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Номер 1, страница 76, 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 1. Свойства и график и тригонометрических функций. 1.2. Определения тригонометрических функций. Упражнения - номер 1, страница 76.
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" => 1119690 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "76" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/2-1-2upr-1" "field_display_title" => "1" "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" => 1119359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Свойства и график и тригонометрических функций" "field_branch_order" => "1" "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" => "74" "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" => 1119359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Свойства и график и тригонометрических функций" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1159} "field_page_start" => "74" "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" => 1119362 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1.2. Определения тригонометрических функций" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1157 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "76" "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" => 1119359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Свойства и график и тригонометрических функций" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1177 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_page_start" => "74" "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" => 1119359 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Свойства и график и тригонометрических функций" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1177} "field_page_start" => "74" "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" => 1119362 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1.2. Определения тригонометрических функций" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1157} "field_page_start" => "76" "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" => 1119363 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1161 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "76" "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" => 1119362 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1.2. Определения тригонометрических функций" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1186 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "76" "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" => 1119362 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1.2. Определения тригонометрических функций" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1186} "field_page_start" => "76" "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" => 1119363 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Упражнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1161} "field_page_start" => "76" "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" => 1300375 "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" => "1119690" "type" => "task" ] "text" => "<p><strong>Упражнение 1</strong></p><p>Отметьте на окружности точки</p><p>$M_{\frac{\pi}{2}}, M_{\frac{2\pi}{3}}, M_{2-\frac{\pi}{3}}, M_{2-\frac{\pi}{6}}, M_{\frac{\pi}{6}}, M_{\frac{\pi}{3}}, M_{\frac{2\pi}{3}}, M_{-\frac{2\pi}{3}}, M_{-2\frac{1}{3}\pi}, M_{-21\frac{1}{2}\pi}$.</p><p>(Отмечать можно и нужно, полагаясь на свой глазомер, например, путь длины $\frac{\pi}{3}$ в 3 раза меньше пути длины $\pi$).