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Номер 18, страница 96, 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 2. Графики функций. 2.6. Примеры. Задачи - номер 18, страница 96.
App\Models\Task {#1024 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "96" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/2-2-6zad-18" "field_display_title" => "18" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 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 {#333 #items: array:1 [ 0 => App\Models\Term {#1033 #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 {#1030 #items: array:1 [ 0 => App\Models\Book {#1032 #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 {#1034 #items: array:1 [ 0 => App\Models\Term {#1035 #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 {#1036 #items: array:1 [ 0 => App\Models\Term {#1037 #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 {#1038 #items: array:1 [ 0 => App\Models\Term {#1039 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almatykitap baspasy" ] #original: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almatykitap baspasy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:3 [ 0 => App\Models\Term {#1041 #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 {#1042 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1043 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1044 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ 0 => App\Models\Term {#1046 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1047 #items: array:1 [ 0 => App\Models\Term {#1048 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1049 #items: array:1 [ 0 => App\Models\Term {#1050 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1056 #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 {#1057 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1060 #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 {#1034} "field_class" => Illuminate\Database\Eloquent\Collection {#1036} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1038} "field_author" => Illuminate\Database\Eloquent\Collection {#1040} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1044} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1045} "field_country" => Illuminate\Database\Eloquent\Collection {#1047} "field_city" => Illuminate\Database\Eloquent\Collection {#1049} "field_series" => Illuminate\Database\Eloquent\Collection {#1051} "field_umk" => Illuminate\Database\Eloquent\Collection {#1052} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1054} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1055} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1056} "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 {#1057} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1058} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1059} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1060} "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 {#1061 #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 {#333} "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 {#1030} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1061} ] #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 {#1062 #items: array:5 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1063 #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 {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #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 {#1066 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1067 #items: array:1 [ 0 => App\Models\Branch {#1068 #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 {#1069 #items: array:1 [ 0 => App\Models\Term {#1070 …30} ] #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 {#1071 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1100 #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 {#1069} "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 [ …4] ] "book" => Illuminate\Database\Eloquent\Collection {#1071} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1100} ] #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" => 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 {#1064} "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 {#1066} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1067} ] #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 {#1101 #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" => 1119364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Графики функций" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ 0 => App\Models\Term {#1103 #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" => "79" "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 {#1104 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:1 [ 0 => App\Models\Branch {#1106 #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 {#1107 #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 {#1108 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1109 #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 {#1107} "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 {#1108} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1109} ] #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" => 1119364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Графики функций" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1102} "field_page_start" => "79" "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 {#1104} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1105} ] #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 {#1110 #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" => 1119371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.6. Примеры" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1111 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "92" "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 {#1112 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => App\Models\Branch {#1114 #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" => 1119364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Графики функций" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1115 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_page_start" => "79" "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 {#1116 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119364 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Графики функций" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1115} "field_page_start" => "79" "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 {#1116} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117} ] #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" => 1119371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.6. Примеры" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1111} "field_page_start" => "92" "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 {#1112} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1113} ] #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 {#1118 #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" => 1119372 "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 {#1119 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "93" "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 {#1120 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Branch {#1122 #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" => 1119371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.6. Примеры" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1123 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "92" "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 {#1124 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.6. Примеры" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1123} "field_page_start" => "92" "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 {#1124} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125} ] #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" => 1119372 "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 {#1119} "field_page_start" => "93" "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 {#1120} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1121} ] #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 {#1126 #items: array:2 [ 0 => App\Models\Element {#1127 #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" => 1300419 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Edition {#1129 #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 {#1130 #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 {#1132 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1133 