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Номер 10, страница 171, часть 1 - гдз по алгебре 10 класс учебник Пак, Ардакулы
Авторы: Пак О. В., Ардакулы Д., Ескендирова Е. В.
Тип: Учебник
Издательство: Алматыкітап баспасы
Год издания: 2019 - 2026
Часть: 1
ISBN: 978-601-01-3958-9
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Часть 1. Глава 3. Тригонометрические уравнения и неравенства. Параграф 3. Простейшие тригонометрические неравенства. 3.3. Неравенства, содержащие tgx и ctgx. Задачи - номер 10, страница 171.
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" => 1119963 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "171" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/3-3-3zad-10" "field_display_title" => "10" "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 {#1242 #items: array:5 [ 0 => App\Models\Branch {#1254 #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 {#1258 #items: array:1 [ 0 => App\Models\Term {#1255 #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 {#1287 #items: array:1 [ 0 => App\Models\Book {#1257 #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 {#1259 #items: array:1 [ 0 => App\Models\Term {#1256 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1260 #items: array:1 [ 0 => App\Models\Term {#1261 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1262 #items: array:1 [ 0 => App\Models\Term {#1263 #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 {#1264 #items: array:3 [ 0 => App\Models\Term {#1265 #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 {#1266 #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 {#1267 #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 {#1268 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1269 #items: array:1 [ 0 => App\Models\Term {#1270 #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 {#1271 #items: array:1 [ 0 => App\Models\Term {#1272 #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 {#1273 #items: array:1 [ 0 => App\Models\Term {#1274 #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 {#1275 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1276 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1277 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1278 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1279 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1280 #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 {#1281 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1282 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1283 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1284 #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 {#1259} "field_class" => Illuminate\Database\Eloquent\Collection {#1260} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1262} "field_author" => Illuminate\Database\Eloquent\Collection {#1264} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1268} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1269} "field_country" => Illuminate\Database\Eloquent\Collection {#1271} "field_city" => Illuminate\Database\Eloquent\Collection {#1273} "field_series" => Illuminate\Database\Eloquent\Collection {#1275} "field_umk" => Illuminate\Database\Eloquent\Collection {#1276} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1277} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1278} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1279} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1280} "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 {#1281} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1282} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1283} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1284} "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 {#1285 #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 {#1258} "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 {#1287} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1285} ] #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 {#1286 #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 {#1291 #items: array:1 [ 0 => App\Models\Term {#1288 #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 {#1290 #items: array:1 [ 0 => App\Models\Book {#1257} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1289 #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 {#1291} "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 {#1290} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1289} ] #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 {#1292 #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 {#1296 #items: array:1 [ 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: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 {#1295 #items: array:1 [ 0 => App\Models\Book {#1257} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1364 #items: array:1 [ 0 => App\Models\Branch {#1294 #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 {#1298 …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 {#1299 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1328 …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 {#1298 …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 {#1299 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1328 …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 {#1296} "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 {#1295} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1364} ] #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 {#1362 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119404 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.3. Неравенства, содержащие tgx и ctgx" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1363 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "168" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1365 #items: array:1 [ 0 => App\Models\Book {#1257} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1439 #items: array:1 [ 0 => App\Models\Branch {#1366 #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 {#1368 …2} "field_page_start" => "157" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1369 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1398 …2} ] #original: array:24 [ "id" => 1119399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Простейшие тригонометрические неравенства" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1368 …2} "field_page_start" => "157" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1369 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1398 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119404 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.3. Неравенства, содержащие tgx и ctgx" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1363} "field_page_start" => "168" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1365} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1439} ] #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 {#1437 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1438 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "168" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1440 #items: array:1 [ 0 => App\Models\Book {#1257} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1449 #items: array:1 [ 0 => App\Models\Branch {#1448 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119404 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.3. Неравенства, содержащие tgx и ctgx" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1450 …2} "field_page_start" => "168" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1451 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1452 …2} ] #original: array:24 [ "id" => 1119404 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "3.3. Неравенства, содержащие tgx и ctgx" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1450 …2} "field_page_start" => "168" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1451 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1452 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119405 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1438} "field_page_start" => "168" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1440} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1449} ] #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 {#1233 #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" => 1300648 