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Номер 6, страница 162, 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. 3.1. Неравенства, содержащие sinx. Задачи - номер 6, страница 162.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119940 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "162" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/3-3-1zad-6" "field_display_title" => "6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ "field_accusative_case" => "Часть" "field_creative_case" => "Частью" "field_dative_case" => "Части" "field_genitive_case" => "Части" "field_nominative_case" => "Часть" "field_prepositional_case" => "Части" ] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1057 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042} "field_class" => Illuminate\Database\Eloquent\Collection {#1044} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046} "field_author" => Illuminate\Database\Eloquent\Collection {#1048} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053} "field_country" => Illuminate\Database\Eloquent\Collection {#1055} "field_city" => Illuminate\Database\Eloquent\Collection {#1057} "field_series" => Illuminate\Database\Eloquent\Collection {#1059} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "6" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_branch_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "book" => Illuminate\Database\Eloquent\Collection {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1270 #items: array:5 [ 0 => App\Models\Branch {#1282 #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 {#1286 #items: array:1 [ 0 => App\Models\Term {#1283 #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 {#1315 #items: array:1 [ 0 => App\Models\Book {#1285 #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 {#1287 #items: array:1 [ 0 => App\Models\Term {#1284 #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 {#1288 #items: array:1 [ 0 => App\Models\Term {#1289 #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 {#1290 #items: array:1 [ 0 => App\Models\Term {#1291 #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 {#1292 #items: array:3 [ 0 => App\Models\Term {#1293 #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 {#1294 #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 {#1295 #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 {#1296 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1297 #items: array:1 [ 0 => App\Models\Term {#1298 #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 {#1299 #items: array:1 [ 0 => App\Models\Term {#1300 #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 {#1301 #items: array:1 [ 0 => App\Models\Term {#1302 #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 {#1303 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1304 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1305 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1306 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1307 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1308 #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 {#1309 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1310 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1311 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1312 #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 {#1287} "field_class" => Illuminate\Database\Eloquent\Collection {#1288} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1290} "field_author" => Illuminate\Database\Eloquent\Collection {#1292} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1296} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1297} "field_country" => Illuminate\Database\Eloquent\Collection {#1299} "field_city" => Illuminate\Database\Eloquent\Collection {#1301} "field_series" => Illuminate\Database\Eloquent\Collection {#1303} "field_umk" => Illuminate\Database\Eloquent\Collection {#1304} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1305} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1306} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1307} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1308} "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 {#1309} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1310} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1311} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1312} "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 {#1313 #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 {#1286} "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 {#1315} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1313} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1314 #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 {#1319 #items: array:1 [ 0 => App\Models\Term {#1316 #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" => "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 {#1318 #items: array:1 [ 0 => App\Models\Book {#1285} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1317 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #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 {#1319} "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 {#1318} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1317} ] #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 {#1320 #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 {#1324 #items: array:1 [ 0 => App\Models\Term {#1321 #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" => "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 {#1323 #items: array:1 [ 0 => App\Models\Book {#1285} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1392 #items: array:1 [ 0 => App\Models\Branch {#1322 #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 {#1326 #items: array:1 [ …1] #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 {#1327 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1356 #items: array:1 [ …1] #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 {#1326} "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 {#1327} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1356} ] #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 {#1324} "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 {#1323} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1392} ] #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 {#1390 #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" => 1119400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.1. Неравенства, содержащие sinx" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1391 #items: [] #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 {#1393 #items: array:1 [ 0 => App\Models\Book {#1285} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1495 #items: array:1 [ 0 => App\Models\Branch {#1394 #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 {#1396 #items: array:1 [ …1] #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 {#1397 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1426 #items: array:1 [ …1] #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 {#1396} "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 {#1397} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1426} ] #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" => 1119400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.1. Неравенства, содержащие sinx" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1391} "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 {#1393} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1495} ] #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 {#1493 #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" => 1119401 "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 {#1494 #items: [] #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 {#1496 #items: array:1 [ 0 => App\Models\Book {#1285} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1505 #items: array:1 [ 0 => App\Models\Branch {#1504 #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" => 1119400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.1. Неравенства, содержащие sinx" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1506 #items: [] #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 {#1507 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1508 #items: array:1 [ …1] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.1. Неравенства, содержащие