</p>" "img" => array:1 [ 0 => array:5 [ "name" => "1-1.jpg" "alt" => null "width" => "1748" "height" => 519 "path" => "/media/algebra_10/baspasy-u19/0-2-1-2upr/1-1.webp?ts=1753534544" ] ] ] #original: array:7 [ "id" => 1300375 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087} "task" => array:2 [ "refs" => "1119690" "type" => "task" ] "text" => "<p><strong>Упражнение 1</strong></p><p>Отметьте на окружности точки</p><p>$M_{\frac{\pi}{2}}, M_{\frac{2\pi}{3}}, M_{2-\frac{\pi}{3}}, M_{2-\frac{\pi}{6}}, M_{\frac{\pi}{6}}, M_{\frac{\pi}{3}}, M_{\frac{2\pi}{3}}, M_{-\frac{2\pi}{3}}, M_{-2\frac{1}{3}\pi}, M_{-21\frac{1}{2}\pi}$.</p><p>(Отмечать можно и нужно, полагаясь на свой глазомер, например, путь длины $\frac{\pi}{3}$ в 3 раза меньше пути длины $\pi$).</p>" "img" => array:1 [ 0 => array:5 [ "name" => "1-1.jpg" "alt" => null "width" => "1748" "height" => 519 "path" => "/media/algebra_10/baspasy-u19/0-2-1-2upr/1-1.webp?ts=1753534544" ] ] ] #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" => 1551411 "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" => "1119690" "type" => "task" ] "text" => "<p>Для того чтобы отметить заданные точки на окружности, необходимо определить их положение, то есть угол, который они образуют с положительным направлением оси Ox. Положительные углы откладываются против часовой стрелки, а отрицательные – по часовой стрелке. Полный оборот по окружности составляет $2\pi$ радиан или $360^\circ$. Если угол по абсолютной величине больше $2\pi$, мы можем прибавлять или вычитать $2\pi$ до тех пор, пока не получим угол в пределах одного оборота (например, от $0$ до $2\pi$), так как точки на окружности будут совпадать.</p><p><strong>$M_{\frac{\pi}{2}}$</strong></p><p>Точка соответствует углу $\frac{\pi}{2}$ радиан, что равно $90^\circ$. Для ее нахождения нужно от начальной точки (соответствующей углу 0) пройти четверть окружности против часовой стрелки. Эта точка находится на положительной части оси OY.</p><p><strong>Ответ:</strong> Точка $M_{\frac{\pi}{2}}$ расположена в верхней части окружности, на пересечении с осью OY, ее координаты (0, 1) на единичной окружности.</p><p><strong>$M_{\frac{2\pi}{3}}$</strong></p><p>Точка соответствует углу $\frac{2\pi}{3}$ радиан, что равно $120^\circ$. Этот угол больше, чем $\frac{\pi}{2}$ ($90^\circ$), но меньше, чем $\pi$ ($180^\circ$), поэтому точка располагается во второй координатной четверти.</p><p><strong>Ответ:</strong> Точка $M_{\frac{2\pi}{3}}$ расположена во второй четверти.</p><p><strong>$M_{2\frac{1}{3}\pi}$</strong></p><p>Сначала преобразуем смешанную дробь в неправильную: $2\frac{1}{3}\pi = \frac{7\pi}{3}$. Чтобы найти положение точки на единичной окружности, вычтем полный оборот $2\pi$: $\frac{7\pi}{3} = \frac{6\pi + \pi}{3} = 2\pi + \frac{\pi}{3}$. Это означает, что точка $M_{2\frac{1}{3}\pi}$ совпадает с точкой $M_{\frac{\pi}{3}}$. Угол $\frac{\pi}{3}$ радиан равен $60^\circ$.</p><p><strong>Ответ:</strong> Точка $M_{2\frac{1}{3}\pi}$ совпадает с точкой $M_{\frac{\pi}{3}}$ и расположена в первой четверти.</p><p><strong>$M_{2\frac{1}{6}\pi}$</strong></p><p>Преобразуем значение угла: $2\frac{1}{6}\pi = \frac{13\pi}{6}$. Найдем основное значение угла, вычитая полный оборот: $\frac{13\pi}{6} = \frac{12\pi + \pi}{6} = 2\pi + \frac{\pi}{6}$. Таким образом, точка $M_{2\frac{1}{6}\pi}$ совпадает с точкой $M_{\frac{\pi}{6}}$. Угол $\frac{\pi}{6}$ радиан равен $30^\circ$.</p><p><strong>Ответ:</strong> Точка $M_{2\frac{1}{6}\pi}$ совпадает с точкой $M_{\frac{\pi}{6}}$ и расположена в первой четверти.