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Номер 2, страница 161, 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 2, страница 161.
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" => 1119936 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "161" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/3-3-1zad-2" "field_display_title" => "2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …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 {#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 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1396 #items: array:5 [ 0 => App\Models\Branch {#1070 #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 {#1073 #items: array:1 [ 0 => App\Models\Term {#1071 #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 {#1074 #items: array:1 [ 0 => App\Models\Book {#1075 #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 {#1076 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1098 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1076 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1098 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1101 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1103 #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 {#1073} "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 {#1074} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1103} ] #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 {#1104 #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 {#1105 #items: array:1 [ 0 => App\Models\Term {#1106 #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 {#1107 #items: array:1 [ 0 => App\Models\Book {#1108 #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 {#1109 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1131 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1109 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1128 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1131 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1136 #items: array:1 [ 0 => App\Models\Branch {#1137 #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 {#1138 …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 {#1140 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1141 …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 {#1138 …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 {#1140 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1141 …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 {#1105} "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 {#1107} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1136} ] #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 {#1142 #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 {#1143 #items: array:1 [ 0 => App\Models\Term {#1144 #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 {#1145 #items: array:1 [ 0 => App\Models\Book {#1146 #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 {#1147 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1157 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1158 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1160 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1164 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1166 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1168 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1169 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1147 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1151 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1157 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1158 