sinx" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1506} "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 {#1507} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1508} ] #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" => 1119401 "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 {#1494} "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 {#1496} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1505} ] #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 {#1170 #items: array:2 [ 0 => App\Models\Element {#1266 #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" => 1300625 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1256 #items: array:1 [ 0 => App\Models\Edition {#1265 #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 {#1263 #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 {#1262 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1261 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1260 #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 {#1263} "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 {#1262} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1261} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1260} "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" => "1119940" "type" => "task" ] "text" => "<p><strong>6. (2) а)</strong> $sin x > -\frac{\sqrt{2}}{2}$;</p><p><strong>б)</strong> $sin \frac{x}{4} \geq \frac{\sqrt{2}}{2}$;</p><p><strong>В)</strong> $sin \frac{3x}{5} < -\frac{\sqrt{2}}{2}$;</p><p><strong>Г)</strong> $sin \left(x + \frac{\pi}{4}\right) \leq \frac{\sqrt{2}}{2}$;</p><p><strong>Д)</strong> $sin \left(\frac{6x}{7} + \frac{6\pi}{7}\right) \leq -\frac{\sqrt{2}}{2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "1484" "height" => 372 "path" => "/media/algebra_10/baspasy-u19/0-3-3-1zad/6-1.webp?ts=1753534676" ] ] ] #original: array:7 [ "id" => 1300625 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1256} "task" => array:2 [ "refs" => "1119940" "type" => "task" ] "text" => "<p><strong>6. (2) а)</strong> $sin x > -\frac{\sqrt{2}}{2}$;</p><p><strong>б)</strong> $sin \frac{x}{4} \geq \frac{\sqrt{2}}{2}$;</p><p><strong>В)</strong> $sin \frac{3x}{5} < -\frac{\sqrt{2}}{2}$;</p><p><strong>Г)</strong> $sin \left(x + \frac{\pi}{4}\right) \leq \frac{\sqrt{2}}{2}$;</p><p><strong>Д)</strong> $sin \left(\frac{6x}{7} + \frac{6\pi}{7}\right) \leq -\frac{\sqrt{2}}{2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "6-1.jpg" "alt" => null "width" => "1484" "height" => 372 "path" => "/media/algebra_10/baspasy-u19/0-3-3-1zad/6-1.webp?ts=1753534676" ] ] ] #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 {#1259 #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" => 1551661 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1250 #items: array:1 [ 0 => App\Models\Edition {#1258 #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 {#1254 #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 {#1255 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1252 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1253 #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 {#1254} "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 {#1255} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1252} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1253} "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" => "1119940" "type" => "task" ] "text" => "<p><strong>а)</strong> $ \sin x > -\frac{\sqrt{2}}{2} $</p><p>Для решения данного тригонометрического неравенства сначала найдем значения $x$, для которых $ \sin x = -\frac{\sqrt{2}}{2} $. На единичной окружности это точки с ординатой (y-координатой), равной $ -\frac{\sqrt{2}}{2} $. Этим точкам соответствуют углы $ x = -\frac{\pi}{4} $ и $ x = -\frac{3\pi}{4} $ (или $ x = \frac{5\pi}{4} $).</p><p>Нам нужны значения $x$, при которых $ \sin x $ строго больше $ -\frac{\sqrt{2}}{2} $. На единичной окружности это соответствует дуге, расположенной выше прямой $ y = -\frac{\sqrt{2}}{2} $. Двигаясь против часовой стрелки, эта дуга начинается в точке, соответствующей углу $ -\frac{\pi}{4} $, и заканчивается в точке, соответствующей углу $ \frac{5\pi}{4} $.