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Номер 9, страница 171, 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 3. Тригонометрические уравнения и неравенства. Параграф 3. Простейшие тригонометрические неравенства. 3.3. Неравенства, содержащие tgx и ctgx. Задачи - номер 9, страница 171.
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" => 1119962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "171" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/3-3-3zad-9" "field_display_title" => "9" "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 [ …6] "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 [ …6] "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 [ …6] "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 [ …6] "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 [ …6] "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 [ …6] "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 [ …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 {#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 [ …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 {#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 [ …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 {#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 [ …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 {#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" => 1119382 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические уравнения и неравенства" "field_branch_order" => "3" "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 [ …6] "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 [ …6] "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" => "122" "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 …2} "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 [ …1] "field_branch_covers" => array:1 [ …1] "book" => Illuminate\Database\Eloquent\Collection {#1071 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1100 …2} ] #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 …2} "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 [ …1] "field_branch_covers" => array:1 [ …1] "book" => Illuminate\Database\Eloquent\Collection {#1071 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1100 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119382 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические уравнения и неравенства" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1064} "field_page_start" => "122" "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" => 1119399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Простейшие тригонометрические неравенства" "field_branch_order" => "3" "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 [ …6] "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 [ …6] "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" => "157" "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" => 1119382 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические уравнения и неравенства" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_page_start" => "122" "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 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1109 …2} ] #original: array:24 [ "id" => 1119382 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические уравнения и неравенства" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_page_start" => "122" "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 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1109 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Простейшие тригонометрические неравенства" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1102} "field_page_start" => "157" "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" => 1119404 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.3. Неравенства, содержащие tgx и