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1160 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1162 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1164 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1166 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1168 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1169 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1174 #items: array:1 [ 0 => App\Models\Branch {#1175 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 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 {#1176 …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 {#1178 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1179 …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 {#1176 …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 {#1178 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1179 …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 {#1143} "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 {#1145} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1174} ] #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 {#1185 #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 {#1186 #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 {#1187 #items: array:1 [ 0 => App\Models\Book {#1188 #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 {#1189 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1202 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1204 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1209 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1211 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] …13 ] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1216 #items: array:1 [ 0 => App\Models\Branch {#1217 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: 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 {#1186} "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 {#1187} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1216} ] #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 {#1288 #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 {#1289 #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 {#1290 #items: array:1 [ 0 => App\Models\Book {#1291 #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 [ …50] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1319 #items: array:1 [ 0 => 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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #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 {#1289} "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 {#1290} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1319} ] #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 {#1395 #items: array:2 [ 0 => App\Models\Element {#1406 #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" => 1300621 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1416 #items: array:1 [ 0 => App\Models\Edition {#1415 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1119936" "type" => "task" ] "text" => "<p><strong>2.(2) a)</strong> $\sin x > -\frac{\sqrt{3}}{2}$;</p><p><strong>б)</strong> $\sin \frac{x}{3} \ge -\frac{\sqrt{3}}{2}$;</p><p><strong>в)</strong> $\sin \frac{2x}{3} < -\frac{\sqrt{3}}{2}$;</p><p><strong>г)</strong> $\sin \left(x+\frac{\pi}{5}\right) \le -\frac{\sqrt{3}}{2}$;</p><p><strong>д)</strong> $\sin \left(\frac{5x}{6}+\frac{6\pi}{7}\right) \le -\frac{\sqrt{3}}{2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1549" "height" => 405 "path" => "/media/algebra_10/baspasy-u19/0-3-3-1zad/2-1.webp?ts=1753534676" ] ] ] #original: array:7 [ "id" => 1300621 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1416} "task" => array:2 [ "refs" => "1119936" "type" => "task" ] "text" => "<p><strong>2.