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Номер 562, страница 166 - гдз по алгебре 9 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2019 - 2026
ISBN: 978-601-317-424-2
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 9 классе
IV. Тригонометрия. 22. Синус, косинус, тангенс и котангенс произвольного угла - номер 562, страница 166.
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" => 1108295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-562" "field_display_title" => "562" "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 {#1071 #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" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "147" "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 {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #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" => 4298 "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 {#1041 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1047 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 #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" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "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 {#1041} "field_class" => Illuminate\Database\Eloquent\Collection {#1043} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045} "field_author" => Illuminate\Database\Eloquent\Collection {#1047} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052} "field_country" => Illuminate\Database\Eloquent\Collection {#1054} "field_city" => Illuminate\Database\Eloquent\Collection {#1056} "field_series" => Illuminate\Database\Eloquent\Collection {#1058} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #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 {#1068 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "147" "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 {#1038} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #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 {#1104 #items: array:2 [ 0 => App\Models\Branch {#1112 #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" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1113 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "147" "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 {#1144 #items: array:1 [ 0 => App\Models\Book {#1114 #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" => 4298 "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 {#1116 #items: array:1 [ 0 => App\Models\Term {#1115 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ 0 => App\Models\Term {#1118 #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" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" …3 ] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #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 {#1119 #items: array:1 [ 0 => App\Models\Term {#1120 #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" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #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 {#1121 #items: array:3 [ 0 => App\Models\Term {#1122 #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" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "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 {#1123 #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" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "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 {#1124 #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" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "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 {#1125 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\Term {#1127 #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 {#1128 #items: array:1 [ 0 => App\Models\Term {#1129 #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 {#1130 #items: array:1 [ 0 => App\Models\Term {#1131 #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 {#1132 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1133 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1134 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1135 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1136 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1137 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1138 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1140 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1141 #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" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "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 {#1116} "field_class" => Illuminate\Database\Eloquent\Collection {#1117} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1119} "field_author" => Illuminate\Database\Eloquent\Collection {#1121} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1125} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1126} "field_country" => Illuminate\Database\Eloquent\Collection {#1128} "field_city" => Illuminate\Database\Eloquent\Collection {#1130} "field_series" => Illuminate\Database\Eloquent\Collection {#1132} "field_umk" => Illuminate\Database\Eloquent\Collection {#1133} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1134} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1135} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1136} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1137} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1138} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1139} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1140} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1141} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #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 {#1142 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1113} "field_page_start" => "147" "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 {#1144} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1142} ] #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 {#1143 #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" => 1107664 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "22. Синус, косинус, тангенс и котангенс произвольного угла" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "158" "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 {#1146 #items: array:1 [ 0 => App\Models\Book {#1114} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1147 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107664 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "22. Синус, косинус, тангенс и котангенс произвольного угла" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1145} "field_page_start" => "158" "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 {#1146} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1147} ] #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 {#1101 #items: array:3 [ 0 => App\Models\Element {#1094 #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" => 1276917 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086 #items: array:1 [ 0 => App\Models\Edition {#1093 #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" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1091} "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 {#1089} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1090} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1087} "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" => "1108295" "type" => "task" ] "text" => "<p><strong>562.</strong> Расположите в порядке возрастания значения:</p><p><strong>а) синусов;</strong></p><p><strong>в) тангенсов;</strong></p><p><strong>б) косинусов;</strong></p><p><strong>г) котангенсов углов</strong> $- \frac{2\pi}{9}, \frac{\pi}{9}, - \frac{5\pi}{3}, \frac{11\pi}{9}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "562-1.jpg" "alt" => null "width" => "1391" "height" => 319 "path" => "/media/algebra_09/soltan-u19/0-1/562-1.webp?ts=1752834575" ] ] ] #original: array:7 [ "id" => 1276917 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086} "task" => array:2 [ "refs" => "1108295" "type" => "task" ] "text" => "<p><strong>562.</strong> Расположите в порядке возрастания значения:</p><p><strong>а) синусов;</strong></p><p><strong>в) тангенсов;</strong></p><p><strong>б) косинусов;</strong></p><p><strong>г) котангенсов углов</strong> $- \frac{2\pi}{9}, \frac{\pi}{9}, - \frac{5\pi}{3}, \frac{11\pi}{9}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "562-1.jpg" "alt" => null "width" => "1391" "height" => 319 "path" => "/media/algebra_09/soltan-u19/0-1/562-1.webp?ts=1752834575" ] ] ] #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 {#1088 #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" => 1278208 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ 0 => App\Models\Edition {#1085 #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" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1080} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1083} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1082} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1081} "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" => "1108295" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "562-1.jpg" "alt" => null "width" => "1291" "height" => 901 "path" => "/media/algebra_09/soltan-u19/1-1/562-1.webp?ts=1752831341" ] ] ] #original: array:6 [ "id" => 1278208 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1076} "task" => array:2 [ "refs" => "1108295" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "562-1.jpg" "alt" => null "width" => "1291" "height" => 901 "path" => "/media/algebra_09/soltan-u19/1-1/562-1.webp?ts=1752831341" ] ] ] #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\Element {#1078 #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" => 1554163 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Edition {#1079 #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" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1074 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1074} "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 {#1075} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1072} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103} "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" => "1108295" "type" => "task" ] "text" => "<p>Для того чтобы расположить значения тригонометрических функций в порядке возрастания, сначала приведем все углы к более удобному для сравнения виду и определим, в какой четверти находится каждый угол. Данные углы: $-\frac{2\pi}{9}$, $\frac{\pi}{9}$, $-\frac{5\pi}{3}$, $\frac{11\pi}{9}$.</p><p>Приведем углы к основному промежутку, используя периодичность и свойства функций:</p><p><strong>1.</strong> Угол $-\frac{2\pi}{9}$ находится в IV четверти.</p><p><strong>2.</strong> Угол $\frac{\pi}{9}$ находится в I четверти.</p><p><strong>3.</strong> Угол $-\frac{5\pi}{3}$ можно привести к углу в первой четверти: $-\frac{5\pi}{3} + 2\pi = \frac{-5\pi + 6\pi}{3} = \frac{\pi}{3}$.</p><p><strong>4.