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:1 [ 0 => App\Models\Edition {#1232 #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 {#1230 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1229 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1228 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1227 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1230 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1229 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1228 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1227 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1119963" "type" => "task" ] "text" => "<p><strong>10.</strong> (2)</p><p><strong>а)</strong> $\tg x > 100$;</p><p><strong>б)</strong> $\tg 20x > -20$;</p><p><strong>в)</strong> $\tg \frac{5x}{8} < \frac{5}{8}$;</p><p><strong>г)</strong> $\tg \left(11x + \frac{\pi}{4}\right) + 100 \le 0$;</p><p><strong>д)</strong> $\tg \left(x + \frac{9\pi}{4}\right) \le \pi$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "10-1.jpg" "alt" => null "width" => "1376" "height" => 340 "path" => "/media/algebra_10/baspasy-u19/0-3-3-3zad/10-1.webp?ts=1753534690" ] ] ] #original: array:7 [ "id" => 1300648 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1223} "task" => array:2 [ "refs" => "1119963" "type" => "task" ] "text" => "<p><strong>10.</strong> (2)</p><p><strong>а)</strong> $\tg x > 100$;</p><p><strong>б)</strong> $\tg 20x > -20$;</p><p><strong>в)</strong> $\tg \frac{5x}{8} < \frac{5}{8}$;</p><p><strong>г)</strong> $\tg \left(11x + \frac{\pi}{4}\right) + 100 \le 0$;</p><p><strong>д)</strong> $\tg \left(x + \frac{9\pi}{4}\right) \le \pi$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "10-1.jpg" "alt" => null "width" => "1376" "height" => 340 "path" => "/media/algebra_10/baspasy-u19/0-3-3-3zad/10-1.webp?ts=1753534690" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1226 #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" => 1551684 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ 0 => App\Models\Edition {#1225 #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 {#1221 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1222 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1220 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1221 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1222 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1220 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1119963" "type" => "task" ] "text" => "<p><strong>а)</strong> Решим неравенство $tg(x) > 100$.</p><p>Общее решение неравенства вида $tg(t) > a$ имеет вид $arctg(a) + \pi n < t < \frac{\pi}{2} + \pi n$, где $n \in \mathbb{Z}$.</p><p>В данном случае $t = x$ и $a = 100$.</p><p>Подставляя эти значения, получаем решение:</p><p>$arctg(100) + \pi n < x < \frac{\pi}{2} + \pi n$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (arctg(100) + \pi n; \frac{\pi}{2} + \pi n)$, $n \in \mathbb{Z}$.</p><p><strong>б)</strong> Решим неравенство $tg(20x) > -20$.</p><p>Пусть $t = 20x$. Неравенство принимает вид $tg(t) > -20$.</p><p>Решение этого неравенства: $arctg(-20) + \pi n < t < \frac{\pi}{2} + \pi n$, где $n \in \mathbb{Z}$.</p><p>Используя свойство арктангенса $arctg(-a) = -arctg(a)$, получаем:</p><p>$-arctg(20) + \pi n < t < \frac{\pi}{2} + \pi n$.</p><p>Теперь произведем обратную замену $t = 20x$:</p><p>$-arctg(20) + \pi n < 20x < \frac{\pi}{2} + \pi n$.</p><p>Разделим все части неравенства на 20, чтобы найти $x$:</p><p>$\frac{-arctg(20) + \pi n}{20} < x < \frac{\frac{\pi}{2} + \pi n}{20}$</p><p>$-\frac{arctg(20)}{20} + \frac{\pi n}{20} < x < \frac{\pi}{40} + \frac{\pi n}{20}$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (-\frac{arctg(20)}{20} + \frac{\pi n}{20}; \frac{\pi}{40} + \frac{\pi n}{20})$, $n \in \mathbb{Z}$.</p><p><strong>в)</strong> Решим неравенство $tg(\frac{5x}{8}) < \frac{5}{8}$.</p><p>Общее решение неравенства вида $tg(t) < a$ имеет вид $-\frac{\pi}{2} + \pi n < t < arctg(a) + \pi n$, где $n \in \mathbb{Z}$.</p><p>В данном случае $t = \frac{5x}{8}$ и $a = \frac{5}{8}$.</p><p>Подставляем эти значения в общее решение:</p><p>$-\frac{\pi}{2} + \pi n < \frac{5x}{8} < arctg(\frac{5}{8}) + \pi n$.