</p><p><strong>$M_{\pi}$</strong></p><p>Точка соответствует углу $\pi$ радиан, или $180^\circ$. Это половина окружности против часовой стрелки от начальной точки. Эта точка находится на отрицательной части оси OX.</p><p><strong>Ответ:</strong> Точка $M_{\pi}$ расположена в левой части окружности, на пересечении с осью OX, ее координаты (-1, 0) на единичной окружности.</p><p><strong>$M_{\frac{\pi}{6}}$</strong></p><p>Точка соответствует углу $\frac{\pi}{6}$ радиан, или $30^\circ$. Движение от начальной точки происходит против часовой стрелки. Точка находится в первой четверти.</p><p><strong>Ответ:</strong> Точка $M_{\frac{\pi}{6}}$ расположена в первой четверти.</p><p><strong>$M_{\frac{\pi}{3}}$</strong></p><p>Точка соответствует углу $\frac{\pi}{3}$ радиан, или $60^\circ$. Движение от начальной точки происходит против часовой стрелки. Точка находится в первой четверти.</p><p><strong>Ответ:</strong> Точка $M_{\frac{\pi}{3}}$ расположена в первой четверти.</p><p><strong>$M_{\frac{2\pi}{3}}$</strong></p><p>Эта точка идентична второй точке в списке. Угол $\frac{2\pi}{3}$ радиан ($120^\circ$) соответствует точке во второй четверти.</p><p><strong>Ответ:</strong> Точка $M_{\frac{2\pi}{3}}$ расположена во второй четверти.</p><p><strong>$M_{-\frac{2\pi}{3}}$</strong></p><p>Отрицательный угол означает движение по часовой стрелке от начальной точки. Угол $-\frac{2\pi}{3}$ радиан равен $-120^\circ$. Эта точка находится в третьей четверти. Ей соответствует положительный угол $-\frac{2\pi}{3} + 2\pi = \frac{4\pi}{3}$ радиан ($240^\circ$).</p><p><strong>Ответ:</strong> Точка $M_{-\frac{2\pi}{3}}$ расположена в третьей четверти.</p><p><strong>$M_{-2\frac{1}{3}\pi}$</strong></p><p>Преобразуем значение угла: $-2\frac{1}{3}\pi = -\frac{7\pi}{3}$. Найдем основное значение угла, прибавив полный оборот $2\pi$: $-\frac{7\pi}{3} = -2\pi - \frac{\pi}{3}$. Это означает, что точка совершает один полный оборот по часовой стрелке и останавливается в положении, соответствующем углу $-\frac{\pi}{3}$. Угол $-\frac{\pi}{3}$ радиан ($-60^\circ$) находится в четвертой четверти.</p><p><strong>Ответ:</strong> Точка $M_{-2\frac{1}{3}\pi}$ совпадает с точкой $M_{-\frac{\pi}{3}}$ и расположена в четвертой четверти.</p><p><strong>$M_{-21\frac{1}{2}\pi}$</strong></p><p>Преобразуем значение угла: $-21\frac{1}{2}\pi = -\frac{43\pi}{2}$. Найдем основное значение угла. Так как $2\pi = \frac{4\pi}{2}$, мы можем добавить кратное $2\pi$ для упрощения: $-\frac{43\pi}{2} = -\frac{44\pi - \pi}{2} = -22\pi + \frac{\pi}{2}$. Это соответствует 11 полным оборотам по часовой стрелке плюс угол $\frac{\pi}{2}$.</p><p><strong>Ответ:</strong> Точка $M_{-21\frac{1}{2}\pi}$ совпадает с точкой $M_{\frac{\pi}{2}}$ и расположена на положительной части оси OY.</p><p>Таким образом, некоторые из указанных точек совпадают. Всего на окружности нужно отметить 7 уникальных положений: <br>$M_{\frac{\pi}{6}}$ (совпадает с $M_{2\frac{1}{6}\pi}$), <br>$M_{\frac{\pi}{3}}$ (совпадает с $M_{2\frac{1}{3}\pi}$), <br>$M_{\frac{\pi}{2}}$ (совпадает с $M_{-21\frac{1}{2}\pi}$), <br>$M_{\frac{2\pi}{3}}$ (указана дважды), <br>$M_{\pi}$, <br>$M_{-\frac{2\pi}{3}}$, <br>$M_{-\frac{\pi}{3}}$ (совпадает с $M_{-2\frac{1}{3}\pi}$). <br>Ниже представлено изображение единичной окружности с отмеченными на ней уникальными точками.</p><svg width="320" height="320" xmlns="http://www.w3.org/2000/svg"> <!-- Axes --> <line x1="10" y1="160" x2="310" y2="160" stroke="#aaa" stroke-width="1"></line> <line x1="160" y1="10" x2="160" y2="310" stroke="#aaa" stroke-width="1"></line> <!