</strong> Угол $\frac{11\pi}{9}$ можно представить как $\pi + \frac{2\pi}{9}$, он находится в III четверти.</p><p><strong>а) синусов</strong></p><p>Нам нужно сравнить значения: $\sin(-\frac{2\pi}{9})$, $\sin(\frac{\pi}{9})$, $\sin(-\frac{5\pi}{3})$ и $\sin(\frac{11\pi}{9})$.</p><p>Используем свойства синуса и приведенные углы:</p><p>• $\sin(-\frac{2\pi}{9}) = -\sin(\frac{2\pi}{9})$, так как синус - нечетная функция. Угол $-\frac{2\pi}{9}$ находится в IV четверти, поэтому его синус отрицателен.</p><p>• $\sin(\frac{\pi}{9})$ – значение положительное, так как угол $\frac{\pi}{9}$ находится в I четверти.</p><p>• $\sin(-\frac{5\pi}{3}) = \sin(\frac{\pi}{3})$ – значение положительное, так как угол $\frac{\pi}{3}$ находится в I четверти.</p><p>• $\sin(\frac{11\pi}{9}) = \sin(\pi + \frac{2\pi}{9}) = -\sin(\frac{2\pi}{9})$. Угол $\frac{11\pi}{9}$ находится в III четверти, поэтому его синус отрицателен.</p><p>Сравниваем отрицательные значения: $\sin(-\frac{2\pi}{9})$ и $\sin(\frac{11\pi}{9})$. Оба они равны $-\sin(\frac{2\pi}{9})$, следовательно, $\sin(-\frac{2\pi}{9}) = \sin(\frac{11\pi}{9})$. Это наименьшие значения.</p><p>Сравниваем положительные значения: $\sin(\frac{\pi}{9})$ и $\sin(\frac{\pi}{3})$. В I четверти (на интервале $(0, \frac{\pi}{2})$) функция синус возрастает. Так как $\frac{\pi}{9} < \frac{\pi}{3}$, то $\sin(\frac{\pi}{9}) < \sin(\frac{\pi}{3})$.</p><p>Объединяя все, получаем следующий порядок возрастания: $\sin(-\frac{2\pi}{9}) = \sin(\frac{11\pi}{9}) < \sin(\frac{\pi}{9}) < \sin(-\frac{5\pi}{3})$.</p><p><strong>Ответ:</strong> $\sin(\frac{11\pi}{9})$, $\sin(-\frac{2\pi}{9})$, $\sin(\frac{\pi}{9})$, $\sin(-\frac{5\pi}{3})$.</p><p><strong>б) косинусов</strong></p><p>Нам нужно сравнить значения: $\cos(-\frac{2\pi}{9})$, $\cos(\frac{\pi}{9})$, $\cos(-\frac{5\pi}{3})$ и $\cos(\frac{11\pi}{9})$.</p><p>Используем свойства косинуса и приведенные углы:</p><p>• $\cos(-\frac{2\pi}{9}) = \cos(\frac{2\pi}{9})$, так как косинус - четная функция. Угол в IV четверти, косинус положителен.</p><p>• $\cos(\frac{\pi}{9})$ – значение положительное, угол в I четверти.</p><p>• $\cos(-\frac{5\pi}{3}) = \cos(\frac{\pi}{3})$ – значение положительное, угол в I четверти.</p><p>• $\cos(\frac{11\pi}{9}) = \cos(\pi + \frac{2\pi}{9}) = -\cos(\frac{2\pi}{9})$. Угол в III четверти, косинус отрицателен.</p><p>Единственное отрицательное значение, $\cos(\frac{11\pi}{9})$, является наименьшим.</p><p>Сравниваем положительные значения: $\cos(\frac{2\pi}{9})$, $\cos(\frac{\pi}{9})$ и $\cos(\frac{\pi}{3})$. В I четверти (на интервале $(0, \frac{\pi}{2})$) функция косинус убывает. Упорядочим углы: $\frac{\pi}{9} < \frac{2\pi}{9} < \frac{\pi}{3}$. Следовательно, для их косинусов порядок будет обратным: $\cos(\frac{\pi}{3}) < \cos(\frac{2\pi}{9}) < \cos(\frac{\pi}{9})$.</p><p>Объединяя все, получаем следующий порядок возрастания: $\cos(\frac{11\pi}{9}) < \cos(-\frac{5\pi}{3}) < \cos(-\frac{2\pi}{9}) < \cos(\frac{\pi}{9})$.</p><p><strong>Ответ:</strong> $\cos(\frac{11\pi}{9})$, $\cos(-\frac{5\pi}{3})$, $\cos(-\frac{2\pi}{9})$, $\cos(\frac{\pi}{9})$.</p><p><strong>в) тангенсов</strong></p><p>Нам нужно сравнить значения: $\tan(-\frac{2\pi}{9})$, $\tan(\frac{\pi}{9})$, $\tan(-\frac{5\pi}{3})$ и $\tan(\frac{11\pi}{9})$.</p><p>Используем свойства тангенса:</p><p>• $\tan(-\frac{2\pi}{9}) = -\tan(\frac{2\pi}{9})$. Угол в IV четверти, тангенс отрицателен.</p><p>• $\tan(\frac{\pi}{9})$ – значение положительное, угол в I четверти.</p><p>• $\tan(-\frac{5\pi}{3}) = \tan(\frac{\pi}{3})$ – значение положительное, угол в I четверти.</p><p>• $\tan(\frac{11\pi}{9}) = \tan(\pi + \frac{2\pi}{9}) = \tan(\frac{2\pi}{9})$. Угол в III четверти, тангенс положителен.</p><p>Единственное отрицательное значение, $\tan(-\frac{2\pi}{9})$, является наименьшим.</p><p>Сравниваем положительные значения: $\tan(\frac{\pi}{9})$, $\tan(\frac{\pi}{3})$ и $\tan(\frac{2\pi}{9})$. В I четверти (на интервале $(-\frac{\pi}{2}, \frac{\pi}{2})$) функция тангенс возрастает. Так как $\frac{\pi}{9} < \frac{2\pi}{9} < \frac{\pi}{3}$, то $\tan(\frac{\pi}{9}) < \tan(\frac{2\pi}{9}) < \tan(\frac{\pi}{3})$.</p><p>Объединяя все, получаем следующий порядок возрастания: $\tan(-\frac{2\pi}{9}) < \tan(\frac{\pi}{9}) < \tan(\frac{11\pi}{9}) < \tan(-\frac{5\pi}{3})$.</p><p><strong>Ответ:</strong> $\tan(-\frac{2\pi}{9})$, $\tan(\frac{\pi}{9})$, $\tan(\frac{11\pi}{9})$, $\tan(-\frac{5\pi}{3})$.</p><p><strong>г) котангенсов</strong></p><p>Нам нужно сравнить значения: $\cot(-\frac{2\pi}{9})$, $\cot(\frac{\pi}{9})$, $\cot(-\frac{5\pi}{3})$ и $\cot(\frac{11\pi}{9})$.</p><p>Используем свойства котангенса:</p><p>• $\cot(-\frac{2\pi}{9}) = -\cot(\frac{2\pi}{9})$. Угол в IV четверти, котангенс отрицателен.</p><p>• $\cot(\frac{\pi}{9})$ – значение положительное, угол в I четверти.</p><p>• $\cot(-\frac{5\pi}{3}) = \cot(\frac{\pi}{3})$ – значение положительное, угол в I четверти.</p><p>• $\cot(\frac{11\pi}{9}) = \cot(\pi + \frac{2\pi}{9}) = \cot(\frac{2\pi}{9})$. Угол в III четверти, котангенс положителен.</p><p>Единственное отрицательное значение, $\cot(-\frac{2\pi}{9})$, является наименьшим.</p><p>Сравниваем положительные значения: $\cot(\frac{\pi}{9})$, $\cot(\frac{\pi}{3})$ и $\cot(\frac{2\pi}{9})$. На интервале $(0, \pi)$ функция котангенс убывает. Упорядочим углы: $\frac{\pi}{9} < \frac{2\pi}{9} < \frac{\pi}{3}$. Следовательно, для их котангенсов порядок будет обратным: $\cot(\frac{\pi}{3}) < \cot(\frac{2\pi}{9}) < \cot(\frac{\pi}{9})$.</p><p>Объединяя все, получаем следующий порядок возрастания: $\cot(-\frac{2\pi}{9}) < \cot(-\frac{5\pi}{3}) < \cot(\frac{11\pi}{9}) < \cot(\frac{\pi}{9})$.