(2) a)</strong> $\sin x > -\frac{\sqrt{3}}{2}$;</p><p><strong>б)</strong> $\sin \frac{x}{3} \ge -\frac{\sqrt{3}}{2}$;</p><p><strong>в)</strong> $\sin \frac{2x}{3} < -\frac{\sqrt{3}}{2}$;</p><p><strong>г)</strong> $\sin \left(x+\frac{\pi}{5}\right) \le -\frac{\sqrt{3}}{2}$;</p><p><strong>д)</strong> $\sin \left(\frac{5x}{6}+\frac{6\pi}{7}\right) \le -\frac{\sqrt{3}}{2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "2-1.jpg" "alt" => null "width" => "1549" "height" => 405 "path" => "/media/algebra_10/baspasy-u19/0-3-3-1zad/2-1.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 {#1408 #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" => 1551657 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1421 #items: array:1 [ 0 => App\Models\Edition {#1409 #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 [ …21] #original: array:21 [ …21] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1119936" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p> Дано неравенство $ \sin x > -\frac{\sqrt{3}}{2} $. <br> Сначала найдем корни уравнения $ \sin x = -\frac{\sqrt{3}}{2} $. На единичной окружности это точки, соответствующие углам $ x = -\frac{\pi}{3} $ и $ x = \pi - (-\frac{\pi}{3}) = \frac{4\pi}{3} $. <br> Нам нужны значения $x$, для которых ордината (синус) на единичной окружности больше $ -\frac{\sqrt{3}}{2} $. Это соответствует дуге, расположенной выше прямой $y = -\frac{\sqrt{3}}{2}$. <br> Двигаясь по окружности против часовой стрелки, мы видим, что искомый интервал начинается в точке $ -\frac{\pi}{3} $ и заканчивается в точке $ \frac{4\pi}{3} $. <br> Учитывая периодичность функции синус, общее решение неравенства имеет вид: <br> $ -\frac{\pi}{3} + 2\pi k < x < \frac{4\pi}{3} + 2\pi k, k \in \mathbb{Z} $. <br> Ответ: $x \in (-\frac{\pi}{3} + 2\pi k; \frac{4\pi}{3} + 2\pi k), k \in \mathbb{Z}$.</p><p><strong>б)</strong></p><p> Дано неравенство $ \sin \frac{x}{3} \ge -\frac{\sqrt{3}}{2} $. <br> Введем замену $ t = \frac{x}{3} $. Неравенство примет вид $ \sin t \ge -\frac{\sqrt{3}}{2} $. <br> Аналогично предыдущему пункту, решением этого неравенства для $t$ будет интервал $ -\frac{\pi}{3} \le t \le \frac{4\pi}{3} $, с учетом периодичности: <br> $ -\frac{\pi}{3} + 2\pi k \le t \le \frac{4\pi}{3} + 2\pi k, k \in \mathbb{Z} $. <br> Теперь выполним обратную замену $ t = \frac{x}{3} $: <br> $ -\frac{\pi}{3} + 2\pi k \le \frac{x}{3} \le \frac{4\pi}{3} + 2\pi k $. <br> Умножим все части двойного неравенства на 3, чтобы выразить $x$: <br> $ 3(-\frac{\pi}{3} + 2\pi k) \le x \le 3(\frac{4\pi}{3} + 2\pi k) $. <br> $ -\pi + 6\pi k \le x \le 4\pi + 6\pi k, k \in \mathbb{Z} $. <br> Ответ: $x \in [-\pi + 6\pi k; 4\pi + 6\pi k], k \in \mathbb{Z}$.</p><p><strong>в)</strong></p><p> Дано неравенство $ \sin \frac{2x}{3} < -\frac{\sqrt{3}}{2} $. <br> Введем замену $ t = \frac{2x}{3} $. Неравенство примет вид $ \sin t < -\frac{\sqrt{3}}{2} $. <br> Нам нужны значения $t$, для которых синус меньше $ -\frac{\sqrt{3}}{2} $. На единичной окружности это дуга, лежащая ниже прямой $y = -\frac{\sqrt{3}}{2}$. Эта дуга заключена между точками $ \frac{4\pi}{3} $ и $ \frac{5\pi}{3} $. Или, что эквивалентно, между $ -\frac{2\pi}{3} $ и $ -\frac{\pi}{3} $. <br> С учетом периодичности, решение для $t$: <br> $ -\frac{2\pi}{3} + 2\pi k < t < -\frac{\pi}{3} + 2\pi k, k \in \mathbb{Z} $. <br> Выполним обратную замену $ t = \frac{2x}{3} $: <br> $ -\frac{2\pi}{3} + 2\pi k < \frac{2x}{3} < -\frac{\pi}{3} + 2\pi k $. <br> Умножим все части на $ \frac{3}{2} $: <br> $ \frac{3}{2}(-\frac{2\pi}{3} + 2\pi k) < x < \frac{3}{2}(-\frac{\pi}{3} + 2\pi k) $. <br> $ -\pi + 3\pi k < x < -\frac{\pi}{2} + 3\pi k, k \in \mathbb{Z} $. <br> Ответ: $x \in (-\pi + 3\pi k; -\frac{\pi}{2} + 3\pi k), k \in \mathbb{Z}$.