-- Axis labels --> <text x="312" y="155" font-family="sans-serif" font-size="14">x</text> <text x="165" y="15" font-family="sans-serif" font-size="14">y</text> <!-- Circle --> <circle cx="160" cy="160" r="120" stroke="black" stroke-width="2" fill="none"></circle> <!-- Points and Labels --> <!-- M_pi/6 --> <circle cx="263.9" cy="100" r="3.5" fill="blue"></circle> <text x="270" y="95" font-family="sans-serif" font-size="12">M<tspan dy="4" font-size="9">π/6</tspan></text> <!-- M_pi/3 --> <circle cx="220" cy="56.1" r="3.5" fill="blue"></circle> <text x="225" y="46" font-family="sans-serif" font-size="12">M<tspan dy="4" font-size="9">π/3</tspan></text> <!-- M_pi/2 --> <circle cx="160" cy="40" r="3.5" fill="blue"></circle> <text x="160" y="30" text-anchor="middle" font-family="sans-serif" font-size="12">M<tspan dy="4" font-size="9">π/2</tspan></text> <!-- M_2pi/3 --> <circle cx="100" cy="56.1" r="3.5" fill="blue"></circle> <text x="95" y="46" text-anchor="end" font-family="sans-serif" font-size="12">M<tspan dy="4" font-size="9">2π/3</tspan></text> <!-- M_pi --> <circle cx="40" cy="160" r="3.5" fill="blue"></circle> <text x="35" y="164" text-anchor="end" font-family="sans-serif" font-size="12">M<tspan dy="4" font-size="9">π</tspan></text> <!-- M_-2pi/3 --> <circle cx="100" cy="263.9" r="3.5" fill="blue"></circle> <text x="95" y="278" text-anchor="end" font-family="sans-serif" font-size="12">M<tspan dy="4" font-size="9">-2π/3</tspan></text> <!-- M_-pi/3 --> <circle cx="220" cy="263.9" r="3.5" fill="blue"></circle> <text x="225" y="278" font-family="sans-serif" font-size="12">M<tspan dy="4" font-size="9">-π/3</tspan></text></svg>" ] #original: array:6 [ "id" => 1551411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081} "task" => array:2 [ "refs" => "1119690" "type" => "task" ] "text" => "<p>Для того чтобы отметить заданные точки на окружности, необходимо определить их положение, то есть угол, который они образуют с положительным направлением оси Ox. Положительные углы откладываются против часовой стрелки, а отрицательные – по часовой стрелке. Полный оборот по окружности составляет $2\pi$ радиан или $360^\circ$. Если угол по абсолютной величине больше $2\pi$, мы можем прибавлять или вычитать $2\pi$ до тех пор, пока не получим угол в пределах одного оборота (например, от $0$ до $2\pi$), так как точки на окружности будут совпадать.</p><p><strong>$M_{\frac{\pi}{2}}$</strong></p><p>Точка соответствует углу $\frac{\pi}{2}$ радиан, что равно $90^\circ$. Для ее нахождения нужно от начальной точки (соответствующей углу 0) пройти четверть окружности против часовой стрелки. Эта точка находится на положительной части оси OY.</p><p><strong>Ответ:</strong> Точка $M_{\frac{\pi}{2}}$ расположена в верхней части окружности, на пересечении с осью OY, ее координаты (0, 1) на единичной окружности.</p><p><strong>$M_{\frac{2\pi}{3}}$</strong></p><p>Точка соответствует углу $\frac{2\pi}{3}$ радиан, что равно $120^\circ$. Этот угол больше, чем $\frac{\pi}{2}$ ($90^\circ$), но меньше, чем $\pi$ ($180^\circ$), поэтому точка располагается во второй координатной четверти.</p><p><strong>Ответ:</strong> Точка $M_{\frac{2\pi}{3}}$ расположена во второй четверти.</p><p><strong>$M_{2\frac{1}{3}\pi}$</strong></p><p>Сначала преобразуем смешанную дробь в неправильную: $2\frac{1}{3}\pi = \frac{7\pi}{3}$. Чтобы найти положение точки на единичной окружности, вычтем полный оборот $2\pi$: $\frac{7\pi}{3} = \frac{6\pi + \pi}{3} = 2\pi + \frac{\pi}{3}$. Это означает, что точка $M_{2\frac{1}{3}\pi}$ совпадает с точкой $M_{\frac{\pi}{3}}$. Угол $\frac{\pi}{3}$ радиан равен $60^\circ$.