</p><p>Таким образом, основной интервал решений: $ -\frac{\pi}{4} < x < \frac{5\pi}{4} $.</p><p>Так как функция синуса периодична с периодом $2\pi$, общее решение неравенства получается добавлением $2\pi k$ к границам найденного интервала, где $k$ — любое целое число.</p><p>$ -\frac{\pi}{4} + 2\pi k < x < \frac{5\pi}{4} + 2\pi k, k \in \mathbb{Z} $.</p><p><strong>Ответ:</strong> $ x \in (-\frac{\pi}{4} + 2\pi k; \frac{5\pi}{4} + 2\pi k), k \in \mathbb{Z} $.</p><p><strong>б)</strong> $ \sin \frac{x}{4} \ge \frac{\sqrt{2}}{2} $</p><p>Введем замену переменной: пусть $ t = \frac{x}{4} $. Неравенство примет вид: $ \sin t \ge \frac{\sqrt{2}}{2} $.</p><p>Решим это неравенство для $t$. На единичной окружности находим точки, где $ \sin t = \frac{\sqrt{2}}{2} $. Это углы $ t = \frac{\pi}{4} $ и $ t = \pi - \frac{\pi}{4} = \frac{3\pi}{4} $. Нам нужны значения $t$, при которых ордината на окружности не меньше $ \frac{\sqrt{2}}{2} $.</p><p>Это соответствует дуге от $ \frac{\pi}{4} $ до $ \frac{3\pi}{4} $. С учетом периодичности, решение для $t$ имеет вид:</p><p>$ \frac{\pi}{4} + 2\pi k \le t \le \frac{3\pi}{4} + 2\pi k, k \in \mathbb{Z} $.</p><p>Теперь выполним обратную замену, подставив $t = \frac{x}{4}$:</p><p>$ \frac{\pi}{4} + 2\pi k \le \frac{x}{4} \le \frac{3\pi}{4} + 2\pi k $.</p><p>Для того чтобы найти $x$, умножим все части двойного неравенства на 4:</p><p>$ 4(\frac{\pi}{4} + 2\pi k) \le x \le 4(\frac{3\pi}{4} + 2\pi k) $.</p><p>$ \pi + 8\pi k \le x \le 3\pi + 8\pi k $.</p><p><strong>Ответ:</strong> $ x \in [\pi + 8\pi k; 3\pi + 8\pi k], k \in \mathbb{Z} $.</p><p><strong>в)</strong> $ \sin \frac{3x}{5} < -\frac{\sqrt{2}}{2} $</p><p>Введем замену переменной: $ t = \frac{3x}{5} $. Неравенство примет вид: $ \sin t < -\frac{\sqrt{2}}{2} $.</p><p>На единичной окружности находим точки, где $ \sin t = -\frac{\sqrt{2}}{2} $. Это углы $ t = -\frac{\pi}{4} $ и $ t = -\frac{3\pi}{4} $. Нам нужны значения $t$, при которых ордината точки на окружности меньше $ -\frac{\sqrt{2}}{2} $.</p><p>Это соответствует дуге между углами $ -\frac{3\pi}{4} $ и $ -\frac{\pi}{4} $. С учетом периода $2\pi$ решение для $t$ будет:</p><p>$ -\frac{3\pi}{4} + 2\pi k < t < -\frac{\pi}{4} + 2\pi k, k \in \mathbb{Z} $.</p><p>Выполним обратную замену $t = \frac{3x}{5}$:</p><p>$ -\frac{3\pi}{4} + 2\pi k < \frac{3x}{5} < -\frac{\pi}{4} + 2\pi k $.</p><p>Чтобы найти $x$, умножим все части неравенства на $ \frac{5}{3} $:</p><p>$ \frac{5}{3}(-\frac{3\pi}{4} + 2\pi k) < x < \frac{5}{3}(-\frac{\pi}{4} + 2\pi k) $.</p><p>$ -\frac{5\pi}{4} + \frac{10\pi k}{3} < x < -\frac{5\pi}{12} + \frac{10\pi k}{3} $.</p><p><strong>Ответ:</strong> $ x \in (-\frac{5\pi}{4} + \frac{10\pi k}{3}; -\frac{5\pi}{12} + \frac{10\pi k}{3}), k \in \mathbb{Z} $.</p><p><strong>г)</strong> $ \sin(x + \frac{\pi}{4}) \le \frac{\sqrt{2}}{2} $</p><p>Выполним замену: $ t = x + \frac{\pi}{4} $. Неравенство станет $ \sin t \le \frac{\sqrt{2}}{2} $.</p><p>На единичной окружности значениям $ \sin t = \frac{\sqrt{2}}{2} $ соответствуют углы $ t = \frac{\pi}{4} $ и $ t = \frac{3\pi}{4} $. Нам нужна дуга, где ордината точек не превышает $ \frac{\sqrt{2}}{2} $. Эта дуга начинается в точке $ \frac{3\pi}{4} $ и, двигаясь против часовой стрелки, заканчивается в точке $ 2\pi + \frac{\pi}{4} = \frac{9\pi}{4} $.</p><p>Следовательно, решение для $t$ с учетом периодичности:</p><p>$ \frac{3\pi}{4} + 2\pi k \le t \le \frac{9\pi}{4} + 2\pi k, k \in \mathbb{Z} $.</p><p>Произведем обратную замену $ t = x + \frac{\pi}{4} $:</p><p>$ \frac{3\pi}{4} + 2\pi k \le x + \frac{\pi}{4} \le \frac{9\pi}{4} + 2\pi k $.</p><p>Вычтем $ \frac{\pi}{4} $ из всех частей неравенства:</p><p>$ \frac{3\pi}{4} - \frac{\pi}{4} + 2\pi k \le x \le \frac{9\pi}{4} - \frac{\pi}{4} + 2\pi k $.</p><p>$ \frac{2\pi}{4} + 2\pi k \le x \le \frac{8\pi}{4} + 2\pi k $.</p><p>$ \frac{\pi}{2} + 2\pi k \le x \le 2\pi + 2\pi k $.</p><p><strong>Ответ:</strong> $ x \in [\frac{\pi}{2} + 2\pi k; 2\pi + 2\pi k], k \in \mathbb{Z} $.</p><p><strong>д)</strong> $ \sin(\frac{6x}{7} + \frac{6\pi}{7}) \le -\frac{\sqrt{2}}{2} $</p><p>Введем замену: $ t = \frac{6x}{7} + \frac{6\pi}{7} $. Неравенство примет вид $ \sin t \le -\frac{\sqrt{2}}{2} $.