ctgx" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1111 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "168" "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" => 1119399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Простейшие тригонометрические неравенства" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_page_start" => "157" "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 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117 …2} ] #original: array:24 [ "id" => 1119399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Простейшие тригонометрические неравенства" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_page_start" => "157" "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 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1117 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119404 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.3. Неравенства, содержащие tgx и ctgx" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1111} "field_page_start" => "168" "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" => 1119405 "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" => "168" "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" => 1119404 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.3. Неравенства, содержащие tgx и ctgx" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_page_start" => "168" "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 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125 …2} ] #original: array:24 [ "id" => 1119404 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.3. Неравенства, содержащие tgx и ctgx" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_page_start" => "168" "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 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119405 "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" => "168" "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" => 1300647 "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 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1134 …2} "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 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1119962" "type" => "task" ] "text" => "<p><strong>9. (2) a)</strong> $ctg x > -\\frac{\\sqrt{3}}{3}$;</p><p><strong>б)</strong> $ctg 3x \\geq \\frac{\\sqrt{3}}{3}$;</p><p><strong>В)</strong> $ctg \\frac{x}{8} < -\\frac{1}{\\sqrt{3}}$;</p><p><strong>Г)</strong> $ctg \\left(x+\\frac{8}{\\pi}\\right) \\leq \\frac{1}{\\sqrt{3}}$;</p><p><strong>Д)</strong> $ctg \\left(x-\\frac{8}{5\\pi}\\right) \\leq -\\frac{1}{\\sqrt{3}}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1478" "height" => 354 "path" => "/media/algebra_10/baspasy-u19/0-3-3-3zad/9-1.webp?ts=1753534690" ] ] ] #original: array:7 [ "id" => 1300647 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128} "task" => array:2 [ "refs" => "1119962" "type" => "task" ] "text" => "<p><strong>9. (2) a)</strong> $ctg x > -\\frac{\\sqrt{3}}{3}$;</p><p><strong>б)</strong> $ctg 3x \\geq \\frac{\\sqrt{3}}{3}$;</p><p><strong>В)</strong> $ctg \\frac{x}{8} < -\\frac{1}{\\sqrt{3}}$;</p><p><strong>Г)</strong> $ctg \\left(x+\\frac{8}{\\pi}\\right) \\leq \\frac{1}{\\sqrt{3}}$;</p><p><strong>Д)</strong> $ctg \\left(x-\\frac{8}{5\\pi}\\right) \\leq -\\frac{1}{\\sqrt{3}}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "9-1.jpg" "alt" => null "width" => "1478" "height" => 354 "path" => "/media/algebra_10/baspasy-u19/0-3-3-3zad/9-1.webp?ts=1753534690" ] ] ] #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" => 1551683 "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 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …2} "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 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1119962" "type" => "task" ] "text" => "<p><strong>а)</strong> Решим неравенство $ctg(x) > -\frac{\sqrt{3}}{3}$.