</p><p><strong>Ответ:</strong> $\cot(-\frac{2\pi}{9})$, $\cot(-\frac{5\pi}{3})$, $\cot(\frac{11\pi}{9})$, $\cot(\frac{\pi}{9})$.</p>" ] #original: array:6 [ "id" => 1554163 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "1108295" "type" => "task" ] "text" => "<p>Для того чтобы расположить значения тригонометрических функций в порядке возрастания, сначала приведем все углы к более удобному для сравнения виду и определим, в какой четверти находится каждый угол. Данные углы: $-\frac{2\pi}{9}$, $\frac{\pi}{9}$, $-\frac{5\pi}{3}$, $\frac{11\pi}{9}$.</p><p>Приведем углы к основному промежутку, используя периодичность и свойства функций:</p><p><strong>1.</strong> Угол $-\frac{2\pi}{9}$ находится в IV четверти.</p><p><strong>2.</strong> Угол $\frac{\pi}{9}$ находится в I четверти.</p><p><strong>3.</strong> Угол $-\frac{5\pi}{3}$ можно привести к углу в первой четверти: $-\frac{5\pi}{3} + 2\pi = \frac{-5\pi + 6\pi}{3} = \frac{\pi}{3}$.</p><p><strong>4.</strong> Угол $\frac{11\pi}{9}$ можно представить как $\pi + \frac{2\pi}{9}$, он находится в III четверти.</p><p><strong>а) синусов</strong></p><p>Нам нужно сравнить значения: $\sin(-\frac{2\pi}{9})$, $\sin(\frac{\pi}{9})$, $\sin(-\frac{5\pi}{3})$ и $\sin(\frac{11\pi}{9})$.</p><p>Используем свойства синуса и приведенные углы:</p><p>• $\sin(-\frac{2\pi}{9}) = -\sin(\frac{2\pi}{9})$, так как синус - нечетная функция. Угол $-\frac{2\pi}{9}$ находится в IV четверти, поэтому его синус отрицателен.</p><p>• $\sin(\frac{\pi}{9})$ – значение положительное, так как угол $\frac{\pi}{9}$ находится в I четверти.</p><p>• $\sin(-\frac{5\pi}{3}) = \sin(\frac{\pi}{3})$ – значение положительное, так как угол $\frac{\pi}{3}$ находится в I четверти.</p><p>• $\sin(\frac{11\pi}{9}) = \sin(\pi + \frac{2\pi}{9}) = -\sin(\frac{2\pi}{9})$. Угол $\frac{11\pi}{9}$ находится в III четверти, поэтому его синус отрицателен.</p><p>Сравниваем отрицательные значения: $\sin(-\frac{2\pi}{9})$ и $\sin(\frac{11\pi}{9})$. Оба они равны $-\sin(\frac{2\pi}{9})$, следовательно, $\sin(-\frac{2\pi}{9}) = \sin(\frac{11\pi}{9})$. Это наименьшие значения.</p><p>Сравниваем положительные значения: $\sin(\frac{\pi}{9})$ и $\sin(\frac{\pi}{3})$. В I четверти (на интервале $(0, \frac{\pi}{2})$) функция синус возрастает. Так как $\frac{\pi}{9} < \frac{\pi}{3}$, то $\sin(\frac{\pi}{9}) < \sin(\frac{\pi}{3})$.</p><p>Объединяя все, получаем следующий порядок возрастания: $\sin(-\frac{2\pi}{9}) = \sin(\frac{11\pi}{9}) < \sin(\frac{\pi}{9}) < \sin(-\frac{5\pi}{3})$.</p><p><strong>Ответ:</strong> $\sin(\frac{11\pi}{9})$, $\sin(-\frac{2\pi}{9})$, $\sin(\frac{\pi}{9})$, $\sin(-\frac{5\pi}{3})$.</p><p><strong>б) косинусов</strong></p><p>Нам нужно сравнить значения: $\cos(-\frac{2\pi}{9})$, $\cos(\frac{\pi}{9})$, $\cos(-\frac{5\pi}{3})$ и $\cos(\frac{11\pi}{9})$.</p><p>Используем свойства косинуса и приведенные углы:</p><p>• $\cos(-\frac{2\pi}{9}) = \cos(\frac{2\pi}{9})$, так как косинус - четная функция. Угол в IV четверти, косинус положителен.</p><p>• $\cos(\frac{\pi}{9})$ – значение положительное, угол в I четверти.</p><p>• $\cos(-\frac{5\pi}{3}) = \cos(\frac{\pi}{3})$ – значение положительное, угол в I четверти.</p><p>• $\cos(\frac{11\pi}{9}) = \cos(\pi + \frac{2\pi}{9}) = -\cos(\frac{2\pi}{9})$. Угол в III четверти, косинус отрицателен.</p><p>Единственное отрицательное значение, $\cos(\frac{11\pi}{9})$, является наименьшим.</p><p>Сравниваем положительные значения: $\cos(\frac{2\pi}{9})$, $\cos(\frac{\pi}{9})$ и $\cos(\frac{\pi}{3})$. В I четверти (на интервале $(0, \frac{\pi}{2})$) функция косинус убывает. Упорядочим углы: $\frac{\pi}{9} < \frac{2\pi}{9} < \frac{\pi}{3}$. Следовательно, для их косинусов порядок будет обратным: $\cos(\frac{\pi}{3}) < \cos(\frac{2\pi}{9}) < \cos(\frac{\pi}{9})$.</p><p>Объединяя все, получаем следующий порядок возрастания: $\cos(\frac{11\pi}{9}) < \cos(-\frac{5\pi}{3}) < \cos(-\frac{2\pi}{9}) < \cos(\frac{\pi}{9})$.</p><p><strong>Ответ:</strong> $\cos(\frac{11\pi}{9})$, $\cos(-\frac{5\pi}{3})$, $\cos(-\frac{2\pi}{9})$, $\cos(\frac{\pi}{9})$.</p><p><strong>в) тангенсов</strong></p><p>Нам нужно сравнить значения: $\tan(-\frac{2\pi}{9})$, $\tan(\frac{\pi}{9})$, $\tan(-\frac{5\pi}{3})$ и $\tan(\frac{11\pi}{9})$.</p><p>Используем свойства тангенса:</p><p>• $\tan(-\frac{2\pi}{9}) = -\tan(\frac{2\pi}{9})$. Угол в IV четверти, тангенс отрицателен.</p><p>• $\tan(\frac{\pi}{9})$ – значение положительное, угол в I четверти.</p><p>• $\tan(-\frac{5\pi}{3}) = \tan(\frac{\pi}{3})$ – значение положительное, угол в I четверти.</p><p>• $\tan(\frac{11\pi}{9}) = \tan(\pi + \frac{2\pi}{9}) = \tan(\frac{2\pi}{9})$. Угол в III четверти, тангенс положителен.</p><p>Единственное отрицательное значение, $\tan(-\frac{2\pi}{9})$, является наименьшим.</p><p>Сравниваем положительные значения: $\tan(\frac{\pi}{9})$, $\tan(\frac{\pi}{3})$ и $\tan(\frac{2\pi}{9})$. В I четверти (на интервале $(-\frac{\pi}{2}, \frac{\pi}{2})$) функция тангенс возрастает. Так как $\frac{\pi}{9} < \frac{2\pi}{9} < \frac{\pi}{3}$, то $\tan(\frac{\pi}{9}) < \tan(\frac{2\pi}{9}) < \tan(\frac{\pi}{3})$.</p><p>Объединяя все, получаем следующий порядок возрастания: $\tan(-\frac{2\pi}{9}) < \tan(\frac{\pi}{9}) < \tan(\frac{11\pi}{9}) < \tan(-\frac{5\pi}{3})$.</p><p><strong>Ответ:</strong> $\tan(-\frac{2\pi}{9})$, $\tan(\frac{\pi}{9})$, $\tan(\frac{11\pi}{9})$, $\tan(-\frac{5\pi}{3})$.</p><p><strong>г) котангенсов</strong></p><p>Нам нужно сравнить значения: $\cot(-\frac{2\pi}{9})$, $\cot(\frac{\pi}{9})$, $\cot(-\frac{5\pi}{3})$ и $\cot(\frac{11\pi}{9})$.