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1134 #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 {#1130} "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 {#1132} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1133} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1134} "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" => "1119736" "type" => "task" ] "text" => "<p><strong>18. (3)</strong> Среди следующих функций определите периодические и укажите их главный период:</p><p><strong>а)</strong> $f_0(x)=\cos \frac{2\pi}{7}\sin 57x+\sin \frac{2\pi}{7}\cos 57x$, $f_1(x)=\cos^2 \frac{2}{3}x+\sin^2 \frac{2}{3}x$,</p><p>$f_2(x)=\left(\cos \frac{2}{3}x+\sin \frac{2}{3}x\right)^2$, $f_3(x)=x^2|\sin x|$;</p><p><strong>б)</strong> $g_0(x)=\frac{\sqrt{3}}{2}\cos\left(\sqrt{7}x-\frac{\pi}{7}\right)$, $g_1(x)=\cos^2 \frac{\pi}{5}x-\sin^2 \frac{\pi}{5}x$, $g_2(x)=x^3-\cos^4 x$,</p><p>$g_3(x)=\cos^4 x$;</p><p><strong>в)</strong> $h_0(x)=\frac{1}{2-\text{ctg }3\pi x}$; $h_1(x)=\frac{x}{2-\text{ctg }3\pi x}$, $h_2(x)=1+\text{ctg}^2 x$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "18-1.jpg" "alt" => null "width" => "1660" "height" => 891 "path" => "/media/algebra_10/baspasy-u19/0-2-2-6zad/18-1.webp?ts=1753534571" ] ] ] #original: array:7 [ "id" => 1300419 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128} "task" => array:2 [ "refs" => "1119736" "type" => "task" ] "text" => "<p><strong>18. (3)</strong> Среди следующих функций определите периодические и укажите их главный период:</p><p><strong>а)</strong> $f_0(x)=\cos \frac{2\pi}{7}\sin 57x+\sin \frac{2\pi}{7}\cos 57x$, $f_1(x)=\cos^2 \frac{2}{3}x+\sin^2 \frac{2}{3}x$,</p><p>$f_2(x)=\left(\cos \frac{2}{3}x+\sin \frac{2}{3}x\right)^2$, $f_3(x)=x^2|\sin x|$;</p><p><strong>б)</strong> $g_0(x)=\frac{\sqrt{3}}{2}\cos\left(\sqrt{7}x-\frac{\pi}{7}\right)$, $g_1(x)=\cos^2 \frac{\pi}{5}x-\sin^2 \frac{\pi}{5}x$, $g_2(x)=x^3-\cos^4 x$,</p><p>$g_3(x)=\cos^4 x$;</p><p><strong>в)</strong> $h_0(x)=\frac{1}{2-\text{ctg }3\pi x}$; $h_1(x)=\frac{x}{2-\text{ctg }3\pi x}$, $h_2(x)=1+\text{ctg}^2 x$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "18-1.jpg" "alt" => null "width" => "1660" "height" => 891 "path" => "/media/algebra_10/baspasy-u19/0-2-2-6zad/18-1.webp?ts=1753534571" ] ] ] #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 {#1135 #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" => 1551455 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136 #items: array:1 [ 0 => App\Models\Edition {#1137 #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 {#1138 #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 {#1140 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 #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 {#1138} "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 {#1140} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142} "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" => "1119736" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Рассмотрим функцию $f_0(x) = \cos\frac{2\pi}{7}\sin57x + \sin\frac{2\pi}{7}\cos57x$. Применим формулу синуса суммы $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$. Полагая $\alpha=57x$ и $\beta=\frac{2\pi}{7}$, получим: $f_0(x)=\sin(57x+\frac{2\pi}{7})$. Это гармоническая функция, ее главный период вычисляется по формуле $T=\frac{2\pi}{|k|}$. Здесь коэффициент $k=57$, следовательно, главный период $T_0 = \frac{2\pi}{57}$. Функция периодическая.</p><p>Рассмотрим функцию $f_1(x) = \cos\frac{2}{3}x + \sin\frac{2}{3}x$. Это сумма двух функций: $\cos\frac{2}{3}x$ и $\sin\frac{2}{3}x$. Обе функции имеют одинаковый главный период $T = \frac{2\pi}{|2/3|} = 3\pi$. Преобразуем выражение: $f_1(x) = \sqrt{2}(\frac{1}{\sqrt{2}}\cos\frac{2}{3}x + \frac{1}{\sqrt{2}}\sin\frac{2}{3}x) = \sqrt{2}(\cos\frac{\pi}{4}\cos\frac{2}{3}x + \sin\frac{\pi}{4}\sin\frac{2}{3}x) = \sqrt{2}\cos(\frac{2}{3}x - \frac{\pi}{4})$. Это гармоническая функция с коэффициентом $k=2/3$, ее главный период $T_1 = \frac{2\pi}{|2/3|} = 3\pi$. Функция периодическая.