</p><p><strong>Ответ:</strong> Точка $M_{2\frac{1}{3}\pi}$ совпадает с точкой $M_{\frac{\pi}{3}}$ и расположена в первой четверти.</p><p><strong>$M_{2\frac{1}{6}\pi}$</strong></p><p>Преобразуем значение угла: $2\frac{1}{6}\pi = \frac{13\pi}{6}$. Найдем основное значение угла, вычитая полный оборот: $\frac{13\pi}{6} = \frac{12\pi + \pi}{6} = 2\pi + \frac{\pi}{6}$. Таким образом, точка $M_{2\frac{1}{6}\pi}$ совпадает с точкой $M_{\frac{\pi}{6}}$. Угол $\frac{\pi}{6}$ радиан равен $30^\circ$.</p><p><strong>Ответ:</strong> Точка $M_{2\frac{1}{6}\pi}$ совпадает с точкой $M_{\frac{\pi}{6}}$ и расположена в первой четверти.</p><p><strong>$M_{\pi}$</strong></p><p>Точка соответствует углу $\pi$ радиан, или $180^\circ$. Это половина окружности против часовой стрелки от начальной точки. Эта точка находится на отрицательной части оси OX.</p><p><strong>Ответ:</strong> Точка $M_{\pi}$ расположена в левой части окружности, на пересечении с осью OX, ее координаты (-1, 0) на единичной окружности.</p><p><strong>$M_{\frac{\pi}{6}}$</strong></p><p>Точка соответствует углу $\frac{\pi}{6}$ радиан, или $30^\circ$. Движение от начальной точки происходит против часовой стрелки. Точка находится в первой четверти.</p><p><strong>Ответ:</strong> Точка $M_{\frac{\pi}{6}}$ расположена в первой четверти.</p><p><strong>$M_{\frac{\pi}{3}}$</strong></p><p>Точка соответствует углу $\frac{\pi}{3}$ радиан, или $60^\circ$. Движение от начальной точки происходит против часовой стрелки. Точка находится в первой четверти.</p><p><strong>Ответ:</strong> Точка $M_{\frac{\pi}{3}}$ расположена в первой четверти.</p><p><strong>$M_{\frac{2\pi}{3}}$</strong></p><p>Эта точка идентична второй точке в списке. Угол $\frac{2\pi}{3}$ радиан ($120^\circ$) соответствует точке во второй четверти.</p><p><strong>Ответ:</strong> Точка $M_{\frac{2\pi}{3}}$ расположена во второй четверти.</p><p><strong>$M_{-\frac{2\pi}{3}}$</strong></p><p>Отрицательный угол означает движение по часовой стрелке от начальной точки. Угол $-\frac{2\pi}{3}$ радиан равен $-120^\circ$. Эта точка находится в третьей четверти. Ей соответствует положительный угол $-\frac{2\pi}{3} + 2\pi = \frac{4\pi}{3}$ радиан ($240^\circ$).</p><p><strong>Ответ:</strong> Точка $M_{-\frac{2\pi}{3}}$ расположена в третьей четверти.</p><p><strong>$M_{-2\frac{1}{3}\pi}$</strong></p><p>Преобразуем значение угла: $-2\frac{1}{3}\pi = -\frac{7\pi}{3}$. Найдем основное значение угла, прибавив полный оборот $2\pi$: $-\frac{7\pi}{3} = -2\pi - \frac{\pi}{3}$. Это означает, что точка совершает один полный оборот по часовой стрелке и останавливается в положении, соответствующем углу $-\frac{\pi}{3}$. Угол $-\frac{\pi}{3}$ радиан ($-60^\circ$) находится в четвертой четверти.</p><p><strong>Ответ:</strong> Точка $M_{-2\frac{1}{3}\pi}$ совпадает с точкой $M_{-\frac{\pi}{3}}$ и расположена в четвертой четверти.</p><p><strong>$M_{-21\frac{1}{2}\pi}$</strong></p><p>Преобразуем значение угла: $-21\frac{1}{2}\pi = -\frac{43\pi}{2}$. Найдем основное значение угла. Так как $2\pi = \frac{4\pi}{2}$, мы можем добавить кратное $2\pi$ для упрощения: $-\frac{43\pi}{2} = -\frac{44\pi - \pi}{2} = -22\pi + \frac{\pi}{2}$. Это соответствует 11 полным оборотам по часовой стрелке плюс угол $\frac{\pi}{2}$.</p><p><strong>Ответ:</strong> Точка $M_{-21\frac{1}{2}\pi}$ совпадает с точкой $M_{\frac{\pi}{2}}$ и расположена на положительной части оси OY.