</p><p>Чтобы выразить $x$, умножим все части неравенства на $\frac{8}{5}$:</p><p>$(-\frac{\pi}{2} + \pi n) \cdot \frac{8}{5} < x < (arctg(\frac{5}{8}) + \pi n) \cdot \frac{8}{5}$</p><p>$-\frac{4\pi}{5} + \frac{8\pi n}{5} < x < \frac{8}{5}arctg(\frac{5}{8}) + \frac{8\pi n}{5}$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (-\frac{4\pi}{5} + \frac{8\pi n}{5}; \frac{8}{5}arctg(\frac{5}{8}) + \frac{8\pi n}{5})$, $n \in \mathbb{Z}$.</p><p><strong>г)</strong> Решим неравенство $tg(11x + \frac{\pi}{4}) + 100 \le 0$.</p><p>Сначала преобразуем неравенство:</p><p>$tg(11x + \frac{\pi}{4}) \le -100$.</p><p>Общее решение неравенства вида $tg(t) \le a$ имеет вид $-\frac{\pi}{2} + \pi n < t \le arctg(a) + \pi n$, где $n \in \mathbb{Z}$.</p><p>В нашем случае $t = 11x + \frac{\pi}{4}$ и $a = -100$.</p><p>Подставляем значения:</p><p>$-\frac{\pi}{2} + \pi n < 11x + \frac{\pi}{4} \le arctg(-100) + \pi n$.</p><p>Используя свойство $arctg(-a) = -arctg(a)$, получаем:</p><p>$-\frac{\pi}{2} + \pi n < 11x + \frac{\pi}{4} \le -arctg(100) + \pi n$.</p><p>Вычтем $\frac{\pi}{4}$ из всех частей неравенства:</p><p>$-\frac{\pi}{2} - \frac{\pi}{4} + \pi n < 11x \le -arctg(100) - \frac{\pi}{4} + \pi n$</p><p>$-\frac{3\pi}{4} + \pi n < 11x \le -arctg(100) - \frac{\pi}{4} + \pi n$.</p><p>Теперь разделим все части на 11:</p><p>$\frac{-\frac{3\pi}{4} + \pi n}{11} < x \le \frac{-arctg(100) - \frac{\pi}{4} + \pi n}{11}$</p><p>$-\frac{3\pi}{44} + \frac{\pi n}{11} < x \le -\frac{arctg(100)}{11} - \frac{\pi}{44} + \frac{\pi n}{11}$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (-\frac{3\pi}{44} + \frac{\pi n}{11}; -\frac{arctg(100)}{11} - \frac{\pi}{44} + \frac{\pi n}{11}]$, $n \in \mathbb{Z}$.</p><p><strong>д)</strong> Решим неравенство $tg(x + \frac{9\pi}{4}) \le \pi$.</p><p>Сначала упростим аргумент тангенса, используя его периодичность с периодом $\pi$. Заметим, что $\frac{9\pi}{4} = 2\pi + \frac{\pi}{4}$.</p><p>$tg(x + \frac{9\pi}{4}) = tg(x + \frac{\pi}{4} + 2\pi) = tg(x + \frac{\pi}{4})$.</p><p>Неравенство принимает вид: $tg(x + \frac{\pi}{4}) \le \pi$.</p><p>Общее решение неравенства вида $tg(t) \le a$ имеет вид $-\frac{\pi}{2} + \pi n < t \le arctg(a) + \pi n$, где $n \in \mathbb{Z}$.</p><p>В данном случае $t = x + \frac{\pi}{4}$ и $a = \pi$.</p><p>Подставляем значения:</p><p>$-\frac{\pi}{2} + \pi n < x + \frac{\pi}{4} \le arctg(\pi) + \pi n$.</p><p>Вычтем $\frac{\pi}{4}$ из всех частей неравенства, чтобы выразить $x$:</p><p>$-\frac{\pi}{2} - \frac{\pi}{4} + \pi n < x \le arctg(\pi) - \frac{\pi}{4} + \pi n$</p><p>$-\frac{3\pi}{4} + \pi n < x \le arctg(\pi) - \frac{\pi}{4} + \pi n$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (-\frac{3\pi}{4} + \pi n; arctg(\pi) - \frac{\pi}{4} + \pi n]$, $n \in \mathbb{Z}$.</p>" ] #original: array:6 [ "id" => 1551684 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1217} "task" => array:2 [ "refs" => "1119963" "type" => "task" ] "text" => "<p><strong>а)</strong> Решим неравенство $tg(x) > 100$.</p><p>Общее решение неравенства вида $tg(t) > a$ имеет вид $arctg(a) + \pi n < t < \frac{\pi}{2} + \pi n$, где $n \in \mathbb{Z}$.</p><p>В данном случае $t = x$ и $a = 100$.</p><p>Подставляя эти значения, получаем решение:</p><p>$arctg(100) + \pi n < x < \frac{\pi}{2} + \pi n$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (arctg(100) + \pi n; \frac{\pi}{2} + \pi n)$, $n \in \mathbb{Z}$.</p><p><strong>б)</strong> Решим неравенство $tg(20x) > -20$.</p><p>Пусть $t = 20x$. Неравенство принимает вид $tg(t) > -20$.</p><p>Решение этого неравенства: $arctg(-20) + \pi n < t < \frac{\pi}{2} + \pi n$, где $n \in \mathbb{Z}$.</p><p>Используя свойство арктангенса $arctg(-a) = -arctg(a)$, получаем:</p><p>$-arctg(20) + \pi n < t < \frac{\pi}{2} + \pi n$.</p><p>Теперь произведем обратную замену $t = 20x$:</p><p>$-arctg(20) + \pi n < 20x < \frac{\pi}{2} + \pi n$.</p><p>Разделим все части неравенства на 20, чтобы найти $x$:</p><p>$\frac{-arctg(20) + \pi n}{20} < x < \frac{\frac{\pi}{2} + \pi n}{20}$</p><p>$-\frac{arctg(20)}{20} + \frac{\pi n}{20} < x < \frac{\pi}{40} + \frac{\pi n}{20}$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (-\frac{arctg(20)}{20} + \frac{\pi n}{20}; \frac{\pi}{40} + \frac{\pi n}{20})$, $n \in \mathbb{Z}$.