</p><p>Рассмотрим функцию $f_2(x) = (\cos\frac{2}{3}x + \sin\frac{2}{3}x)^2$. Раскроем скобки: $f_2(x) = \cos^2\frac{2}{3}x + 2\cos\frac{2}{3}x\sin\frac{2}{3}x + \sin^2\frac{2}{3}x$. Используя основное тригонометрическое тождество $\sin^2\alpha+\cos^2\alpha=1$ и формулу синуса двойного угла $\sin(2\alpha)=2\sin\alpha\cos\alpha$, получаем: $f_2(x) = 1 + \sin(2 \cdot \frac{2}{3}x) = 1 + \sin(\frac{4}{3}x)$. Это гармоническая функция, сдвинутая по оси ординат. Ее период определяется слагаемым $\sin(\frac{4}{3}x)$. Главный период $T_2 = \frac{2\pi}{|4/3|} = \frac{6\pi}{4} = \frac{3\pi}{2}$. Функция периодическая.</p><p>Рассмотрим функцию $f_3(x) = x^2|\sin x|$. Функция $|\sin x|$ является периодической с главным периодом $\pi$. Однако множитель $x^2$ не является периодической функцией. Предположим, что функция $f_3(x)$ периодическая с периодом $T>0$. Тогда должно выполняться равенство $f_3(x+T) = f_3(x)$ для всех $x$, то есть $(x+T)^2|\sin(x+T)| = x^2|\sin x|$. Если выбрать $T=k\pi$ для целого $k\neq0$, то $|\sin(x+k\pi)|=|\sin x|$. Тогда равенство примет вид $(x+k\pi)^2|\sin x| = x^2|\sin x|$. Для $x$, где $|\sin x|\neq0$, получаем $(x+k\pi)^2=x^2$, что неверно для всех $x$. Следовательно, функция $f_3(x)$ не является периодической.</p><p><strong>Ответ:</strong> периодические функции: $f_0(x)$ с главным периодом $T_0=\frac{2\pi}{57}$, $f_1(x)$ с главным периодом $T_1=3\pi$, $f_2(x)$ с главным периодом $T_2=\frac{3\pi}{2}$; функция $f_3(x)$ не является периодической.</p><p><strong>б)</strong></p><p>Рассмотрим функцию $g_0(x) = \frac{\sqrt{3}}{2}\cos(\sqrt{7}x - \frac{\pi}{7})$. Это гармоническая функция вида $y=A\cos(kx+\phi)$. Ее главный период равен $T=\frac{2\pi}{|k|}$. В данном случае $k=\sqrt{7}$, следовательно, главный период $T_0 = \frac{2\pi}{\sqrt{7}}$. Функция периодическая.</p><p>Рассмотрим функцию $g_1(x) = \cos^2\frac{\pi}{5}x - \sin^2\frac{\pi}{5}x$. Применим формулу косинуса двойного угла $\cos(2\alpha) = \cos^2\alpha - \sin^2\alpha$. Полагая $\alpha=\frac{\pi}{5}x$, получим $g_1(x) = \cos(2 \cdot \frac{\pi}{5}x) = \cos(\frac{2\pi}{5}x)$. Это гармоническая функция с коэффициентом $k=\frac{2\pi}{5}$. Ее главный период $T_1 = \frac{2\pi}{|2\pi/5|} = 5$. Функция периодическая.</p><p>Рассмотрим функцию $g_2(x) = x^3 - \cos^4x$. Эта функция является разностью непериодической функции $y_1=x^3$ и периодической функции $y_2=\cos^4x$ (период $\cos^4x$ равен $\pi$). Сумма или разность периодической и непериодической функции является непериодической функцией. Предположим, что $g_2(x)$ периодична с периодом $T>0$. Тогда $g_2(x+T) = g_2(x)$. Возьмем $T=k\pi$ для целого $k\neq0$. Тогда $\cos^4(x+k\pi)=\cos^4x$, и равенство примет вид $(x+k\pi)^3-\cos^4x = x^3-\cos^4x$, откуда $(x+k\pi)^3=x^3$, что неверно для всех $x$. Следовательно, функция $g_2(x)$ не является периодической.</p><p>Рассмотрим функцию $g_3(x) = \cos^4x$. Так как $\cos(x+\pi)=-\cos x$, то $g_3(x+\pi) = \cos^4(x+\pi) = (-\cos x)^4 = \cos^4x = g_3(x)$. Таким образом, $\pi$ является периодом. Можно показать, что это главный период, используя формулы понижения степени: $\cos^4x = (\frac{1+\cos(2x)}{2})^2 = \frac{1+2\cos(2x)+\cos^2(2x)}{4} = \frac{1+2\cos(2x)+\frac{1+\cos(4x)}{2}}{4} = \frac{3}{8}+\frac{1}{2}\cos(2x)+\frac{1}{8}\cos(4x)$. Периоды слагаемых: $T_1=\frac{2\pi}{2}=\pi$ и $T_2=\frac{2\pi}{4}=\frac{\pi}{2}$. Главный период суммы равен наименьшему общему кратному периодов слагаемых: $T_3=\text{НОК}(\pi, \frac{\pi}{2})=\pi$. Функция периодическая.</p><p><strong>Ответ:</strong> периодические функции: $g_0(x)$ с главным периодом $T_0=\frac{2\pi}{\sqrt{7}}$, $g_1(x)$ с главным периодом $T_1=5$, $g_3(x)$ с главным периодом $T_3=\pi$; функция $g_2(x)$ не является периодической.</p><p><strong>в)</strong></p><p>Рассмотрим функцию $h_0(x) = \frac{1}{2-\text{ctg}(3\pi x)}$. Периодичность функции определяется периодичностью функции $\text{ctg}(3\pi x)$. Главный период функции $\text{ctg}(y)$ равен $\pi$. Для функции $\text{ctg}(kx)$ главный период равен $T=\frac{\pi}{|k|}$. Здесь $k=3\pi$, поэтому главный период $T_0 = \frac{\pi}{|3\pi|} = \frac{1}{3}$. Функция периодическая.</p><p>Рассмотрим функцию $h_1(x) = \frac{x}{2-\text{ctg}(3\pi x)}$. Эта функция является частным от деления непериодической функции $y_1=x$ на периодическую функцию $y_2=2-\text{ctg}(3\pi x)$ (с периодом $1/3$). Предположим, что $h_1(x)$ периодична с периодом $T>0$. Тогда $h_1(x+T)=h_1(x)$. Возьмем $T=k/3$ для целого $k\neq0$. Тогда $\text{ctg}(3\pi(x+k/3)) = \text{ctg}(3\pi x+k\pi)=\text{ctg}(3\pi x)$, и равенство примет вид $\frac{x+k/3}{2-\text{ctg}(3\pi x)} = \frac{x}{2-\text{ctg}(3\pi x)}$. Отсюда $x+k/3=x$, что неверно, так как $k\neq0$. Следовательно, функция $h_1(x)$ не является периодической.