</p><p>Это неравенство выполняется для тех $t$, ордината которых на единичной окружности не больше $ -\frac{\sqrt{2}}{2} $. Граничные точки $ \sin t = -\frac{\sqrt{2}}{2} $ соответствуют углам $ t = -\frac{3\pi}{4} $ и $ t = -\frac{\pi}{4} $.</p><p>Решение для $t$ с учетом периодичности:</p><p>$ -\frac{3\pi}{4} + 2\pi k \le t \le -\frac{\pi}{4} + 2\pi k, k \in \mathbb{Z} $.</p><p>Выполним обратную замену $ t = \frac{6}{7}(x+\pi) $:</p><p>$ -\frac{3\pi}{4} + 2\pi k \le \frac{6}{7}(x+\pi) \le -\frac{\pi}{4} + 2\pi k $.</p><p>Умножим все части неравенства на $ \frac{7}{6} $:</p><p>$ \frac{7}{6}(-\frac{3\pi}{4}) + \frac{7}{6}(2\pi k) \le x+\pi \le \frac{7}{6}(-\frac{\pi}{4}) + \frac{7}{6}(2\pi k) $.</p><p>$ -\frac{7\pi}{8} + \frac{7\pi k}{3} \le x+\pi \le -\frac{7\pi}{24} + \frac{7\pi k}{3} $.</p><p>Вычтем $ \pi $ из всех частей неравенства:</p><p>$ -\frac{7\pi}{8} - \pi + \frac{7\pi k}{3} \le x \le -\frac{7\pi}{24} - \pi + \frac{7\pi k}{3} $.</p><p>$ -\frac{7\pi+8\pi}{8} + \frac{7\pi k}{3} \le x \le -\frac{7\pi+24\pi}{24} + \frac{7\pi k}{3} $.</p><p>$ -\frac{15\pi}{8} + \frac{7\pi k}{3} \le x \le -\frac{31\pi}{24} + \frac{7\pi k}{3} $.</p><p><strong>Ответ:</strong> $ x \in [-\frac{15\pi}{8} + \frac{7\pi k}{3}; -\frac{31\pi}{24} + \frac{7\pi k}{3}], k \in \mathbb{Z} $.</p>" ] #original: array:6 [ "id" => 1551661 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1250} "task" => array:2 [ "refs" => "1119940" "type" => "task" ] "text" => "<p><strong>а)</strong> $ \sin x > -\frac{\sqrt{2}}{2} $</p><p>Для решения данного тригонометрического неравенства сначала найдем значения $x$, для которых $ \sin x = -\frac{\sqrt{2}}{2} $. На единичной окружности это точки с ординатой (y-координатой), равной $ -\frac{\sqrt{2}}{2} $. Этим точкам соответствуют углы $ x = -\frac{\pi}{4} $ и $ x = -\frac{3\pi}{4} $ (или $ x = \frac{5\pi}{4} $).</p><p>Нам нужны значения $x$, при которых $ \sin x $ строго больше $ -\frac{\sqrt{2}}{2} $. На единичной окружности это соответствует дуге, расположенной выше прямой $ y = -\frac{\sqrt{2}}{2} $. Двигаясь против часовой стрелки, эта дуга начинается в точке, соответствующей углу $ -\frac{\pi}{4} $, и заканчивается в точке, соответствующей углу $ \frac{5\pi}{4} $.</p><p>Таким образом, основной интервал решений: $ -\frac{\pi}{4} < x < \frac{5\pi}{4} $.</p><p>Так как функция синуса периодична с периодом $2\pi$, общее решение неравенства получается добавлением $2\pi k$ к границам найденного интервала, где $k$ — любое целое число.</p><p>$ -\frac{\pi}{4} + 2\pi k < x < \frac{5\pi}{4} + 2\pi k, k \in \mathbb{Z} $.</p><p><strong>Ответ:</strong> $ x \in (-\frac{\pi}{4} + 2\pi k; \frac{5\pi}{4} + 2\pi k), k \in \mathbb{Z} $.</p><p><strong>б)</strong> $ \sin \frac{x}{4} \ge \frac{\sqrt{2}}{2} $</p><p>Введем замену переменной: пусть $ t = \frac{x}{4} $. Неравенство примет вид: $ \sin t \ge \frac{\sqrt{2}}{2} $.</p><p>Решим это неравенство для $t$. На единичной окружности находим точки, где $ \sin t = \frac{\sqrt{2}}{2} $. Это углы $ t = \frac{\pi}{4} $ и $ t = \pi - \frac{\pi}{4} = \frac{3\pi}{4} $. Нам нужны значения $t$, при которых ордината на окружности не меньше $ \frac{\sqrt{2}}{2} $.</p><p>Это соответствует дуге от $ \frac{\pi}{4} $ до $ \frac{3\pi}{4} $. С учетом периодичности, решение для $t$ имеет вид:</p><p>$ \frac{\pi}{4} + 2\pi k \le t \le \frac{3\pi}{4} + 2\pi k, k \in \mathbb{Z} $.</p><p>Теперь выполним обратную замену, подставив $t = \frac{x}{4}$:</p><p>$ \frac{\pi}{4} + 2\pi k \le \frac{x}{4} \le \frac{3\pi}{4} + 2\pi k $.</p><p>Для того чтобы найти $x$, умножим все части двойного неравенства на 4:</p><p>$ 4(\frac{\pi}{4} + 2\pi k) \le x \le 4(\frac{3\pi}{4} + 2\pi k) $.</p><p>$ \pi + 8\pi k \le x \le 3\pi + 8\pi k $.</p><p><strong>Ответ:</strong> $ x \in [\pi + 8\pi k; 3\pi + 8\pi k], k \in \mathbb{Z} $.</p><p><strong>в)</strong> $ \sin \frac{3x}{5} < -\frac{\sqrt{2}}{2} $</p><p>Введем замену переменной: $ t = \frac{3x}{5} $. Неравенство примет вид: $ \sin t < -\frac{\sqrt{2}}{2} $.</p><p>На единичной окружности находим точки, где $ \sin t = -\frac{\sqrt{2}}{2} $. Это углы $ t = -\frac{\pi}{4} $ и $ t = -\frac{3\pi}{4} $. Нам нужны значения $t$, при которых ордината точки на окружности меньше $ -\frac{\sqrt{2}}{2} $.