</p><p><strong>в)</strong> Решим неравенство $tg(\frac{5x}{8}) < \frac{5}{8}$.</p><p>Общее решение неравенства вида $tg(t) < a$ имеет вид $-\frac{\pi}{2} + \pi n < t < arctg(a) + \pi n$, где $n \in \mathbb{Z}$.</p><p>В данном случае $t = \frac{5x}{8}$ и $a = \frac{5}{8}$.</p><p>Подставляем эти значения в общее решение:</p><p>$-\frac{\pi}{2} + \pi n < \frac{5x}{8} < arctg(\frac{5}{8}) + \pi n$.</p><p>Чтобы выразить $x$, умножим все части неравенства на $\frac{8}{5}$:</p><p>$(-\frac{\pi}{2} + \pi n) \cdot \frac{8}{5} < x < (arctg(\frac{5}{8}) + \pi n) \cdot \frac{8}{5}$</p><p>$-\frac{4\pi}{5} + \frac{8\pi n}{5} < x < \frac{8}{5}arctg(\frac{5}{8}) + \frac{8\pi n}{5}$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (-\frac{4\pi}{5} + \frac{8\pi n}{5}; \frac{8}{5}arctg(\frac{5}{8}) + \frac{8\pi n}{5})$, $n \in \mathbb{Z}$.</p><p><strong>г)</strong> Решим неравенство $tg(11x + \frac{\pi}{4}) + 100 \le 0$.</p><p>Сначала преобразуем неравенство:</p><p>$tg(11x + \frac{\pi}{4}) \le -100$.</p><p>Общее решение неравенства вида $tg(t) \le a$ имеет вид $-\frac{\pi}{2} + \pi n < t \le arctg(a) + \pi n$, где $n \in \mathbb{Z}$.</p><p>В нашем случае $t = 11x + \frac{\pi}{4}$ и $a = -100$.</p><p>Подставляем значения:</p><p>$-\frac{\pi}{2} + \pi n < 11x + \frac{\pi}{4} \le arctg(-100) + \pi n$.</p><p>Используя свойство $arctg(-a) = -arctg(a)$, получаем:</p><p>$-\frac{\pi}{2} + \pi n < 11x + \frac{\pi}{4} \le -arctg(100) + \pi n$.</p><p>Вычтем $\frac{\pi}{4}$ из всех частей неравенства:</p><p>$-\frac{\pi}{2} - \frac{\pi}{4} + \pi n < 11x \le -arctg(100) - \frac{\pi}{4} + \pi n$</p><p>$-\frac{3\pi}{4} + \pi n < 11x \le -arctg(100) - \frac{\pi}{4} + \pi n$.</p><p>Теперь разделим все части на 11:</p><p>$\frac{-\frac{3\pi}{4} + \pi n}{11} < x \le \frac{-arctg(100) - \frac{\pi}{4} + \pi n}{11}$</p><p>$-\frac{3\pi}{44} + \frac{\pi n}{11} < x \le -\frac{arctg(100)}{11} - \frac{\pi}{44} + \frac{\pi n}{11}$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (-\frac{3\pi}{44} + \frac{\pi n}{11}; -\frac{arctg(100)}{11} - \frac{\pi}{44} + \frac{\pi n}{11}]$, $n \in \mathbb{Z}$.</p><p><strong>д)</strong> Решим неравенство $tg(x + \frac{9\pi}{4}) \le \pi$.</p><p>Сначала упростим аргумент тангенса, используя его периодичность с периодом $\pi$. Заметим, что $\frac{9\pi}{4} = 2\pi + \frac{\pi}{4}$.</p><p>$tg(x + \frac{9\pi}{4}) = tg(x + \frac{\pi}{4} + 2\pi) = tg(x + \frac{\pi}{4})$.</p><p>Неравенство принимает вид: $tg(x + \frac{\pi}{4}) \le \pi$.</p><p>Общее решение неравенства вида $tg(t) \le a$ имеет вид $-\frac{\pi}{2} + \pi n < t \le arctg(a) + \pi n$, где $n \in \mathbb{Z}$.</p><p>В данном случае $t = x + \frac{\pi}{4}$ и $a = \pi$.</p><p>Подставляем значения:</p><p>$-\frac{\pi}{2} + \pi n < x + \frac{\pi}{4} \le arctg(\pi) + \pi n$.</p><p>Вычтем $\frac{\pi}{4}$ из всех частей неравенства, чтобы выразить $x$:</p><p>$-\frac{\pi}{2} - \frac{\pi}{4} + \pi n < x \le arctg(\pi) - \frac{\pi}{4} + \pi n$</p><p>$-\frac{3\pi}{4} + \pi n < x \le arctg(\pi) - \frac{\pi}{4} + \pi n$, где $n \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $x \in (-\frac{3\pi}{4} + \pi n; arctg(\pi) - \frac{\pi}{4} + \pi n]$, $n \in \mathbb{Z}$.</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" => "1119964" "type" => "task" ] "previous" => array:2 [ "refs" => "1119962" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1239 #items: array:1 [ 0 => App\Models\Book {#1257} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:1 [ 0 => App\Models\BookPage {#1240 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1099084 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "171" "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/page-1-171" "field_display_title" => "171" "field_folder" => "1" "field_image_name" => "171" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:1 [ 0 => App\Models\Branch {#1254} ] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1214 #items: array:1 [ 0 => App\Models\Book {#1257} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1216 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1206 #items: array:1 [ 0 => App\Models\Element {#1238 #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" => 