</p><p>Рассмотрим функцию $h_2(x) = 1 + \text{ctg}^2x$. Период функции $\text{ctg} x$ равен $\pi$. Так как $\text{ctg}(x+\pi)=\text{ctg} x$, то $h_2(x+\pi) = 1+\text{ctg}^2(x+\pi) = 1+(\text{ctg} x)^2 = h_2(x)$. Период $\pi$ является главным, так как для $T=\pi/2$, $h_2(x+\pi/2) = 1+\text{ctg}^2(x+\pi/2) = 1+(-\text{tg} x)^2 = 1+\text{tg}^2x \neq h_2(x)$ в общем случае. Также можно использовать тождество $1+\text{ctg}^2x = \frac{1}{\sin^2x}$. Главный период функции $\sin^2x$ равен $\pi$, следовательно, главный период функции $h_2(x)$ также равен $\pi$. Функция периодическая.</p><p><strong>Ответ:</strong> периодические функции: $h_0(x)$ с главным периодом $T_0=\frac{1}{3}$, $h_2(x)$ с главным периодом $T_2=\pi$; функция $h_1(x)$ не является периодической.</p>" ] #original: array:6 [ "id" => 1551455 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136} "task" => array:2 [ "refs" => "1119736" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Рассмотрим функцию $f_0(x) = \cos\frac{2\pi}{7}\sin57x + \sin\frac{2\pi}{7}\cos57x$. Применим формулу синуса суммы $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$. Полагая $\alpha=57x$ и $\beta=\frac{2\pi}{7}$, получим: $f_0(x)=\sin(57x+\frac{2\pi}{7})$. Это гармоническая функция, ее главный период вычисляется по формуле $T=\frac{2\pi}{|k|}$. Здесь коэффициент $k=57$, следовательно, главный период $T_0 = \frac{2\pi}{57}$. Функция периодическая.</p><p>Рассмотрим функцию $f_1(x) = \cos\frac{2}{3}x + \sin\frac{2}{3}x$. Это сумма двух функций: $\cos\frac{2}{3}x$ и $\sin\frac{2}{3}x$. Обе функции имеют одинаковый главный период $T = \frac{2\pi}{|2/3|} = 3\pi$. Преобразуем выражение: $f_1(x) = \sqrt{2}(\frac{1}{\sqrt{2}}\cos\frac{2}{3}x + \frac{1}{\sqrt{2}}\sin\frac{2}{3}x) = \sqrt{2}(\cos\frac{\pi}{4}\cos\frac{2}{3}x + \sin\frac{\pi}{4}\sin\frac{2}{3}x) = \sqrt{2}\cos(\frac{2}{3}x - \frac{\pi}{4})$. Это гармоническая функция с коэффициентом $k=2/3$, ее главный период $T_1 = \frac{2\pi}{|2/3|} = 3\pi$. Функция периодическая.</p><p>Рассмотрим функцию $f_2(x) = (\cos\frac{2}{3}x + \sin\frac{2}{3}x)^2$. Раскроем скобки: $f_2(x) = \cos^2\frac{2}{3}x + 2\cos\frac{2}{3}x\sin\frac{2}{3}x + \sin^2\frac{2}{3}x$. Используя основное тригонометрическое тождество $\sin^2\alpha+\cos^2\alpha=1$ и формулу синуса двойного угла $\sin(2\alpha)=2\sin\alpha\cos\alpha$, получаем: $f_2(x) = 1 + \sin(2 \cdot \frac{2}{3}x) = 1 + \sin(\frac{4}{3}x)$. Это гармоническая функция, сдвинутая по оси ординат. Ее период определяется слагаемым $\sin(\frac{4}{3}x)$. Главный период $T_2 = \frac{2\pi}{|4/3|} = \frac{6\pi}{4} = \frac{3\pi}{2}$. Функция периодическая.</p><p>Рассмотрим функцию $f_3(x) = x^2|\sin x|$. Функция $|\sin x|$ является периодической с главным периодом $\pi$. Однако множитель $x^2$ не является периодической функцией. Предположим, что функция $f_3(x)$ периодическая с периодом $T>0$. Тогда должно выполняться равенство $f_3(x+T) = f_3(x)$ для всех $x$, то есть $(x+T)^2|\sin(x+T)| = x^2|\sin x|$. Если выбрать $T=k\pi$ для целого $k\neq0$, то $|\sin(x+k\pi)|=|\sin x|$. Тогда равенство примет вид $(x+k\pi)^2|\sin x| = x^2|\sin x|$. Для $x$, где $|\sin x|\neq0$, получаем $(x+k\pi)^2=x^2$, что неверно для всех $x$. Следовательно, функция $f_3(x)$ не является периодической.</p><p><strong>Ответ:</strong> периодические функции: $f_0(x)$ с главным периодом $T_0=\frac{2\pi}{57}$, $f_1(x)$ с главным периодом $T_1=3\pi$, $f_2(x)$ с главным периодом $T_2=\frac{3\pi}{2}$; функция $f_3(x)$ не является периодической.</p><p><strong>б)</strong></p><p>Рассмотрим функцию $g_0(x) = \frac{\sqrt{3}}{2}\cos(\sqrt{7}x - \frac{\pi}{7})$. Это гармоническая функция вида $y=A\cos(kx+\phi)$. Ее главный период равен $T=\frac{2\pi}{|k|}$. В данном случае $k=\sqrt{7}$, следовательно, главный период $T_0 = \frac{2\pi}{\sqrt{7}}$. Функция периодическая.</p><p>Рассмотрим функцию $g_1(x) = \cos^2\frac{\pi}{5}x - \sin^2\frac{\pi}{5}x$. Применим формулу косинуса двойного угла $\cos(2\alpha) = \cos^2\alpha - \sin^2\alpha$. Полагая $\alpha=\frac{\pi}{5}x$, получим $g_1(x) = \cos(2 \cdot \frac{\pi}{5}x) = \cos(\frac{2\pi}{5}x)$. Это гармоническая функция с коэффициентом $k=\frac{2\pi}{5}$. Ее главный период $T_1 = \frac{2\pi}{|2\pi/5|} = 5$. Функция периодическая.