</p><p>Используем свойства котангенса:</p><p>• $\cot(-\frac{2\pi}{9}) = -\cot(\frac{2\pi}{9})$. Угол в IV четверти, котангенс отрицателен.</p><p>• $\cot(\frac{\pi}{9})$ – значение положительное, угол в I четверти.</p><p>• $\cot(-\frac{5\pi}{3}) = \cot(\frac{\pi}{3})$ – значение положительное, угол в I четверти.</p><p>• $\cot(\frac{11\pi}{9}) = \cot(\pi + \frac{2\pi}{9}) = \cot(\frac{2\pi}{9})$. Угол в III четверти, котангенс положителен.</p><p>Единственное отрицательное значение, $\cot(-\frac{2\pi}{9})$, является наименьшим.</p><p>Сравниваем положительные значения: $\cot(\frac{\pi}{9})$, $\cot(\frac{\pi}{3})$ и $\cot(\frac{2\pi}{9})$. На интервале $(0, \pi)$ функция котангенс убывает. Упорядочим углы: $\frac{\pi}{9} < \frac{2\pi}{9} < \frac{\pi}{3}$. Следовательно, для их котангенсов порядок будет обратным: $\cot(\frac{\pi}{3}) < \cot(\frac{2\pi}{9}) < \cot(\frac{\pi}{9})$.</p><p>Объединяя все, получаем следующий порядок возрастания: $\cot(-\frac{2\pi}{9}) < \cot(-\frac{5\pi}{3}) < \cot(\frac{11\pi}{9}) < \cot(\frac{\pi}{9})$.</p><p><strong>Ответ:</strong> $\cot(-\frac{2\pi}{9})$, $\cot(-\frac{5\pi}{3})$, $\cot(\frac{11\pi}{9})$, $\cot(\frac{\pi}{9})$.</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" => "1108296" "type" => "task" ] "previous" => array:2 [ "refs" => "1108294" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1097 #items: array:1 [ 0 => App\Models\Book {#1114} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1099 #items: array:1 [ 0 => App\Models\BookPage {#1070 #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" => 1097908 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-166" "field_display_title" => "166" "field_folder" => "folder1" "field_image_name" => "166" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ 0 => App\Models\Book {#1114} ] #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 {#1111 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1154 #items: array:1 [ 0 => App\Models\Element {#1153 #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" => 1261059 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261059 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155 …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" => "1097909" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097907" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1250 #items: array:9 [ 0 => App\Models\Task {#1266 #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" => 1108288 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-555" "field_display_title" => "555" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1267 …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 {#1268 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1269 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1270 …2} "content" => Illuminate\Database\Eloquent\Collection {#1273 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1279 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108288 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-555" "field_display_title" => "555" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1267 …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 {#1268 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1269 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1270 …2} "content" => Illuminate\Database\Eloquent\Collection {#1273 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1279 …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 {#1272 #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" => 1108289 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-556" "field_display_title" => "556" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1271 …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 {#1278 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1280 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1276 …2} "content" => Illuminate\Database\Eloquent\Collection {#1275 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1293 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108289 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-556" "field_display_title" => "556" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1271 …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 {#1278 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1280 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1276 …2} "content" => Illuminate\Database\Eloquent\Collection {#1275 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1293 …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 => "*" ] } 2 => App\Models\Task {#1277 #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" => 1108290 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-557" "field_display_title" => "557" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1274 …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 {#1292 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1294 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1290 …2} "content" => Illuminate\Database\Eloquent\Collection {#1289 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1307 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108290 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-557" "field_display_title" => "557" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1274 …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 {#1292 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1294 