</p><p>Функция $y=ctg(x)$ является периодической с периодом $\pi$ и убывающей на каждом интервале своей области определения $(\pi n, \pi(n+1))$, где $n \in \mathbb{Z}$.</p><p>Сначала найдем значение $x$, для которого выполняется равенство $ctg(x) = -\frac{\sqrt{3}}{3}$.</p><p>Основное решение этого уравнения: $x = arcctg(-\frac{\sqrt{3}}{3}) = \pi - arcctg(\frac{\sqrt{3}}{3}) = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$.</p><p>Неравенство $ctg(x) > a$ равносильно двойному неравенству $\pi k < x < arcctg(a) + \pi k$ для любого целого $k$.</p><p>Подставляя наше значение $a = -\frac{\sqrt{3}}{3}$, получаем:</p><p>$\pi n < x < \frac{2\pi}{3} + \pi n$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (\pi n; \frac{2\pi}{3} + \pi n), n \in \mathbb{Z}$.</p><p><strong>б)</strong> Решим неравенство $ctg(3x) \geq \frac{\sqrt{3}}{3}$.</p><p>Введем замену переменной $t = 3x$. Неравенство примет вид $ctg(t) \geq \frac{\sqrt{3}}{3}$.</p><p>Сначала решим уравнение $ctg(t) = \frac{\sqrt{3}}{3}$.</p><p>Основное решение: $t = arcctg(\frac{\sqrt{3}}{3}) = \frac{\pi}{3}$.</p><p>Неравенство $ctg(t) \geq a$ равносильно двойному неравенству $\pi n < t \leq arcctg(a) + \pi n$ для любого целого $n$.</p><p>Подставляя $a = \frac{\sqrt{3}}{3}$, получаем:</p><p>$\pi n < t \leq \frac{\pi}{3} + \pi n$, где $n \in \mathbb{Z}$.</p><p>Теперь выполним обратную замену $t = 3x$:</p><p>$\pi n < 3x \leq \frac{\pi}{3} + \pi n$.</p><p>Разделим все части неравенства на 3:</p><p>$\frac{\pi n}{3} < x \leq \frac{\pi}{9} + \frac{\pi n}{3}$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (\frac{\pi n}{3}; \frac{\pi}{9} + \frac{\pi n}{3}], n \in \mathbb{Z}$.</p><p><strong>в)</strong> Решим неравенство $ctg(\frac{x}{8}) < -\frac{1}{\sqrt{3}}$.</p><p>Введем замену переменной $t = \frac{x}{8}$. Неравенство примет вид $ctg(t) < -\frac{1}{\sqrt{3}}$.</p><p>Сначала решим уравнение $ctg(t) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$.</p><p>Основное решение: $t = arcctg(-\frac{\sqrt{3}}{3}) = \frac{2\pi}{3}$.</p><p>Неравенство $ctg(t) < a$ равносильно двойному неравенству $arcctg(a) + \pi n < t < \pi + \pi n$ для любого целого $n$.</p><p>Подставляя наше значение $a$, получаем:</p><p>$\frac{2\pi}{3} + \pi n < t < \pi + \pi n$, где $n \in \mathbb{Z}$.</p><p>Выполним обратную замену $t = \frac{x}{8}$:</p><p>$\frac{2\pi}{3} + \pi n < \frac{x}{8} < \pi(n+1)$.</p><p>Умножим все части неравенства на 8:</p><p>$\frac{16\pi}{3} + 8\pi n < x < 8\pi(n+1)$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (\frac{16\pi}{3} + 8\pi n; 8\pi + 8\pi n), n \in \mathbb{Z}$.</p><p><strong>г)</strong> Решим неравенство $ctg(x + \frac{8}{\pi}) \leq \frac{1}{\sqrt{3}}$.</p><p>Введем замену переменной $t = x + \frac{8}{\pi}$. Неравенство примет вид $ctg(t) \leq \frac{1}{\sqrt{3}}$.</p><p>Сначала решим уравнение $ctg(t) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$.</p><p>Основное решение: $t = arcctg(\frac{\sqrt{3}}{3}) = \frac{\pi}{3}$.</p><p>Неравенство $ctg(t) \leq a$ равносильно двойному неравенству $arcctg(a) + \pi n \leq t < \pi + \pi n$ для любого целого $n$.</p><p>Подставляя наше значение $a$, получаем:</p><p>$\frac{\pi}{3} + \pi n \leq t < \pi + \pi n$, где $n \in \mathbb{Z}$.</p><p>Выполним обратную замену $t = x + \frac{8}{\pi}$:</p><p>$\frac{\pi}{3} + \pi n \leq x + \frac{8}{\pi} < \pi + \pi n$.</p><p>Вычтем $\frac{8}{\pi}$ из всех частей неравенства:</p><p>$\frac{\pi}{3} - \frac{8}{\pi} + \pi n \leq x < \pi - \frac{8}{\pi} + \pi n$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in [\frac{\pi}{3} - \frac{8}{\pi} + \pi n; \pi - \frac{8}{\pi} + \pi n), n \in \mathbb{Z}$.</p><p><strong>д)</strong> Решим неравенство $ctg(x - \frac{8}{5\pi}) \leq -\frac{1}{\sqrt{3}}$.</p><p>Введем замену переменной $t = x - \frac{8}{5\pi}$. Неравенство примет вид $ctg(t) \leq -\frac{1}{\sqrt{3}}$.