</p><p>Рассмотрим функцию $g_2(x) = x^3 - \cos^4x$. Эта функция является разностью непериодической функции $y_1=x^3$ и периодической функции $y_2=\cos^4x$ (период $\cos^4x$ равен $\pi$). Сумма или разность периодической и непериодической функции является непериодической функцией. Предположим, что $g_2(x)$ периодична с периодом $T>0$. Тогда $g_2(x+T) = g_2(x)$. Возьмем $T=k\pi$ для целого $k\neq0$. Тогда $\cos^4(x+k\pi)=\cos^4x$, и равенство примет вид $(x+k\pi)^3-\cos^4x = x^3-\cos^4x$, откуда $(x+k\pi)^3=x^3$, что неверно для всех $x$. Следовательно, функция $g_2(x)$ не является периодической.</p><p>Рассмотрим функцию $g_3(x) = \cos^4x$. Так как $\cos(x+\pi)=-\cos x$, то $g_3(x+\pi) = \cos^4(x+\pi) = (-\cos x)^4 = \cos^4x = g_3(x)$. Таким образом, $\pi$ является периодом. Можно показать, что это главный период, используя формулы понижения степени: $\cos^4x = (\frac{1+\cos(2x)}{2})^2 = \frac{1+2\cos(2x)+\cos^2(2x)}{4} = \frac{1+2\cos(2x)+\frac{1+\cos(4x)}{2}}{4} = \frac{3}{8}+\frac{1}{2}\cos(2x)+\frac{1}{8}\cos(4x)$. Периоды слагаемых: $T_1=\frac{2\pi}{2}=\pi$ и $T_2=\frac{2\pi}{4}=\frac{\pi}{2}$. Главный период суммы равен наименьшему общему кратному периодов слагаемых: $T_3=\text{НОК}(\pi, \frac{\pi}{2})=\pi$. Функция периодическая.</p><p><strong>Ответ:</strong> периодические функции: $g_0(x)$ с главным периодом $T_0=\frac{2\pi}{\sqrt{7}}$, $g_1(x)$ с главным периодом $T_1=5$, $g_3(x)$ с главным периодом $T_3=\pi$; функция $g_2(x)$ не является периодической.</p><p><strong>в)</strong></p><p>Рассмотрим функцию $h_0(x) = \frac{1}{2-\text{ctg}(3\pi x)}$. Периодичность функции определяется периодичностью функции $\text{ctg}(3\pi x)$. Главный период функции $\text{ctg}(y)$ равен $\pi$. Для функции $\text{ctg}(kx)$ главный период равен $T=\frac{\pi}{|k|}$. Здесь $k=3\pi$, поэтому главный период $T_0 = \frac{\pi}{|3\pi|} = \frac{1}{3}$. Функция периодическая.</p><p>Рассмотрим функцию $h_1(x) = \frac{x}{2-\text{ctg}(3\pi x)}$. Эта функция является частным от деления непериодической функции $y_1=x$ на периодическую функцию $y_2=2-\text{ctg}(3\pi x)$ (с периодом $1/3$). Предположим, что $h_1(x)$ периодична с периодом $T>0$. Тогда $h_1(x+T)=h_1(x)$. Возьмем $T=k/3$ для целого $k\neq0$. Тогда $\text{ctg}(3\pi(x+k/3)) = \text{ctg}(3\pi x+k\pi)=\text{ctg}(3\pi x)$, и равенство примет вид $\frac{x+k/3}{2-\text{ctg}(3\pi x)} = \frac{x}{2-\text{ctg}(3\pi x)}$. Отсюда $x+k/3=x$, что неверно, так как $k\neq0$. Следовательно, функция $h_1(x)$ не является периодической.</p><p>Рассмотрим функцию $h_2(x) = 1 + \text{ctg}^2x$. Период функции $\text{ctg} x$ равен $\pi$. Так как $\text{ctg}(x+\pi)=\text{ctg} x$, то $h_2(x+\pi) = 1+\text{ctg}^2(x+\pi) = 1+(\text{ctg} x)^2 = h_2(x)$. Период $\pi$ является главным, так как для $T=\pi/2$, $h_2(x+\pi/2) = 1+\text{ctg}^2(x+\pi/2) = 1+(-\text{tg} x)^2 = 1+\text{tg}^2x \neq h_2(x)$ в общем случае. Также можно использовать тождество $1+\text{ctg}^2x = \frac{1}{\sin^2x}$. Главный период функции $\sin^2x$ равен $\pi$, следовательно, главный период функции $h_2(x)$ также равен $\pi$. Функция периодическая.</p><p><strong>Ответ:</strong> периодические функции: $h_0(x)$ с главным периодом $T_0=\frac{1}{3}$, $h_2(x)$ с главным периодом $T_2=\pi$; функция $h_1(x)$ не является периодической.</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" => "1119737" "type" => "task" ] "previous" => array:2 [ "refs" => "1119735" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\BookPage {#1157 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true 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№18 (с. 96)
Условие. №18 (с. 96)
скриншот условия
18. (3) Среди следующих функций определите периодические и укажите их главный период:
а) $f_0(x)=\cos \frac{2\pi}{7}\sin 57x+\sin \frac{2\pi}{7}\cos 57x$, $f_1(x)=\cos^2 \frac{2}{3}x+\sin^2 \frac{2}{3}x$,
$f_2(x)=\left(\cos \frac{2}{3}x+\sin \frac{2}{3}x\right)^2$, $f_3(x)=x^2|\sin x|$;
б) $g_0(x)=\frac{\sqrt{3}}{2}\cos\left(\sqrt{7}x-\frac{\pi}{7}\right)$, $g_1(x)=\cos^2 \frac{\pi}{5}x-\sin^2 \frac{\pi}{5}x$, $g_2(x)=x^3-\cos^4 x$,
$g_3(x)=\cos^4 x$;
в) $h_0(x)=\frac{1}{2-\text{ctg }3\pi x}$; $h_1(x)=\frac{x}{2-\text{ctg }3\pi x}$, $h_2(x)=1+\text{ctg}^2 x$.