1262509 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1204 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1262509 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1204 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #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" => "1099085" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1099083" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1549 #items: array:3 [ 0 => App\Models\Task {#1559 #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" => 1119961 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "171" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/3-3-3zad-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1560 …2} "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 {#1561 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1562 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1563 …2} "content" => Illuminate\Database\Eloquent\Collection {#1565 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1571 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1119961 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "171" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/3-3-3zad-8" "field_display_title" => "8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1560 …2} "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 {#1561 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1562 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1563 …2} "content" => Illuminate\Database\Eloquent\Collection {#1565 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1571 …2} "page" => array:2 [ …2] ] #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\Task {#1566 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "171" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/3-3-3zad-9" "field_display_title" => "9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1564 …2} "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 {#1573 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1569 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1567 …2} "content" => Illuminate\Database\Eloquent\Collection {#1568 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1583 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1119962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "171" "field_page_end" => null "field_url" 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№10 (с. 171)
Условие. №10 (с. 171)
скриншот условия
10. (2)
а) $\tg x > 100$;
б) $\tg 20x > -20$;
в) $\tg \frac{5x}{8} < \frac{5}{8}$;
г) $\tg \left(11x + \frac{\pi}{4}\right) + 100 \le 0$;
д) $\tg \left(x + \frac{9\pi}{4}\right) \le \pi$.
Решение 2 (rus). №10 (с. 171)
а) Решим неравенство $tg(x) > 100$.
Общее решение неравенства вида $tg(t) > a$ имеет вид $arctg(a) + \pi n < t < \frac{\pi}{2} + \pi n$, где $n \in \mathbb{Z}$.
В данном случае $t = x$ и $a = 100$.
Подставляя эти значения, получаем решение:
$arctg(100) + \pi n < x < \frac{\pi}{2} + \pi n$, где $n \in \mathbb{Z}$.
Ответ: $x \in (arctg(100) + \pi n; \frac{\pi}{2} + \pi n)$, $n \in \mathbb{Z}$.
б) Решим неравенство $tg(20x) > -20$.
Пусть $t = 20x$. Неравенство принимает вид $tg(t) > -20$.
Решение этого неравенства: $arctg(-20) + \pi n < t < \frac{\pi}{2} + \pi n$, где $n \in \mathbb{Z}$.
Используя свойство арктангенса $arctg(-a) = -arctg(a)$, получаем:
$-arctg(20) + \pi n < t < \frac{\pi}{2} + \pi n$.
Теперь произведем обратную замену $t = 20x$:
$-arctg(20) + \pi n < 20x < \frac{\pi}{2} + \pi n$.
Разделим все части неравенства на 20, чтобы найти $x$:
$\frac{-arctg(20) + \pi n}{20} < x < \frac{\frac{\pi}{2} + \pi n}{20}$
$-\frac{arctg(20)}{20} + \frac{\pi n}{20} < x < \frac{\pi}{40} + \frac{\pi n}{20}$, где $n \in \mathbb{Z}$.
Ответ: $x \in (-\frac{arctg(20)}{20} + \frac{\pi n}{20}; \frac{\pi}{40} + \frac{\pi n}{20})$, $n \in \mathbb{Z}$.
в) Решим неравенство $tg(\frac{5x}{8}) < \frac{5}{8}$.
Общее решение неравенства вида $tg(t) < a$ имеет вид $-\frac{\pi}{2} + \pi n < t < arctg(a) + \pi n$, где $n \in \mathbb{Z}$.
В данном случае $t = \frac{5x}{8}$ и $a = \frac{5}{8}$.
Подставляем эти значения в общее решение:
$-\frac{\pi}{2} + \pi n < \frac{5x}{8} < arctg(\frac{5}{8}) + \pi n$.