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1290 …2} "content" => Illuminate\Database\Eloquent\Collection {#1289 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1307 …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 => "*" ] } 3 => App\Models\Task {#1291 #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" => 1108291 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-558" "field_display_title" => "558" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1288 …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 {#1306 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1308 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1304 …2} "content" => Illuminate\Database\Eloquent\Collection {#1303 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1321 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108291 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-558" "field_display_title" => "558" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1288 …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 {#1306 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1308 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1304 …2} "content" => Illuminate\Database\Eloquent\Collection {#1303 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1321 …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 => "*" ] } 4 => App\Models\Task {#1305 #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" => 1108292 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-559" "field_display_title" => "559" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1302 …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 {#1320 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1322 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1318 …2} "content" => Illuminate\Database\Eloquent\Collection {#1317 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1335 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108292 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-559" "field_display_title" => "559" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1302 …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 {#1320 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1322 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1318 …2} "content" => Illuminate\Database\Eloquent\Collection {#1317 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1335 …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 => "*" ] } 5 => App\Models\Task {#1319 #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" => 1108293 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-560" "field_display_title" => "560" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1316 …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 {#1334 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1336 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1332 …2} "content" => Illuminate\Database\Eloquent\Collection {#1331 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1349 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108293 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-560" "field_display_title" => "560" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1316 …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 {#1334 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1336 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1332 …2} "content" => Illuminate\Database\Eloquent\Collection {#1331 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1349 …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 => "*" ] } 6 => App\Models\Task {#1333 #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" => 1108294 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-561" "field_display_title" => "561" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1330 …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 {#1348 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1350 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1346 …2} "content" => Illuminate\Database\Eloquent\Collection {#1345 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1363 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108294 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-561" "field_display_title" => "561" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1330 …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 {#1348 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1350 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1346 …2} "content" => Illuminate\Database\Eloquent\Collection {#1345 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1363 …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 => "*" ] } 7 => App\Models\Task {#1347 #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" => 1108295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "166" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-562" "field_display_title" => "562" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1344 …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 {#1362 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1364 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1360 …2} "content" => 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№562 (с. 166)
Условие. №562 (с. 166)
скриншот условия
562. Расположите в порядке возрастания значения:
а) синусов;
в) тангенсов;
б) косинусов;
г) котангенсов углов $- \frac{2\pi}{9}, \frac{\pi}{9}, - \frac{5\pi}{3}, \frac{11\pi}{9}$.