Решение 2 (rus). №18 (с. 96)
а)
Рассмотрим функцию $f_0(x) = \cos\frac{2\pi}{7}\sin57x + \sin\frac{2\pi}{7}\cos57x$. Применим формулу синуса суммы $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$. Полагая $\alpha=57x$ и $\beta=\frac{2\pi}{7}$, получим: $f_0(x)=\sin(57x+\frac{2\pi}{7})$. Это гармоническая функция, ее главный период вычисляется по формуле $T=\frac{2\pi}{|k|}$. Здесь коэффициент $k=57$, следовательно, главный период $T_0 = \frac{2\pi}{57}$. Функция периодическая.
Рассмотрим функцию $f_1(x) = \cos\frac{2}{3}x + \sin\frac{2}{3}x$. Это сумма двух функций: $\cos\frac{2}{3}x$ и $\sin\frac{2}{3}x$. Обе функции имеют одинаковый главный период $T = \frac{2\pi}{|2/3|} = 3\pi$. Преобразуем выражение: $f_1(x) = \sqrt{2}(\frac{1}{\sqrt{2}}\cos\frac{2}{3}x + \frac{1}{\sqrt{2}}\sin\frac{2}{3}x) = \sqrt{2}(\cos\frac{\pi}{4}\cos\frac{2}{3}x + \sin\frac{\pi}{4}\sin\frac{2}{3}x) = \sqrt{2}\cos(\frac{2}{3}x - \frac{\pi}{4})$. Это гармоническая функция с коэффициентом $k=2/3$, ее главный период $T_1 = \frac{2\pi}{|2/3|} = 3\pi$. Функция периодическая.
Рассмотрим функцию $f_2(x) = (\cos\frac{2}{3}x + \sin\frac{2}{3}x)^2$. Раскроем скобки: $f_2(x) = \cos^2\frac{2}{3}x + 2\cos\frac{2}{3}x\sin\frac{2}{3}x + \sin^2\frac{2}{3}x$. Используя основное тригонометрическое тождество $\sin^2\alpha+\cos^2\alpha=1$ и формулу синуса двойного угла $\sin(2\alpha)=2\sin\alpha\cos\alpha$, получаем: $f_2(x) = 1 + \sin(2 \cdot \frac{2}{3}x) = 1 + \sin(\frac{4}{3}x)$. Это гармоническая функция, сдвинутая по оси ординат. Ее период определяется слагаемым $\sin(\frac{4}{3}x)$. Главный период $T_2 = \frac{2\pi}{|4/3|} = \frac{6\pi}{4} = \frac{3\pi}{2}$. Функция периодическая.
Рассмотрим функцию $f_3(x) = x^2|\sin x|$. Функция $|\sin x|$ является периодической с главным периодом $\pi$. Однако множитель $x^2$ не является периодической функцией. Предположим, что функция $f_3(x)$ периодическая с периодом $T>0$. Тогда должно выполняться равенство $f_3(x+T) = f_3(x)$ для всех $x$, то есть $(x+T)^2|\sin(x+T)| = x^2|\sin x|$. Если выбрать $T=k\pi$ для целого $k\neq0$, то $|\sin(x+k\pi)|=|\sin x|$. Тогда равенство примет вид $(x+k\pi)^2|\sin x| = x^2|\sin x|$. Для $x$, где $|\sin x|\neq0$, получаем $(x+k\pi)^2=x^2$, что неверно для всех $x$. Следовательно, функция $f_3(x)$ не является периодической.
Ответ: периодические функции: $f_0(x)$ с главным периодом $T_0=\frac{2\pi}{57}$, $f_1(x)$ с главным периодом $T_1=3\pi$, $f_2(x)$ с главным периодом $T_2=\frac{3\pi}{2}$; функция $f_3(x)$ не является периодической.
б)
Рассмотрим функцию $g_0(x) = \frac{\sqrt{3}}{2}\cos(\sqrt{7}x - \frac{\pi}{7})$. Это гармоническая функция вида $y=A\cos(kx+\phi)$. Ее главный период равен $T=\frac{2\pi}{|k|}$. В данном случае $k=\sqrt{7}$, следовательно, главный период $T_0 = \frac{2\pi}{\sqrt{7}}$. Функция периодическая.
Рассмотрим функцию $g_1(x) = \cos^2\frac{\pi}{5}x - \sin^2\frac{\pi}{5}x$. Применим формулу косинуса двойного угла $\cos(2\alpha) = \cos^2\alpha - \sin^2\alpha$. Полагая $\alpha=\frac{\pi}{5}x$, получим $g_1(x) = \cos(2 \cdot \frac{\pi}{5}x) = \cos(\frac{2\pi}{5}x)$. Это гармоническая функция с коэффициентом $k=\frac{2\pi}{5}$. Ее главный период $T_1 = \frac{2\pi}{|2\pi/5|} = 5$. Функция периодическая.