</p><p><strong>г)</strong></p><p> Дано неравенство $ \sin(x + \frac{\pi}{5}) \le \frac{\sqrt{3}}{2} $. <br> Введем замену $ t = x + \frac{\pi}{5} $. Неравенство примет вид $ \sin t \le \frac{\sqrt{3}}{2} $. <br> Решениями уравнения $ \sin t = \frac{\sqrt{3}}{2} $ являются $ t = \frac{\pi}{3} $ и $ t = \frac{2\pi}{3} $. <br> Неравенству $ \sin t \le \frac{\sqrt{3}}{2} $ соответствуют все точки на единичной окружности, кроме дуги, где $ \sin t > \frac{\sqrt{3}}{2} $ (то есть дуги между $ \frac{\pi}{3} $ и $ \frac{2\pi}{3} $). <br> Следовательно, решение для $t$ можно записать как $ \frac{2\pi}{3} + 2\pi k \le t \le \frac{\pi}{3} + 2\pi(k+1) $, что то же самое, что и $ \frac{2\pi}{3} + 2\pi k \le t \le \frac{7\pi}{3} + 2\pi k, k \in \mathbb{Z} $. <br> Выполним обратную замену $ t = x + \frac{\pi}{5} $: <br> $ \frac{2\pi}{3} + 2\pi k \le x + \frac{\pi}{5} \le \frac{7\pi}{3} + 2\pi k $. <br> Вычтем $ \frac{\pi}{5} $ из всех частей неравенства: <br> $ \frac{2\pi}{3} - \frac{\pi}{5} + 2\pi k \le x \le \frac{7\pi}{3} - \frac{\pi}{5} + 2\pi k $. <br> Приведем дроби к общему знаменателю: <br> $ \frac{10\pi - 3\pi}{15} + 2\pi k \le x \le \frac{35\pi - 3\pi}{15} + 2\pi k $. <br> $ \frac{7\pi}{15} + 2\pi k \le x \le \frac{32\pi}{15} + 2\pi k, k \in \mathbb{Z} $. <br> Ответ: $x \in [\frac{7\pi}{15} + 2\pi k; \frac{32\pi}{15} + 2\pi k], k \in \mathbb{Z}$.</p><p><strong>д)</strong></p><p> Дано неравенство $ \sin(\frac{5x}{6} + \frac{6\pi}{7}) \le -\frac{\sqrt{3}}{2} $. <br> Введем замену $ t = \frac{5x}{6} + \frac{6\pi}{7} $. Неравенство примет вид $ \sin t \le -\frac{\sqrt{3}}{2} $. <br> Как и в пункте в), но с нестрогим неравенством, решение для $t$ будет: <br> $ -\frac{2\pi}{3} + 2\pi k \le t \le -\frac{\pi}{3} + 2\pi k, k \in \mathbb{Z} $. <br> Выполним обратную замену: <br> $ -\frac{2\pi}{3} + 2\pi k \le \frac{5x}{6} + \frac{6\pi}{7} \le -\frac{\pi}{3} + 2\pi k $. <br> Вычтем $ \frac{6\pi}{7} $ из всех частей: <br> $ -\frac{2\pi}{3} - \frac{6\pi}{7} + 2\pi k \le \frac{5x}{6} \le -\frac{\pi}{3} - \frac{6\pi}{7} + 2\pi k $. <br> Приведем дроби к общему знаменателю 21: <br> $ \frac{-14\pi - 18\pi}{21} + 2\pi k \le \frac{5x}{6} \le \frac{-7\pi - 18\pi}{21} + 2\pi k $. <br> $ -\frac{32\pi}{21} + 2\pi k \le \frac{5x}{6} \le -\frac{25\pi}{21} + 2\pi k $. <br> Умножим все части на $ \frac{6}{5} $: <br> $ \frac{6}{5} \cdot (-\frac{32\pi}{21}) + \frac{6}{5} \cdot 2\pi k \le x \le \frac{6}{5} \cdot (-\frac{25\pi}{21}) + \frac{6}{5} \cdot 2\pi k $. <br> Упростим выражения: <br> $ -\frac{64\pi}{35} + \frac{12\pi k}{5} \le x \le -\frac{10\pi}{7} + \frac{12\pi k}{5}, k \in \mathbb{Z} $. <br> Ответ: $x \in [-\frac{64\pi}{35} + \frac{12\pi k}{5}; -\frac{10\pi}{7} + \frac{12\pi k}{5}], k \in \mathbb{Z}$.</p>" ] #original: array:6 [ "id" => 1551657 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1421} "task" => array:2 [ "refs" => "1119936" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p> Дано неравенство $ \sin x > -\frac{\sqrt{3}}{2} $. <br> Сначала найдем корни уравнения $ \sin x = -\frac{\sqrt{3}}{2} $. На единичной окружности это точки, соответствующие углам $ x = -\frac{\pi}{3} $ и $ x = \pi - (-\frac{\pi}{3}) = \frac{4\pi}{3} $. <br> Нам нужны значения $x$, для которых ордината (синус) на единичной окружности больше $ -\frac{\sqrt{3}}{2} $. Это соответствует дуге, расположенной выше прямой $y = -\frac{\sqrt{3}}{2}$. <br> Двигаясь по окружности против часовой стрелки, мы видим, что искомый интервал начинается в точке $ -\frac{\pi}{3} $ и заканчивается в точке $ \frac{4\pi}{3} $. <br> Учитывая периодичность функции синус, общее решение неравенства имеет вид: <br> $ -\frac{\pi}{3} + 2\pi k < x < \frac{4\pi}{3} + 2\pi k, k \in \mathbb{Z} $. <br> Ответ: $x \in (-\frac{\pi}{3} + 2\pi k; \frac{4\pi}{3} + 2\pi k), k \in \mathbb{Z}$.