</p><p>Сначала решим уравнение $ctg(t) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$.</p><p>Основное решение: $t = arcctg(-\frac{\sqrt{3}}{3}) = \frac{2\pi}{3}$.</p><p>Неравенство $ctg(t) \leq a$ равносильно двойному неравенству $arcctg(a) + \pi n \leq t < \pi + \pi n$ для любого целого $n$.</p><p>Подставляя наше значение $a$, получаем:</p><p>$\frac{2\pi}{3} + \pi n \leq t < \pi + \pi n$, где $n \in \mathbb{Z}$.</p><p>Выполним обратную замену $t = x - \frac{8}{5\pi}$:</p><p>$\frac{2\pi}{3} + \pi n \leq x - \frac{8}{5\pi} < \pi + \pi n$.</p><p>Прибавим $\frac{8}{5\pi}$ ко всем частям неравенства:</p><p>$\frac{2\pi}{3} + \frac{8}{5\pi} + \pi n \leq x < \pi + \frac{8}{5\pi} + \pi n$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in [\frac{2\pi}{3} + \frac{8}{5\pi} + \pi n; \pi + \frac{8}{5\pi} + \pi n), n \in \mathbb{Z}$.</p>" ] #original: array:6 [ "id" => 1551683 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136} "task" => array:2 [ "refs" => "1119962" "type" => "task" ] "text" => "<p><strong>а)</strong> Решим неравенство $ctg(x) > -\frac{\sqrt{3}}{3}$.</p><p>Функция $y=ctg(x)$ является периодической с периодом $\pi$ и убывающей на каждом интервале своей области определения $(\pi n, \pi(n+1))$, где $n \in \mathbb{Z}$.</p><p>Сначала найдем значение $x$, для которого выполняется равенство $ctg(x) = -\frac{\sqrt{3}}{3}$.</p><p>Основное решение этого уравнения: $x = arcctg(-\frac{\sqrt{3}}{3}) = \pi - arcctg(\frac{\sqrt{3}}{3}) = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$.</p><p>Неравенство $ctg(x) > a$ равносильно двойному неравенству $\pi k < x < arcctg(a) + \pi k$ для любого целого $k$.</p><p>Подставляя наше значение $a = -\frac{\sqrt{3}}{3}$, получаем:</p><p>$\pi n < x < \frac{2\pi}{3} + \pi n$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (\pi n; \frac{2\pi}{3} + \pi n), n \in \mathbb{Z}$.</p><p><strong>б)</strong> Решим неравенство $ctg(3x) \geq \frac{\sqrt{3}}{3}$.</p><p>Введем замену переменной $t = 3x$. Неравенство примет вид $ctg(t) \geq \frac{\sqrt{3}}{3}$.</p><p>Сначала решим уравнение $ctg(t) = \frac{\sqrt{3}}{3}$.</p><p>Основное решение: $t = arcctg(\frac{\sqrt{3}}{3}) = \frac{\pi}{3}$.</p><p>Неравенство $ctg(t) \geq a$ равносильно двойному неравенству $\pi n < t \leq arcctg(a) + \pi n$ для любого целого $n$.</p><p>Подставляя $a = \frac{\sqrt{3}}{3}$, получаем:</p><p>$\pi n < t \leq \frac{\pi}{3} + \pi n$, где $n \in \mathbb{Z}$.</p><p>Теперь выполним обратную замену $t = 3x$:</p><p>$\pi n < 3x \leq \frac{\pi}{3} + \pi n$.</p><p>Разделим все части неравенства на 3:</p><p>$\frac{\pi n}{3} < x \leq \frac{\pi}{9} + \frac{\pi n}{3}$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (\frac{\pi n}{3}; \frac{\pi}{9} + \frac{\pi n}{3}], n \in \mathbb{Z}$.</p><p><strong>в)</strong> Решим неравенство $ctg(\frac{x}{8}) < -\frac{1}{\sqrt{3}}$.</p><p>Введем замену переменной $t = \frac{x}{8}$. Неравенство примет вид $ctg(t) < -\frac{1}{\sqrt{3}}$.</p><p>Сначала решим уравнение $ctg(t) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$.</p><p>Основное решение: $t = arcctg(-\frac{\sqrt{3}}{3}) = \frac{2\pi}{3}$.</p><p>Неравенство $ctg(t) < a$ равносильно двойному неравенству $arcctg(a) + \pi n < t < \pi + \pi n$ для любого целого $n$.</p><p>Подставляя наше значение $a$, получаем:</p><p>$\frac{2\pi}{3} + \pi n < t < \pi + \pi n$, где $n \in \mathbb{Z}$.</p><p>Выполним обратную замену $t = \frac{x}{8}$:</p><p>$\frac{2\pi}{3} + \pi n < \frac{x}{8} < \pi(n+1)$.</p><p>Умножим все части неравенства на 8:</p><p>$\frac{16\pi}{3} + 8\pi n < x < 8\pi(n+1)$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (\frac{16\pi}{3} + 8\pi n; 8\pi + 8\pi n), n \in \mathbb{Z}$.</p><p><strong>г)</strong> Решим неравенство $ctg(x + \frac{8}{\pi}) \leq \frac{1}{\sqrt{3}}$.</p><p>Введем замену переменной $t = x + \frac{8}{\pi}$. Неравенство примет вид $ctg(t) \leq \frac{1}{\sqrt{3}}$.