</p><p>Это соответствует дуге между углами $ -\frac{3\pi}{4} $ и $ -\frac{\pi}{4} $. С учетом периода $2\pi$ решение для $t$ будет:</p><p>$ -\frac{3\pi}{4} + 2\pi k < t < -\frac{\pi}{4} + 2\pi k, k \in \mathbb{Z} $.</p><p>Выполним обратную замену $t = \frac{3x}{5}$:</p><p>$ -\frac{3\pi}{4} + 2\pi k < \frac{3x}{5} < -\frac{\pi}{4} + 2\pi k $.</p><p>Чтобы найти $x$, умножим все части неравенства на $ \frac{5}{3} $:</p><p>$ \frac{5}{3}(-\frac{3\pi}{4} + 2\pi k) < x < \frac{5}{3}(-\frac{\pi}{4} + 2\pi k) $.</p><p>$ -\frac{5\pi}{4} + \frac{10\pi k}{3} < x < -\frac{5\pi}{12} + \frac{10\pi k}{3} $.</p><p><strong>Ответ:</strong> $ x \in (-\frac{5\pi}{4} + \frac{10\pi k}{3}; -\frac{5\pi}{12} + \frac{10\pi k}{3}), k \in \mathbb{Z} $.</p><p><strong>г)</strong> $ \sin(x + \frac{\pi}{4}) \le \frac{\sqrt{2}}{2} $</p><p>Выполним замену: $ t = x + \frac{\pi}{4} $. Неравенство станет $ \sin t \le \frac{\sqrt{2}}{2} $.</p><p>На единичной окружности значениям $ \sin t = \frac{\sqrt{2}}{2} $ соответствуют углы $ t = \frac{\pi}{4} $ и $ t = \frac{3\pi}{4} $. Нам нужна дуга, где ордината точек не превышает $ \frac{\sqrt{2}}{2} $. Эта дуга начинается в точке $ \frac{3\pi}{4} $ и, двигаясь против часовой стрелки, заканчивается в точке $ 2\pi + \frac{\pi}{4} = \frac{9\pi}{4} $.</p><p>Следовательно, решение для $t$ с учетом периодичности:</p><p>$ \frac{3\pi}{4} + 2\pi k \le t \le \frac{9\pi}{4} + 2\pi k, k \in \mathbb{Z} $.</p><p>Произведем обратную замену $ t = x + \frac{\pi}{4} $:</p><p>$ \frac{3\pi}{4} + 2\pi k \le x + \frac{\pi}{4} \le \frac{9\pi}{4} + 2\pi k $.</p><p>Вычтем $ \frac{\pi}{4} $ из всех частей неравенства:</p><p>$ \frac{3\pi}{4} - \frac{\pi}{4} + 2\pi k \le x \le \frac{9\pi}{4} - \frac{\pi}{4} + 2\pi k $.</p><p>$ \frac{2\pi}{4} + 2\pi k \le x \le \frac{8\pi}{4} + 2\pi k $.</p><p>$ \frac{\pi}{2} + 2\pi k \le x \le 2\pi + 2\pi k $.</p><p><strong>Ответ:</strong> $ x \in [\frac{\pi}{2} + 2\pi k; 2\pi + 2\pi k], k \in \mathbb{Z} $.</p><p><strong>д)</strong> $ \sin(\frac{6x}{7} + \frac{6\pi}{7}) \le -\frac{\sqrt{2}}{2} $</p><p>Введем замену: $ t = \frac{6x}{7} + \frac{6\pi}{7} $. Неравенство примет вид $ \sin t \le -\frac{\sqrt{2}}{2} $.</p><p>Это неравенство выполняется для тех $t$, ордината которых на единичной окружности не больше $ -\frac{\sqrt{2}}{2} $. Граничные точки $ \sin t = -\frac{\sqrt{2}}{2} $ соответствуют углам $ t = -\frac{3\pi}{4} $ и $ t = -\frac{\pi}{4} $.</p><p>Решение для $t$ с учетом периодичности:</p><p>$ -\frac{3\pi}{4} + 2\pi k \le t \le -\frac{\pi}{4} + 2\pi k, k \in \mathbb{Z} $.</p><p>Выполним обратную замену $ t = \frac{6}{7}(x+\pi) $:</p><p>$ -\frac{3\pi}{4} + 2\pi k \le \frac{6}{7}(x+\pi) \le -\frac{\pi}{4} + 2\pi k $.</p><p>Умножим все части неравенства на $ \frac{7}{6} $:</p><p>$ \frac{7}{6}(-\frac{3\pi}{4}) + \frac{7}{6}(2\pi k) \le x+\pi \le \frac{7}{6}(-\frac{\pi}{4}) + \frac{7}{6}(2\pi k) $.</p><p>$ -\frac{7\pi}{8} + \frac{7\pi k}{3} \le x+\pi \le -\frac{7\pi}{24} + \frac{7\pi k}{3} $.</p><p>Вычтем $ \pi $ из всех частей неравенства:</p><p>$ -\frac{7\pi}{8} - \pi + \frac{7\pi k}{3} \le x \le -\frac{7\pi}{24} - \pi + \frac{7\pi k}{3} $.</p><p>$ -\frac{7\pi+8\pi}{8} + \frac{7\pi k}{3} \le x \le -\frac{7\pi+24\pi}{24} + \frac{7\pi k}{3} $.</p><p>$ -\frac{15\pi}{8} + \frac{7\pi k}{3} \le x \le -\frac{31\pi}{24} + \frac{7\pi k}{3} $.</p><p><strong>Ответ:</strong> $ x \in [-\frac{15\pi}{8} + \frac{7\pi k}{3}; -\frac{31\pi}{24} + \frac{7\pi k}{3}], k \in \mathbb{Z} $.</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" => "1119941" "type" => "task" ] "previous" => array:2 [ "refs" => "1119939" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1239 #items: array:1 [ 0 => App\Models\Book {#1285} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:1 [ 0 => App\Models\BookPage {#1268 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] 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[] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№6 (с. 162)
Условие. №6 (с. 162)
скриншот условия
6. (2) а) $sin x > -\frac{\sqrt{2}}{2}$;
б) $sin \frac{x}{4} \geq \frac{\sqrt{2}}{2}$;
В) $sin \frac{3x}{5} < -\frac{\sqrt{2}}{2}$;
Г) $sin \left(x + \frac{\pi}{4}\right) \leq \frac{\sqrt{2}}{2}$;
Д) $sin \left(\frac{6x}{7} + \frac{6\pi}{7}\right) \leq -\frac{\sqrt{2}}{2}$.