</p><p><strong>б)</strong></p><p> Дано неравенство $ \sin \frac{x}{3} \ge -\frac{\sqrt{3}}{2} $. <br> Введем замену $ t = \frac{x}{3} $. Неравенство примет вид $ \sin t \ge -\frac{\sqrt{3}}{2} $. <br> Аналогично предыдущему пункту, решением этого неравенства для $t$ будет интервал $ -\frac{\pi}{3} \le t \le \frac{4\pi}{3} $, с учетом периодичности: <br> $ -\frac{\pi}{3} + 2\pi k \le t \le \frac{4\pi}{3} + 2\pi k, k \in \mathbb{Z} $. <br> Теперь выполним обратную замену $ t = \frac{x}{3} $: <br> $ -\frac{\pi}{3} + 2\pi k \le \frac{x}{3} \le \frac{4\pi}{3} + 2\pi k $. <br> Умножим все части двойного неравенства на 3, чтобы выразить $x$: <br> $ 3(-\frac{\pi}{3} + 2\pi k) \le x \le 3(\frac{4\pi}{3} + 2\pi k) $. <br> $ -\pi + 6\pi k \le x \le 4\pi + 6\pi k, k \in \mathbb{Z} $. <br> Ответ: $x \in [-\pi + 6\pi k; 4\pi + 6\pi k], k \in \mathbb{Z}$.</p><p><strong>в)</strong></p><p> Дано неравенство $ \sin \frac{2x}{3} < -\frac{\sqrt{3}}{2} $. <br> Введем замену $ t = \frac{2x}{3} $. Неравенство примет вид $ \sin t < -\frac{\sqrt{3}}{2} $. <br> Нам нужны значения $t$, для которых синус меньше $ -\frac{\sqrt{3}}{2} $. На единичной окружности это дуга, лежащая ниже прямой $y = -\frac{\sqrt{3}}{2}$. Эта дуга заключена между точками $ \frac{4\pi}{3} $ и $ \frac{5\pi}{3} $. Или, что эквивалентно, между $ -\frac{2\pi}{3} $ и $ -\frac{\pi}{3} $. <br> С учетом периодичности, решение для $t$: <br> $ -\frac{2\pi}{3} + 2\pi k < t < -\frac{\pi}{3} + 2\pi k, k \in \mathbb{Z} $. <br> Выполним обратную замену $ t = \frac{2x}{3} $: <br> $ -\frac{2\pi}{3} + 2\pi k < \frac{2x}{3} < -\frac{\pi}{3} + 2\pi k $. <br> Умножим все части на $ \frac{3}{2} $: <br> $ \frac{3}{2}(-\frac{2\pi}{3} + 2\pi k) < x < \frac{3}{2}(-\frac{\pi}{3} + 2\pi k) $. <br> $ -\pi + 3\pi k < x < -\frac{\pi}{2} + 3\pi k, k \in \mathbb{Z} $. <br> Ответ: $x \in (-\pi + 3\pi k; -\frac{\pi}{2} + 3\pi k), k \in \mathbb{Z}$.</p><p><strong>г)</strong></p><p> Дано неравенство $ \sin(x + \frac{\pi}{5}) \le \frac{\sqrt{3}}{2} $. <br> Введем замену $ t = x + \frac{\pi}{5} $. Неравенство примет вид $ \sin t \le \frac{\sqrt{3}}{2} $. <br> Решениями уравнения $ \sin t = \frac{\sqrt{3}}{2} $ являются $ t = \frac{\pi}{3} $ и $ t = \frac{2\pi}{3} $. <br> Неравенству $ \sin t \le \frac{\sqrt{3}}{2} $ соответствуют все точки на единичной окружности, кроме дуги, где $ \sin t > \frac{\sqrt{3}}{2} $ (то есть дуги между $ \frac{\pi}{3} $ и $ \frac{2\pi}{3} $). <br> Следовательно, решение для $t$ можно записать как $ \frac{2\pi}{3} + 2\pi k \le t \le \frac{\pi}{3} + 2\pi(k+1) $, что то же самое, что и $ \frac{2\pi}{3} + 2\pi k \le t \le \frac{7\pi}{3} + 2\pi k, k \in \mathbb{Z} $. <br> Выполним обратную замену $ t = x + \frac{\pi}{5} $: <br> $ \frac{2\pi}{3} + 2\pi k \le x + \frac{\pi}{5} \le \frac{7\pi}{3} + 2\pi k $. <br> Вычтем $ \frac{\pi}{5} $ из всех частей неравенства: <br> $ \frac{2\pi}{3} - \frac{\pi}{5} + 2\pi k \le x \le \frac{7\pi}{3} - \frac{\pi}{5} + 2\pi k $. <br> Приведем дроби к общему знаменателю: <br> $ \frac{10\pi - 3\pi}{15} + 2\pi k \le x \le \frac{35\pi - 3\pi}{15} + 2\pi k $. <br> $ \frac{7\pi}{15} + 2\pi k \le x \le \frac{32\pi}{15} + 2\pi k, k \in \mathbb{Z} $. <br> Ответ: $x \in [\frac{7\pi}{15} + 2\pi k; \frac{32\pi}{15} + 2\pi k], k \in \mathbb{Z}$.