</p><p>Сначала решим уравнение $ctg(t) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$.</p><p>Основное решение: $t = arcctg(\frac{\sqrt{3}}{3}) = \frac{\pi}{3}$.</p><p>Неравенство $ctg(t) \leq a$ равносильно двойному неравенству $arcctg(a) + \pi n \leq t < \pi + \pi n$ для любого целого $n$.</p><p>Подставляя наше значение $a$, получаем:</p><p>$\frac{\pi}{3} + \pi n \leq t < \pi + \pi n$, где $n \in \mathbb{Z}$.</p><p>Выполним обратную замену $t = x + \frac{8}{\pi}$:</p><p>$\frac{\pi}{3} + \pi n \leq x + \frac{8}{\pi} < \pi + \pi n$.</p><p>Вычтем $\frac{8}{\pi}$ из всех частей неравенства:</p><p>$\frac{\pi}{3} - \frac{8}{\pi} + \pi n \leq x < \pi - \frac{8}{\pi} + \pi n$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in [\frac{\pi}{3} - \frac{8}{\pi} + \pi n; \pi - \frac{8}{\pi} + \pi n), n \in \mathbb{Z}$.</p><p><strong>д)</strong> Решим неравенство $ctg(x - \frac{8}{5\pi}) \leq -\frac{1}{\sqrt{3}}$.</p><p>Введем замену переменной $t = x - \frac{8}{5\pi}$. Неравенство примет вид $ctg(t) \leq -\frac{1}{\sqrt{3}}$.</p><p>Сначала решим уравнение $ctg(t) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$.</p><p>Основное решение: $t = arcctg(-\frac{\sqrt{3}}{3}) = \frac{2\pi}{3}$.</p><p>Неравенство $ctg(t) \leq a$ равносильно двойному неравенству $arcctg(a) + \pi n \leq t < \pi + \pi n$ для любого целого $n$.</p><p>Подставляя наше значение $a$, получаем:</p><p>$\frac{2\pi}{3} + \pi n \leq t < \pi + \pi n$, где $n \in \mathbb{Z}$.</p><p>Выполним обратную замену $t = x - \frac{8}{5\pi}$:</p><p>$\frac{2\pi}{3} + \pi n \leq x - \frac{8}{5\pi} < \pi + \pi n$.</p><p>Прибавим $\frac{8}{5\pi}$ ко всем частям неравенства:</p><p>$\frac{2\pi}{3} + \frac{8}{5\pi} + \pi n \leq x < \pi + \frac{8}{5\pi} + \pi n$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in [\frac{2\pi}{3} + \frac{8}{5\pi} + \pi n; \pi + \frac{8}{5\pi} + \pi n), n \in \mathbb{Z}$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null 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№9 (с. 171)
Условие. №9 (с. 171)
скриншот условия
9. (2) a) $ctg x > -\\frac{\\sqrt{3}}{3}$;
б) $ctg 3x \\geq \\frac{\\sqrt{3}}{3}$;
В) $ctg \\frac{x}{8} < -\\frac{1}{\\sqrt{3}}$;
Г) $ctg \\left(x+\\frac{8}{\\pi}\\right) \\leq \\frac{1}{\\sqrt{3}}$;
Д) $ctg \\left(x-\\frac{8}{5\\pi}\\right) \\leq -\\frac{1}{\\sqrt{3}}$.
Решение 2 (rus). №9 (с. 171)
а) Решим неравенство $ctg(x) > -\frac{\sqrt{3}}{3}$.
Функция $y=ctg(x)$ является периодической с периодом $\pi$ и убывающей на каждом интервале своей области определения $(\pi n, \pi(n+1))$, где $n \in \mathbb{Z}$.
Сначала найдем значение $x$, для которого выполняется равенство $ctg(x) = -\frac{\sqrt{3}}{3}$.
Основное решение этого уравнения: $x = arcctg(-\frac{\sqrt{3}}{3}) = \pi - arcctg(\frac{\sqrt{3}}{3}) = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$.
Неравенство $ctg(x) > a$ равносильно двойному неравенству $\pi k < x < arcctg(a) + \pi k$ для любого целого $k$.
Подставляя наше значение $a = -\frac{\sqrt{3}}{3}$, получаем:
$\pi n < x < \frac{2\pi}{3} + \pi n$, где $n \in \mathbb{Z}$.
Ответ: $x \in (\pi n; \frac{2\pi}{3} + \pi n), n \in \mathbb{Z}$.
б) Решим неравенство $ctg(3x) \geq \frac{\sqrt{3}}{3}$.
Введем замену переменной $t = 3x$. Неравенство примет вид $ctg(t) \geq \frac{\sqrt{3}}{3}$.
Сначала решим уравнение $ctg(t) = \frac{\sqrt{3}}{3}$.
Основное решение: $t = arcctg(\frac{\sqrt{3}}{3}) = \frac{\pi}{3}$.
Неравенство $ctg(t) \geq a$ равносильно двойному неравенству $\pi n < t \leq arcctg(a) + \pi n$ для любого целого $n$.
Подставляя $a = \frac{\sqrt{3}}{3}$, получаем:
$\pi n < t \leq \frac{\pi}{3} + \pi n$, где $n \in \mathbb{Z}$.
Теперь выполним обратную замену $t = 3x$:
$\pi n < 3x \leq \frac{\pi}{3} + \pi n$.
Разделим все части неравенства на 3:
$\frac{\pi n}{3} < x \leq \frac{\pi}{9} + \frac{\pi n}{3}$, где $n \in \mathbb{Z}$.
Ответ: $x \in (\frac{\pi n}{3}; \frac{\pi}{9} + \frac{\pi n}{3}], n \in \mathbb{Z}$.