</p><p>Таким образом, некоторые из указанных точек совпадают. Всего на окружности нужно отметить 7 уникальных положений: <br>$M_{\frac{\pi}{6}}$ (совпадает с $M_{2\frac{1}{6}\pi}$), <br>$M_{\frac{\pi}{3}}$ (совпадает с $M_{2\frac{1}{3}\pi}$), <br>$M_{\frac{\pi}{2}}$ (совпадает с $M_{-21\frac{1}{2}\pi}$), <br>$M_{\frac{2\pi}{3}}$ (указана дважды), <br>$M_{\pi}$, <br>$M_{-\frac{2\pi}{3}}$, <br>$M_{-\frac{\pi}{3}}$ (совпадает с $M_{-2\frac{1}{3}\pi}$). <br>Ниже представлено изображение единичной окружности с отмеченными на ней уникальными точками.</p><svg width="320" height="320" xmlns="http://www.w3.org/2000/svg"> <!-- Axes --> <line x1="10" y1="160" x2="310" y2="160" stroke="#aaa" stroke-width="1"></line> <line x1="160" y1="10" x2="160" y2="310" stroke="#aaa" stroke-width="1"></line> <!-- Axis labels --> <text x="312" y="155" font-family="sans-serif" font-size="14">x</text> <text x="165" y="15" font-family="sans-serif" font-size="14">y</text> <!-- Circle --> <circle cx="160" cy="160" r="120" stroke="black" stroke-width="2" fill="none"></circle> 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№1 (с. 76)
Условие. №1 (с. 76)
скриншот условия
Упражнение 1
Отметьте на окружности точки
$M_{\frac{\pi}{2}}, M_{\frac{2\pi}{3}}, M_{2-\frac{\pi}{3}}, M_{2-\frac{\pi}{6}}, M_{\frac{\pi}{6}}, M_{\frac{\pi}{3}}, M_{\frac{2\pi}{3}}, M_{-\frac{2\pi}{3}}, M_{-2\frac{1}{3}\pi}, M_{-21\frac{1}{2}\pi}$.
(Отмечать можно и нужно, полагаясь на свой глазомер, например, путь длины $\frac{\pi}{3}$ в 3 раза меньше пути длины $\pi$).
Решение 2 (rus). №1 (с. 76)
Для того чтобы отметить заданные точки на окружности, необходимо определить их положение, то есть угол, который они образуют с положительным направлением оси Ox. Положительные углы откладываются против часовой стрелки, а отрицательные – по часовой стрелке. Полный оборот по окружности составляет $2\pi$ радиан или $360^\circ$. Если угол по абсолютной величине больше $2\pi$, мы можем прибавлять или вычитать $2\pi$ до тех пор, пока не получим угол в пределах одного оборота (например, от $0$ до $2\pi$), так как точки на окружности будут совпадать.
$M_{\frac{\pi}{2}}$
Точка соответствует углу $\frac{\pi}{2}$ радиан, что равно $90^\circ$. Для ее нахождения нужно от начальной точки (соответствующей углу 0) пройти четверть окружности против часовой стрелки. Эта точка находится на положительной части оси OY.
Ответ: Точка $M_{\frac{\pi}{2}}$ расположена в верхней части окружности, на пересечении с осью OY, ее координаты (0, 1) на единичной окружности.
$M_{\frac{2\pi}{3}}$
Точка соответствует углу $\frac{2\pi}{3}$ радиан, что равно $120^\circ$. Этот угол больше, чем $\frac{\pi}{2}$ ($90^\circ$), но меньше, чем $\pi$ ($180^\circ$), поэтому точка располагается во второй координатной четверти.
Ответ: Точка $M_{\frac{2\pi}{3}}$ расположена во второй четверти.
$M_{2\frac{1}{3}\pi}$
Сначала преобразуем смешанную дробь в неправильную: $2\frac{1}{3}\pi = \frac{7\pi}{3}$. Чтобы найти положение точки на единичной окружности, вычтем полный оборот $2\pi$: $\frac{7\pi}{3} = \frac{6\pi + \pi}{3} = 2\pi + \frac{\pi}{3}$. Это означает, что точка $M_{2\frac{1}{3}\pi}$ совпадает с точкой $M_{\frac{\pi}{3}}$. Угол $\frac{\pi}{3}$ радиан равен $60^\circ$.
Ответ: Точка $M_{2\frac{1}{3}\pi}$ совпадает с точкой $M_{\frac{\pi}{3}}$ и расположена в первой четверти.
$M_{2\frac{1}{6}\pi}$
Преобразуем значение угла: $2\frac{1}{6}\pi = \frac{13\pi}{6}$. Найдем основное значение угла, вычитая полный оборот: $\frac{13\pi}{6} = \frac{12\pi + \pi}{6} = 2\pi + \frac{\pi}{6}$. Таким образом, точка $M_{2\frac{1}{6}\pi}$ совпадает с точкой $M_{\frac{\pi}{6}}$. Угол $\frac{\pi}{6}$ радиан равен $30^\circ$.