</p><p><strong>д)</strong></p><p> Дано неравенство $ \sin(\frac{5x}{6} + \frac{6\pi}{7}) \le -\frac{\sqrt{3}}{2} $. <br> Введем замену $ t = \frac{5x}{6} + \frac{6\pi}{7} $. Неравенство примет вид $ \sin t \le -\frac{\sqrt{3}}{2} $. <br> Как и в пункте в), но с нестрогим неравенством, решение для $t$ будет: <br> $ -\frac{2\pi}{3} + 2\pi k \le t \le -\frac{\pi}{3} + 2\pi k, k \in \mathbb{Z} $. <br> Выполним обратную замену: <br> $ -\frac{2\pi}{3} + 2\pi k \le \frac{5x}{6} + \frac{6\pi}{7} \le -\frac{\pi}{3} + 2\pi k $. <br> Вычтем $ \frac{6\pi}{7} $ из всех частей: <br> $ -\frac{2\pi}{3} - \frac{6\pi}{7} + 2\pi k \le \frac{5x}{6} \le -\frac{\pi}{3} - \frac{6\pi}{7} + 2\pi k $. <br> Приведем дроби к общему знаменателю 21: <br> $ \frac{-14\pi - 18\pi}{21} + 2\pi k \le \frac{5x}{6} \le \frac{-7\pi - 18\pi}{21} + 2\pi k $. <br> $ -\frac{32\pi}{21} + 2\pi k \le \frac{5x}{6} \le -\frac{25\pi}{21} + 2\pi k $. <br> Умножим все части на $ \frac{6}{5} $: <br> $ \frac{6}{5} \cdot (-\frac{32\pi}{21}) + \frac{6}{5} \cdot 2\pi k \le x \le \frac{6}{5} \cdot (-\frac{25\pi}{21}) + \frac{6}{5} \cdot 2\pi k $. <br> Упростим выражения: <br> $ -\frac{64\pi}{35} + \frac{12\pi k}{5} \le x \le -\frac{10\pi}{7} + \frac{12\pi k}{5}, k \in \mathbb{Z} $. <br> Ответ: $x \in [-\frac{64\pi}{35} + \frac{12\pi k}{5}; -\frac{10\pi}{7} + \frac{12\pi k}{5}], k \in \mathbb{Z}$.</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" => "1119937" "type" => "task" ] "previous" => array:2 [ "refs" => "1119935" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1447 #items: array:1 [ 0 => App\Models\Book {#1403 #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 {#1397 #items: array:1 [ 0 => App\Models\Term {#1398 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1405 #items: array:1 [ 0 => App\Models\Term {#1401 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1399 #items: array:1 [ 0 => App\Models\Term {#1402 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1400 #items: array:3 [ 0 => App\Models\Term {#1418 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 1 => App\Models\Term {#1422 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Term {#1420 #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 [ …12] #original: array:12 [ …12] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1427 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1428 #items: array:1 [ 0 => App\Models\Term {#1430 #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 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1431 #items: array:1 [ 0 => App\Models\Term {#1432 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1433 #items: array:1 [ 0 => App\Models\Term {#1434 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true 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№2 (с. 161)
Условие. №2 (с. 161)
скриншот условия
2.(2) a) $\sin x > -\frac{\sqrt{3}}{2}$;
б) $\sin \frac{x}{3} \ge -\frac{\sqrt{3}}{2}$;
в) $\sin \frac{2x}{3} < -\frac{\sqrt{3}}{2}$;
г) $\sin \left(x+\frac{\pi}{5}\right) \le -\frac{\sqrt{3}}{2}$;
д) $\sin \left(\frac{5x}{6}+\frac{6\pi}{7}\right) \le -\frac{\sqrt{3}}{2}$.
Решение 2 (rus). №2 (с. 161)
а)
Дано неравенство $ \sin x > -\frac{\sqrt{3}}{2} $.
Сначала найдем корни уравнения $ \sin x = -\frac{\sqrt{3}}{2} $. На единичной окружности это точки, соответствующие углам $ x = -\frac{\pi}{3} $ и $ x = \pi - (-\frac{\pi}{3}) = \frac{4\pi}{3} $.