в) Решим неравенство $ctg(\frac{x}{8}) < -\frac{1}{\sqrt{3}}$.
Введем замену переменной $t = \frac{x}{8}$. Неравенство примет вид $ctg(t) < -\frac{1}{\sqrt{3}}$.
Сначала решим уравнение $ctg(t) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$.
Основное решение: $t = arcctg(-\frac{\sqrt{3}}{3}) = \frac{2\pi}{3}$.
Неравенство $ctg(t) < a$ равносильно двойному неравенству $arcctg(a) + \pi n < t < \pi + \pi n$ для любого целого $n$.
Подставляя наше значение $a$, получаем:
$\frac{2\pi}{3} + \pi n < t < \pi + \pi n$, где $n \in \mathbb{Z}$.
Выполним обратную замену $t = \frac{x}{8}$:
$\frac{2\pi}{3} + \pi n < \frac{x}{8} < \pi(n+1)$.
Умножим все части неравенства на 8:
$\frac{16\pi}{3} + 8\pi n < x < 8\pi(n+1)$, где $n \in \mathbb{Z}$.
Ответ: $x \in (\frac{16\pi}{3} + 8\pi n; 8\pi + 8\pi n), n \in \mathbb{Z}$.
г) Решим неравенство $ctg(x + \frac{8}{\pi}) \leq \frac{1}{\sqrt{3}}$.
Введем замену переменной $t = x + \frac{8}{\pi}$. Неравенство примет вид $ctg(t) \leq \frac{1}{\sqrt{3}}$.
Сначала решим уравнение $ctg(t) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$.
Основное решение: $t = arcctg(\frac{\sqrt{3}}{3}) = \frac{\pi}{3}$.
Неравенство $ctg(t) \leq a$ равносильно двойному неравенству $arcctg(a) + \pi n \leq t < \pi + \pi n$ для любого целого $n$.
Подставляя наше значение $a$, получаем:
$\frac{\pi}{3} + \pi n \leq t < \pi + \pi n$, где $n \in \mathbb{Z}$.
Выполним обратную замену $t = x + \frac{8}{\pi}$:
$\frac{\pi}{3} + \pi n \leq x + \frac{8}{\pi} < \pi + \pi n$.
Вычтем $\frac{8}{\pi}$ из всех частей неравенства:
$\frac{\pi}{3} - \frac{8}{\pi} + \pi n \leq x < \pi - \frac{8}{\pi} + \pi n$, где $n \in \mathbb{Z}$.
Ответ: $x \in [\frac{\pi}{3} - \frac{8}{\pi} + \pi n; \pi - \frac{8}{\pi} + \pi n), n \in \mathbb{Z}$.
д) Решим неравенство $ctg(x - \frac{8}{5\pi}) \leq -\frac{1}{\sqrt{3}}$.
Введем замену переменной $t = x - \frac{8}{5\pi}$. Неравенство примет вид $ctg(t) \leq -\frac{1}{\sqrt{3}}$.
Сначала решим уравнение $ctg(t) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$.
Основное решение: $t = arcctg(-\frac{\sqrt{3}}{3}) = \frac{2\pi}{3}$.
Неравенство $ctg(t) \leq a$ равносильно двойному неравенству $arcctg(a) + \pi n \leq t < \pi + \pi n$ для любого целого $n$.
Подставляя наше значение $a$, получаем:
$\frac{2\pi}{3} + \pi n \leq t < \pi + \pi n$, где $n \in \mathbb{Z}$.
Выполним обратную замену $t = x - \frac{8}{5\pi}$:
$\frac{2\pi}{3} + \pi n \leq x - \frac{8}{5\pi} < \pi + \pi n$.
Прибавим $\frac{8}{5\pi}$ ко всем частям неравенства:
$\frac{2\pi}{3} + \frac{8}{5\pi} + \pi n \leq x < \pi + \frac{8}{5\pi} + \pi n$, где $n \in \mathbb{Z}$.
Ответ: $x \in [\frac{2\pi}{3} + \frac{8}{5\pi} + \pi n; \pi + \frac{8}{5\pi} + \pi n), n \in \mathbb{Z}$.
Другие задания:
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