Чтобы выразить $x$, умножим все части неравенства на $\frac{8}{5}$:
$(-\frac{\pi}{2} + \pi n) \cdot \frac{8}{5} < x < (arctg(\frac{5}{8}) + \pi n) \cdot \frac{8}{5}$
$-\frac{4\pi}{5} + \frac{8\pi n}{5} < x < \frac{8}{5}arctg(\frac{5}{8}) + \frac{8\pi n}{5}$, где $n \in \mathbb{Z}$.
Ответ: $x \in (-\frac{4\pi}{5} + \frac{8\pi n}{5}; \frac{8}{5}arctg(\frac{5}{8}) + \frac{8\pi n}{5})$, $n \in \mathbb{Z}$.
г) Решим неравенство $tg(11x + \frac{\pi}{4}) + 100 \le 0$.
Сначала преобразуем неравенство:
$tg(11x + \frac{\pi}{4}) \le -100$.
Общее решение неравенства вида $tg(t) \le a$ имеет вид $-\frac{\pi}{2} + \pi n < t \le arctg(a) + \pi n$, где $n \in \mathbb{Z}$.
В нашем случае $t = 11x + \frac{\pi}{4}$ и $a = -100$.
Подставляем значения:
$-\frac{\pi}{2} + \pi n < 11x + \frac{\pi}{4} \le arctg(-100) + \pi n$.
Используя свойство $arctg(-a) = -arctg(a)$, получаем:
$-\frac{\pi}{2} + \pi n < 11x + \frac{\pi}{4} \le -arctg(100) + \pi n$.
Вычтем $\frac{\pi}{4}$ из всех частей неравенства:
$-\frac{\pi}{2} - \frac{\pi}{4} + \pi n < 11x \le -arctg(100) - \frac{\pi}{4} + \pi n$
$-\frac{3\pi}{4} + \pi n < 11x \le -arctg(100) - \frac{\pi}{4} + \pi n$.
Теперь разделим все части на 11:
$\frac{-\frac{3\pi}{4} + \pi n}{11} < x \le \frac{-arctg(100) - \frac{\pi}{4} + \pi n}{11}$
$-\frac{3\pi}{44} + \frac{\pi n}{11} < x \le -\frac{arctg(100)}{11} - \frac{\pi}{44} + \frac{\pi n}{11}$, где $n \in \mathbb{Z}$.
Ответ: $x \in (-\frac{3\pi}{44} + \frac{\pi n}{11}; -\frac{arctg(100)}{11} - \frac{\pi}{44} + \frac{\pi n}{11}]$, $n \in \mathbb{Z}$.
д) Решим неравенство $tg(x + \frac{9\pi}{4}) \le \pi$.
Сначала упростим аргумент тангенса, используя его периодичность с периодом $\pi$. Заметим, что $\frac{9\pi}{4} = 2\pi + \frac{\pi}{4}$.
$tg(x + \frac{9\pi}{4}) = tg(x + \frac{\pi}{4} + 2\pi) = tg(x + \frac{\pi}{4})$.
Неравенство принимает вид: $tg(x + \frac{\pi}{4}) \le \pi$.
Общее решение неравенства вида $tg(t) \le a$ имеет вид $-\frac{\pi}{2} + \pi n < t \le arctg(a) + \pi n$, где $n \in \mathbb{Z}$.
В данном случае $t = x + \frac{\pi}{4}$ и $a = \pi$.
Подставляем значения:
$-\frac{\pi}{2} + \pi n < x + \frac{\pi}{4} \le arctg(\pi) + \pi n$.
Вычтем $\frac{\pi}{4}$ из всех частей неравенства, чтобы выразить $x$:
$-\frac{\pi}{2} - \frac{\pi}{4} + \pi n < x \le arctg(\pi) - \frac{\pi}{4} + \pi n$
$-\frac{3\pi}{4} + \pi n < x \le arctg(\pi) - \frac{\pi}{4} + \pi n$, где $n \in \mathbb{Z}$.
Ответ: $x \in (-\frac{3\pi}{4} + \pi n; arctg(\pi) - \frac{\pi}{4} + \pi n]$, $n \in \mathbb{Z}$.
Другие задания:
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