Решение. №562 (с. 166)
Решение 2 (rus). №562 (с. 166)
Для того чтобы расположить значения тригонометрических функций в порядке возрастания, сначала приведем все углы к более удобному для сравнения виду и определим, в какой четверти находится каждый угол. Данные углы: $-\frac{2\pi}{9}$, $\frac{\pi}{9}$, $-\frac{5\pi}{3}$, $\frac{11\pi}{9}$.
Приведем углы к основному промежутку, используя периодичность и свойства функций:
1. Угол $-\frac{2\pi}{9}$ находится в IV четверти.
2. Угол $\frac{\pi}{9}$ находится в I четверти.
3. Угол $-\frac{5\pi}{3}$ можно привести к углу в первой четверти: $-\frac{5\pi}{3} + 2\pi = \frac{-5\pi + 6\pi}{3} = \frac{\pi}{3}$.
4. Угол $\frac{11\pi}{9}$ можно представить как $\pi + \frac{2\pi}{9}$, он находится в III четверти.
а) синусов
Нам нужно сравнить значения: $\sin(-\frac{2\pi}{9})$, $\sin(\frac{\pi}{9})$, $\sin(-\frac{5\pi}{3})$ и $\sin(\frac{11\pi}{9})$.
Используем свойства синуса и приведенные углы:
• $\sin(-\frac{2\pi}{9}) = -\sin(\frac{2\pi}{9})$, так как синус - нечетная функция. Угол $-\frac{2\pi}{9}$ находится в IV четверти, поэтому его синус отрицателен.
• $\sin(\frac{\pi}{9})$ – значение положительное, так как угол $\frac{\pi}{9}$ находится в I четверти.
• $\sin(-\frac{5\pi}{3}) = \sin(\frac{\pi}{3})$ – значение положительное, так как угол $\frac{\pi}{3}$ находится в I четверти.
• $\sin(\frac{11\pi}{9}) = \sin(\pi + \frac{2\pi}{9}) = -\sin(\frac{2\pi}{9})$. Угол $\frac{11\pi}{9}$ находится в III четверти, поэтому его синус отрицателен.
Сравниваем отрицательные значения: $\sin(-\frac{2\pi}{9})$ и $\sin(\frac{11\pi}{9})$. Оба они равны $-\sin(\frac{2\pi}{9})$, следовательно, $\sin(-\frac{2\pi}{9}) = \sin(\frac{11\pi}{9})$. Это наименьшие значения.
Сравниваем положительные значения: $\sin(\frac{\pi}{9})$ и $\sin(\frac{\pi}{3})$. В I четверти (на интервале $(0, \frac{\pi}{2})$) функция синус возрастает. Так как $\frac{\pi}{9} < \frac{\pi}{3}$, то $\sin(\frac{\pi}{9}) < \sin(\frac{\pi}{3})$.
Объединяя все, получаем следующий порядок возрастания: $\sin(-\frac{2\pi}{9}) = \sin(\frac{11\pi}{9}) < \sin(\frac{\pi}{9}) < \sin(-\frac{5\pi}{3})$.
Ответ: $\sin(\frac{11\pi}{9})$, $\sin(-\frac{2\pi}{9})$, $\sin(\frac{\pi}{9})$, $\sin(-\frac{5\pi}{3})$.
б) косинусов
Нам нужно сравнить значения: $\cos(-\frac{2\pi}{9})$, $\cos(\frac{\pi}{9})$, $\cos(-\frac{5\pi}{3})$ и $\cos(\frac{11\pi}{9})$.
Используем свойства косинуса и приведенные углы:
• $\cos(-\frac{2\pi}{9}) = \cos(\frac{2\pi}{9})$, так как косинус - четная функция. Угол в IV четверти, косинус положителен.