Решение 2 (rus). №6 (с. 162)
а) $ \sin x > -\frac{\sqrt{2}}{2} $
Для решения данного тригонометрического неравенства сначала найдем значения $x$, для которых $ \sin x = -\frac{\sqrt{2}}{2} $. На единичной окружности это точки с ординатой (y-координатой), равной $ -\frac{\sqrt{2}}{2} $. Этим точкам соответствуют углы $ x = -\frac{\pi}{4} $ и $ x = -\frac{3\pi}{4} $ (или $ x = \frac{5\pi}{4} $).
Нам нужны значения $x$, при которых $ \sin x $ строго больше $ -\frac{\sqrt{2}}{2} $. На единичной окружности это соответствует дуге, расположенной выше прямой $ y = -\frac{\sqrt{2}}{2} $. Двигаясь против часовой стрелки, эта дуга начинается в точке, соответствующей углу $ -\frac{\pi}{4} $, и заканчивается в точке, соответствующей углу $ \frac{5\pi}{4} $.
Таким образом, основной интервал решений: $ -\frac{\pi}{4} < x < \frac{5\pi}{4} $.
Так как функция синуса периодична с периодом $2\pi$, общее решение неравенства получается добавлением $2\pi k$ к границам найденного интервала, где $k$ — любое целое число.
$ -\frac{\pi}{4} + 2\pi k < x < \frac{5\pi}{4} + 2\pi k, k \in \mathbb{Z} $.
Ответ: $ x \in (-\frac{\pi}{4} + 2\pi k; \frac{5\pi}{4} + 2\pi k), k \in \mathbb{Z} $.
б) $ \sin \frac{x}{4} \ge \frac{\sqrt{2}}{2} $
Введем замену переменной: пусть $ t = \frac{x}{4} $. Неравенство примет вид: $ \sin t \ge \frac{\sqrt{2}}{2} $.
Решим это неравенство для $t$. На единичной окружности находим точки, где $ \sin t = \frac{\sqrt{2}}{2} $. Это углы $ t = \frac{\pi}{4} $ и $ t = \pi - \frac{\pi}{4} = \frac{3\pi}{4} $. Нам нужны значения $t$, при которых ордината на окружности не меньше $ \frac{\sqrt{2}}{2} $.
Это соответствует дуге от $ \frac{\pi}{4} $ до $ \frac{3\pi}{4} $. С учетом периодичности, решение для $t$ имеет вид:
$ \frac{\pi}{4} + 2\pi k \le t \le \frac{3\pi}{4} + 2\pi k, k \in \mathbb{Z} $.
Теперь выполним обратную замену, подставив $t = \frac{x}{4}$:
$ \frac{\pi}{4} + 2\pi k \le \frac{x}{4} \le \frac{3\pi}{4} + 2\pi k $.
Для того чтобы найти $x$, умножим все части двойного неравенства на 4:
$ 4(\frac{\pi}{4} + 2\pi k) \le x \le 4(\frac{3\pi}{4} + 2\pi k) $.
$ \pi + 8\pi k \le x \le 3\pi + 8\pi k $.
Ответ: $ x \in [\pi + 8\pi k; 3\pi + 8\pi k], k \in \mathbb{Z} $.
в) $ \sin \frac{3x}{5} < -\frac{\sqrt{2}}{2} $
Введем замену переменной: $ t = \frac{3x}{5} $. Неравенство примет вид: $ \sin t < -\frac{\sqrt{2}}{2} $.
На единичной окружности находим точки, где $ \sin t = -\frac{\sqrt{2}}{2} $. Это углы $ t = -\frac{\pi}{4} $ и $ t = -\frac{3\pi}{4} $. Нам нужны значения $t$, при которых ордината точки на окружности меньше $ -\frac{\sqrt{2}}{2} $.
Это соответствует дуге между углами $ -\frac{3\pi}{4} $ и $ -\frac{\pi}{4} $. С учетом периода $2\pi$ решение для $t$ будет:
$ -\frac{3\pi}{4} + 2\pi k < t < -\frac{\pi}{4} + 2\pi k, k \in \mathbb{Z} $.
Выполним обратную замену $t = \frac{3x}{5}$:
$ -\frac{3\pi}{4} + 2\pi k < \frac{3x}{5} < -\frac{\pi}{4} + 2\pi k $.
Чтобы найти $x$, умножим все части неравенства на $ \frac{5}{3} $:
$ \frac{5}{3}(-\frac{3\pi}{4} + 2\pi k) < x < \frac{5}{3}(-\frac{\pi}{4} + 2\pi k) $.