Нам нужны значения $x$, для которых ордината (синус) на единичной окружности больше $ -\frac{\sqrt{3}}{2} $. Это соответствует дуге, расположенной выше прямой $y = -\frac{\sqrt{3}}{2}$.
Двигаясь по окружности против часовой стрелки, мы видим, что искомый интервал начинается в точке $ -\frac{\pi}{3} $ и заканчивается в точке $ \frac{4\pi}{3} $.
Учитывая периодичность функции синус, общее решение неравенства имеет вид:
$ -\frac{\pi}{3} + 2\pi k < x < \frac{4\pi}{3} + 2\pi k, k \in \mathbb{Z} $.
Ответ: $x \in (-\frac{\pi}{3} + 2\pi k; \frac{4\pi}{3} + 2\pi k), k \in \mathbb{Z}$.
б)
Дано неравенство $ \sin \frac{x}{3} \ge -\frac{\sqrt{3}}{2} $.
Введем замену $ t = \frac{x}{3} $. Неравенство примет вид $ \sin t \ge -\frac{\sqrt{3}}{2} $.
Аналогично предыдущему пункту, решением этого неравенства для $t$ будет интервал $ -\frac{\pi}{3} \le t \le \frac{4\pi}{3} $, с учетом периодичности:
$ -\frac{\pi}{3} + 2\pi k \le t \le \frac{4\pi}{3} + 2\pi k, k \in \mathbb{Z} $.
Теперь выполним обратную замену $ t = \frac{x}{3} $:
$ -\frac{\pi}{3} + 2\pi k \le \frac{x}{3} \le \frac{4\pi}{3} + 2\pi k $.
Умножим все части двойного неравенства на 3, чтобы выразить $x$:
$ 3(-\frac{\pi}{3} + 2\pi k) \le x \le 3(\frac{4\pi}{3} + 2\pi k) $.
$ -\pi + 6\pi k \le x \le 4\pi + 6\pi k, k \in \mathbb{Z} $.
Ответ: $x \in [-\pi + 6\pi k; 4\pi + 6\pi k], k \in \mathbb{Z}$.
в)
Дано неравенство $ \sin \frac{2x}{3} < -\frac{\sqrt{3}}{2} $.
Введем замену $ t = \frac{2x}{3} $. Неравенство примет вид $ \sin t < -\frac{\sqrt{3}}{2} $.
Нам нужны значения $t$, для которых синус меньше $ -\frac{\sqrt{3}}{2} $. На единичной окружности это дуга, лежащая ниже прямой $y = -\frac{\sqrt{3}}{2}$. Эта дуга заключена между точками $ \frac{4\pi}{3} $ и $ \frac{5\pi}{3} $. Или, что эквивалентно, между $ -\frac{2\pi}{3} $ и $ -\frac{\pi}{3} $.
С учетом периодичности, решение для $t$:
$ -\frac{2\pi}{3} + 2\pi k < t < -\frac{\pi}{3} + 2\pi k, k \in \mathbb{Z} $.
Выполним обратную замену $ t = \frac{2x}{3} $:
$ -\frac{2\pi}{3} + 2\pi k < \frac{2x}{3} < -\frac{\pi}{3} + 2\pi k $.
Умножим все части на $ \frac{3}{2} $:
$ \frac{3}{2}(-\frac{2\pi}{3} + 2\pi k) < x < \frac{3}{2}(-\frac{\pi}{3} + 2\pi k) $.
$ -\pi + 3\pi k < x < -\frac{\pi}{2} + 3\pi k, k \in \mathbb{Z} $.
Ответ: $x \in (-\pi + 3\pi k; -\frac{\pi}{2} + 3\pi k), k \in \mathbb{Z}$.
г)
Дано неравенство $ \sin(x + \frac{\pi}{5}) \le \frac{\sqrt{3}}{2} $.
Введем замену $ t = x + \frac{\pi}{5} $. Неравенство примет вид $ \sin t \le \frac{\sqrt{3}}{2} $.