Рассмотрим функцию $g_2(x) = x^3 - \cos^4x$. Эта функция является разностью непериодической функции $y_1=x^3$ и периодической функции $y_2=\cos^4x$ (период $\cos^4x$ равен $\pi$). Сумма или разность периодической и непериодической функции является непериодической функцией. Предположим, что $g_2(x)$ периодична с периодом $T>0$. Тогда $g_2(x+T) = g_2(x)$. Возьмем $T=k\pi$ для целого $k\neq0$. Тогда $\cos^4(x+k\pi)=\cos^4x$, и равенство примет вид $(x+k\pi)^3-\cos^4x = x^3-\cos^4x$, откуда $(x+k\pi)^3=x^3$, что неверно для всех $x$. Следовательно, функция $g_2(x)$ не является периодической.
Рассмотрим функцию $g_3(x) = \cos^4x$. Так как $\cos(x+\pi)=-\cos x$, то $g_3(x+\pi) = \cos^4(x+\pi) = (-\cos x)^4 = \cos^4x = g_3(x)$. Таким образом, $\pi$ является периодом. Можно показать, что это главный период, используя формулы понижения степени: $\cos^4x = (\frac{1+\cos(2x)}{2})^2 = \frac{1+2\cos(2x)+\cos^2(2x)}{4} = \frac{1+2\cos(2x)+\frac{1+\cos(4x)}{2}}{4} = \frac{3}{8}+\frac{1}{2}\cos(2x)+\frac{1}{8}\cos(4x)$. Периоды слагаемых: $T_1=\frac{2\pi}{2}=\pi$ и $T_2=\frac{2\pi}{4}=\frac{\pi}{2}$. Главный период суммы равен наименьшему общему кратному периодов слагаемых: $T_3=\text{НОК}(\pi, \frac{\pi}{2})=\pi$. Функция периодическая.
Ответ: периодические функции: $g_0(x)$ с главным периодом $T_0=\frac{2\pi}{\sqrt{7}}$, $g_1(x)$ с главным периодом $T_1=5$, $g_3(x)$ с главным периодом $T_3=\pi$; функция $g_2(x)$ не является периодической.
в)
Рассмотрим функцию $h_0(x) = \frac{1}{2-\text{ctg}(3\pi x)}$. Периодичность функции определяется периодичностью функции $\text{ctg}(3\pi x)$. Главный период функции $\text{ctg}(y)$ равен $\pi$. Для функции $\text{ctg}(kx)$ главный период равен $T=\frac{\pi}{|k|}$. Здесь $k=3\pi$, поэтому главный период $T_0 = \frac{\pi}{|3\pi|} = \frac{1}{3}$. Функция периодическая.
Рассмотрим функцию $h_1(x) = \frac{x}{2-\text{ctg}(3\pi x)}$. Эта функция является частным от деления непериодической функции $y_1=x$ на периодическую функцию $y_2=2-\text{ctg}(3\pi x)$ (с периодом $1/3$). Предположим, что $h_1(x)$ периодична с периодом $T>0$. Тогда $h_1(x+T)=h_1(x)$. Возьмем $T=k/3$ для целого $k\neq0$. Тогда $\text{ctg}(3\pi(x+k/3)) = \text{ctg}(3\pi x+k\pi)=\text{ctg}(3\pi x)$, и равенство примет вид $\frac{x+k/3}{2-\text{ctg}(3\pi x)} = \frac{x}{2-\text{ctg}(3\pi x)}$. Отсюда $x+k/3=x$, что неверно, так как $k\neq0$. Следовательно, функция $h_1(x)$ не является периодической.
Рассмотрим функцию $h_2(x) = 1 + \text{ctg}^2x$. Период функции $\text{ctg} x$ равен $\pi$. Так как $\text{ctg}(x+\pi)=\text{ctg} x$, то $h_2(x+\pi) = 1+\text{ctg}^2(x+\pi) = 1+(\text{ctg} x)^2 = h_2(x)$. Период $\pi$ является главным, так как для $T=\pi/2$, $h_2(x+\pi/2) = 1+\text{ctg}^2(x+\pi/2) = 1+(-\text{tg} x)^2 = 1+\text{tg}^2x \neq h_2(x)$ в общем случае. Также можно использовать тождество $1+\text{ctg}^2x = \frac{1}{\sin^2x}$. Главный период функции $\sin^2x$ равен $\pi$, следовательно, главный период функции $h_2(x)$ также равен $\pi$. Функция периодическая.
Ответ: периодические функции: $h_0(x)$ с главным периодом $T_0=\frac{1}{3}$, $h_2(x)$ с главным периодом $T_2=\pi$; функция $h_1(x)$ не является периодической.
Другие задания:
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