$ -\frac{5\pi}{4} + \frac{10\pi k}{3} < x < -\frac{5\pi}{12} + \frac{10\pi k}{3} $.
Ответ: $ x \in (-\frac{5\pi}{4} + \frac{10\pi k}{3}; -\frac{5\pi}{12} + \frac{10\pi k}{3}), k \in \mathbb{Z} $.
г) $ \sin(x + \frac{\pi}{4}) \le \frac{\sqrt{2}}{2} $
Выполним замену: $ t = x + \frac{\pi}{4} $. Неравенство станет $ \sin t \le \frac{\sqrt{2}}{2} $.
На единичной окружности значениям $ \sin t = \frac{\sqrt{2}}{2} $ соответствуют углы $ t = \frac{\pi}{4} $ и $ t = \frac{3\pi}{4} $. Нам нужна дуга, где ордината точек не превышает $ \frac{\sqrt{2}}{2} $. Эта дуга начинается в точке $ \frac{3\pi}{4} $ и, двигаясь против часовой стрелки, заканчивается в точке $ 2\pi + \frac{\pi}{4} = \frac{9\pi}{4} $.
Следовательно, решение для $t$ с учетом периодичности:
$ \frac{3\pi}{4} + 2\pi k \le t \le \frac{9\pi}{4} + 2\pi k, k \in \mathbb{Z} $.
Произведем обратную замену $ t = x + \frac{\pi}{4} $:
$ \frac{3\pi}{4} + 2\pi k \le x + \frac{\pi}{4} \le \frac{9\pi}{4} + 2\pi k $.
Вычтем $ \frac{\pi}{4} $ из всех частей неравенства:
$ \frac{3\pi}{4} - \frac{\pi}{4} + 2\pi k \le x \le \frac{9\pi}{4} - \frac{\pi}{4} + 2\pi k $.
$ \frac{2\pi}{4} + 2\pi k \le x \le \frac{8\pi}{4} + 2\pi k $.
$ \frac{\pi}{2} + 2\pi k \le x \le 2\pi + 2\pi k $.
Ответ: $ x \in [\frac{\pi}{2} + 2\pi k; 2\pi + 2\pi k], k \in \mathbb{Z} $.
д) $ \sin(\frac{6x}{7} + \frac{6\pi}{7}) \le -\frac{\sqrt{2}}{2} $
Введем замену: $ t = \frac{6x}{7} + \frac{6\pi}{7} $. Неравенство примет вид $ \sin t \le -\frac{\sqrt{2}}{2} $.
Это неравенство выполняется для тех $t$, ордината которых на единичной окружности не больше $ -\frac{\sqrt{2}}{2} $. Граничные точки $ \sin t = -\frac{\sqrt{2}}{2} $ соответствуют углам $ t = -\frac{3\pi}{4} $ и $ t = -\frac{\pi}{4} $.
Решение для $t$ с учетом периодичности:
$ -\frac{3\pi}{4} + 2\pi k \le t \le -\frac{\pi}{4} + 2\pi k, k \in \mathbb{Z} $.
Выполним обратную замену $ t = \frac{6}{7}(x+\pi) $:
$ -\frac{3\pi}{4} + 2\pi k \le \frac{6}{7}(x+\pi) \le -\frac{\pi}{4} + 2\pi k $.
Умножим все части неравенства на $ \frac{7}{6} $:
$ \frac{7}{6}(-\frac{3\pi}{4}) + \frac{7}{6}(2\pi k) \le x+\pi \le \frac{7}{6}(-\frac{\pi}{4}) + \frac{7}{6}(2\pi k) $.
$ -\frac{7\pi}{8} + \frac{7\pi k}{3} \le x+\pi \le -\frac{7\pi}{24} + \frac{7\pi k}{3} $.
Вычтем $ \pi $ из всех частей неравенства:
$ -\frac{7\pi}{8} - \pi + \frac{7\pi k}{3} \le x \le -\frac{7\pi}{24} - \pi + \frac{7\pi k}{3} $.
$ -\frac{7\pi+8\pi}{8} + \frac{7\pi k}{3} \le x \le -\frac{7\pi+24\pi}{24} + \frac{7\pi k}{3} $.
$ -\frac{15\pi}{8} + \frac{7\pi k}{3} \le x \le -\frac{31\pi}{24} + \frac{7\pi k}{3} $.
Ответ: $ x \in [-\frac{15\pi}{8} + \frac{7\pi k}{3}; -\frac{31\pi}{24} + \frac{7\pi k}{3}], k \in \mathbb{Z} $.
Другие задания:
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