Решениями уравнения $ \sin t = \frac{\sqrt{3}}{2} $ являются $ t = \frac{\pi}{3} $ и $ t = \frac{2\pi}{3} $.
Неравенству $ \sin t \le \frac{\sqrt{3}}{2} $ соответствуют все точки на единичной окружности, кроме дуги, где $ \sin t > \frac{\sqrt{3}}{2} $ (то есть дуги между $ \frac{\pi}{3} $ и $ \frac{2\pi}{3} $).
Следовательно, решение для $t$ можно записать как $ \frac{2\pi}{3} + 2\pi k \le t \le \frac{\pi}{3} + 2\pi(k+1) $, что то же самое, что и $ \frac{2\pi}{3} + 2\pi k \le t \le \frac{7\pi}{3} + 2\pi k, k \in \mathbb{Z} $.
Выполним обратную замену $ t = x + \frac{\pi}{5} $:
$ \frac{2\pi}{3} + 2\pi k \le x + \frac{\pi}{5} \le \frac{7\pi}{3} + 2\pi k $.
Вычтем $ \frac{\pi}{5} $ из всех частей неравенства:
$ \frac{2\pi}{3} - \frac{\pi}{5} + 2\pi k \le x \le \frac{7\pi}{3} - \frac{\pi}{5} + 2\pi k $.
Приведем дроби к общему знаменателю:
$ \frac{10\pi - 3\pi}{15} + 2\pi k \le x \le \frac{35\pi - 3\pi}{15} + 2\pi k $.
$ \frac{7\pi}{15} + 2\pi k \le x \le \frac{32\pi}{15} + 2\pi k, k \in \mathbb{Z} $.
Ответ: $x \in [\frac{7\pi}{15} + 2\pi k; \frac{32\pi}{15} + 2\pi k], k \in \mathbb{Z}$.
д)
Дано неравенство $ \sin(\frac{5x}{6} + \frac{6\pi}{7}) \le -\frac{\sqrt{3}}{2} $.
Введем замену $ t = \frac{5x}{6} + \frac{6\pi}{7} $. Неравенство примет вид $ \sin t \le -\frac{\sqrt{3}}{2} $.
Как и в пункте в), но с нестрогим неравенством, решение для $t$ будет:
$ -\frac{2\pi}{3} + 2\pi k \le t \le -\frac{\pi}{3} + 2\pi k, k \in \mathbb{Z} $.
Выполним обратную замену:
$ -\frac{2\pi}{3} + 2\pi k \le \frac{5x}{6} + \frac{6\pi}{7} \le -\frac{\pi}{3} + 2\pi k $.
Вычтем $ \frac{6\pi}{7} $ из всех частей:
$ -\frac{2\pi}{3} - \frac{6\pi}{7} + 2\pi k \le \frac{5x}{6} \le -\frac{\pi}{3} - \frac{6\pi}{7} + 2\pi k $.
Приведем дроби к общему знаменателю 21:
$ \frac{-14\pi - 18\pi}{21} + 2\pi k \le \frac{5x}{6} \le \frac{-7\pi - 18\pi}{21} + 2\pi k $.
$ -\frac{32\pi}{21} + 2\pi k \le \frac{5x}{6} \le -\frac{25\pi}{21} + 2\pi k $.
Умножим все части на $ \frac{6}{5} $:
$ \frac{6}{5} \cdot (-\frac{32\pi}{21}) + \frac{6}{5} \cdot 2\pi k \le x \le \frac{6}{5} \cdot (-\frac{25\pi}{21}) + \frac{6}{5} \cdot 2\pi k $.
Упростим выражения:
$ -\frac{64\pi}{35} + \frac{12\pi k}{5} \le x \le -\frac{10\pi}{7} + \frac{12\pi k}{5}, k \in \mathbb{Z} $.
Ответ: $x \in [-\frac{64\pi}{35} + \frac{12\pi k}{5}; -\frac{10\pi}{7} + \frac{12\pi k}{5}], k \in \mathbb{Z}$.
Другие задания:
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