ: $\frac{\pi}{4}$.
$arcctg(\frac{cos93^\circ}{sin3^\circ})$. По формуле приведения $cos93^\circ = cos(90^\circ+3^\circ) = -sin3^\circ$. Тогда аргумент равен $\frac{-sin3^\circ}{sin3^\circ} = -1$. Получаем $arcctg(-1) = \pi - arcctg(1) = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$.
Ответ: $\frac{3\pi}{4}$.
$arcctg\sqrt{\frac{3(1-cos14^\circ)}{2sin^27^\circ}}$. По формуле двойного угла $1-cos14^\circ = 2sin^27^\circ$. Тогда выражение под корнем равно $\frac{3(2sin^27^\circ)}{2sin^27^\circ} = 3$. Получаем $arcctg(\sqrt{3}) = \frac{\pi}{6}$.
Ответ: $\frac{\pi}{6}$.
$arcctg(-\sqrt{3})$. Используем свойство арккотангенса: $arcctg(-x) = \pi - arcctg(x)$. Получаем $arcctg(-\sqrt{3}) = \pi - arcctg(\sqrt{3}) = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$.
Ответ: $\frac{5\pi}{6}$.
Другие задания:
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