Ответ: Точка $M_{2\frac{1}{6}\pi}$ совпадает с точкой $M_{\frac{\pi}{6}}$ и расположена в первой четверти.
$M_{\pi}$
Точка соответствует углу $\pi$ радиан, или $180^\circ$. Это половина окружности против часовой стрелки от начальной точки. Эта точка находится на отрицательной части оси OX.
Ответ: Точка $M_{\pi}$ расположена в левой части окружности, на пересечении с осью OX, ее координаты (-1, 0) на единичной окружности.
$M_{\frac{\pi}{6}}$
Точка соответствует углу $\frac{\pi}{6}$ радиан, или $30^\circ$. Движение от начальной точки происходит против часовой стрелки. Точка находится в первой четверти.
Ответ: Точка $M_{\frac{\pi}{6}}$ расположена в первой четверти.
$M_{\frac{\pi}{3}}$
Точка соответствует углу $\frac{\pi}{3}$ радиан, или $60^\circ$. Движение от начальной точки происходит против часовой стрелки. Точка находится в первой четверти.
Ответ: Точка $M_{\frac{\pi}{3}}$ расположена в первой четверти.
$M_{\frac{2\pi}{3}}$
Эта точка идентична второй точке в списке. Угол $\frac{2\pi}{3}$ радиан ($120^\circ$) соответствует точке во второй четверти.
Ответ: Точка $M_{\frac{2\pi}{3}}$ расположена во второй четверти.
$M_{-\frac{2\pi}{3}}$
Отрицательный угол означает движение по часовой стрелке от начальной точки. Угол $-\frac{2\pi}{3}$ радиан равен $-120^\circ$. Эта точка находится в третьей четверти. Ей соответствует положительный угол $-\frac{2\pi}{3} + 2\pi = \frac{4\pi}{3}$ радиан ($240^\circ$).
Ответ: Точка $M_{-\frac{2\pi}{3}}$ расположена в третьей четверти.
$M_{-2\frac{1}{3}\pi}$
Преобразуем значение угла: $-2\frac{1}{3}\pi = -\frac{7\pi}{3}$. Найдем основное значение угла, прибавив полный оборот $2\pi$: $-\frac{7\pi}{3} = -2\pi - \frac{\pi}{3}$. Это означает, что точка совершает один полный оборот по часовой стрелке и останавливается в положении, соответствующем углу $-\frac{\pi}{3}$. Угол $-\frac{\pi}{3}$ радиан ($-60^\circ$) находится в четвертой четверти.
Ответ: Точка $M_{-2\frac{1}{3}\pi}$ совпадает с точкой $M_{-\frac{\pi}{3}}$ и расположена в четвертой четверти.
$M_{-21\frac{1}{2}\pi}$
Преобразуем значение угла: $-21\frac{1}{2}\pi = -\frac{43\pi}{2}$. Найдем основное значение угла. Так как $2\pi = \frac{4\pi}{2}$, мы можем добавить кратное $2\pi$ для упрощения: $-\frac{43\pi}{2} = -\frac{44\pi - \pi}{2} = -22\pi + \frac{\pi}{2}$. Это соответствует 11 полным оборотам по часовой стрелке плюс угол $\frac{\pi}{2}$.
Ответ: Точка $M_{-21\frac{1}{2}\pi}$ совпадает с точкой $M_{\frac{\pi}{2}}$ и расположена на положительной части оси OY.
Таким образом, некоторые из указанных точек совпадают. Всего на окружности нужно отметить 7 уникальных положений:
$M_{\frac{\pi}{6}}$ (совпадает с $M_{2\frac{1}{6}\pi}$),
$M_{\frac{\pi}{3}}$ (совпадает с $M_{2\frac{1}{3}\pi}$),
$M_{\frac{\pi}{2}}$ (совпадает с $M_{-21\frac{1}{2}\pi}$),
$M_{\frac{2\pi}{3}}$ (указана дважды),
$M_{\pi}$,
$M_{-\frac{2\pi}{3}}$,
$M_{-\frac{\pi}{3}}$ (совпадает с $M_{-2\frac{1}{3}\pi}$).
Ниже представлено изображение единичной окружности с отмеченными на ней уникальными точками.
Другие задания:
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