• $\cos(\frac{\pi}{9})$ – значение положительное, угол в I четверти.
• $\cos(-\frac{5\pi}{3}) = \cos(\frac{\pi}{3})$ – значение положительное, угол в I четверти.
• $\cos(\frac{11\pi}{9}) = \cos(\pi + \frac{2\pi}{9}) = -\cos(\frac{2\pi}{9})$. Угол в III четверти, косинус отрицателен.
Единственное отрицательное значение, $\cos(\frac{11\pi}{9})$, является наименьшим.
Сравниваем положительные значения: $\cos(\frac{2\pi}{9})$, $\cos(\frac{\pi}{9})$ и $\cos(\frac{\pi}{3})$. В I четверти (на интервале $(0, \frac{\pi}{2})$) функция косинус убывает. Упорядочим углы: $\frac{\pi}{9} < \frac{2\pi}{9} < \frac{\pi}{3}$. Следовательно, для их косинусов порядок будет обратным: $\cos(\frac{\pi}{3}) < \cos(\frac{2\pi}{9}) < \cos(\frac{\pi}{9})$.
Объединяя все, получаем следующий порядок возрастания: $\cos(\frac{11\pi}{9}) < \cos(-\frac{5\pi}{3}) < \cos(-\frac{2\pi}{9}) < \cos(\frac{\pi}{9})$.
Ответ: $\cos(\frac{11\pi}{9})$, $\cos(-\frac{5\pi}{3})$, $\cos(-\frac{2\pi}{9})$, $\cos(\frac{\pi}{9})$.
в) тангенсов
Нам нужно сравнить значения: $\tan(-\frac{2\pi}{9})$, $\tan(\frac{\pi}{9})$, $\tan(-\frac{5\pi}{3})$ и $\tan(\frac{11\pi}{9})$.
Используем свойства тангенса:
• $\tan(-\frac{2\pi}{9}) = -\tan(\frac{2\pi}{9})$. Угол в IV четверти, тангенс отрицателен.
• $\tan(\frac{\pi}{9})$ – значение положительное, угол в I четверти.
• $\tan(-\frac{5\pi}{3}) = \tan(\frac{\pi}{3})$ – значение положительное, угол в I четверти.
• $\tan(\frac{11\pi}{9}) = \tan(\pi + \frac{2\pi}{9}) = \tan(\frac{2\pi}{9})$. Угол в III четверти, тангенс положителен.
Единственное отрицательное значение, $\tan(-\frac{2\pi}{9})$, является наименьшим.
Сравниваем положительные значения: $\tan(\frac{\pi}{9})$, $\tan(\frac{\pi}{3})$ и $\tan(\frac{2\pi}{9})$. В I четверти (на интервале $(-\frac{\pi}{2}, \frac{\pi}{2})$) функция тангенс возрастает. Так как $\frac{\pi}{9} < \frac{2\pi}{9} < \frac{\pi}{3}$, то $\tan(\frac{\pi}{9}) < \tan(\frac{2\pi}{9}) < \tan(\frac{\pi}{3})$.
Объединяя все, получаем следующий порядок возрастания: $\tan(-\frac{2\pi}{9}) < \tan(\frac{\pi}{9}) < \tan(\frac{11\pi}{9}) < \tan(-\frac{5\pi}{3})$.
Ответ: $\tan(-\frac{2\pi}{9})$, $\tan(\frac{\pi}{9})$, $\tan(\frac{11\pi}{9})$, $\tan(-\frac{5\pi}{3})$.
г) котангенсов
Нам нужно сравнить значения: $\cot(-\frac{2\pi}{9})$, $\cot(\frac{\pi}{9})$, $\cot(-\frac{5\pi}{3})$ и $\cot(\frac{11\pi}{9})$.
Используем свойства котангенса:
• $\cot(-\frac{2\pi}{9}) = -\cot(\frac{2\pi}{9})$. Угол в IV четверти, котангенс отрицателен.
• $\cot(\frac{\pi}{9})$ – значение положительное, угол в I четверти.
• $\cot(-\frac{5\pi}{3}) = \cot(\frac{\pi}{3})$ – значение положительное, угол в I четверти.
• $\cot(\frac{11\pi}{9}) = \cot(\pi + \frac{2\pi}{9}) = \cot(\frac{2\pi}{9})$. Угол в III четверти, котангенс положителен.
Единственное отрицательное значение, $\cot(-\frac{2\pi}{9})$, является наименьшим.
Сравниваем положительные значения: $\cot(\frac{\pi}{9})$, $\cot(\frac{\pi}{3})$ и $\cot(\frac{2\pi}{9})$. На интервале $(0, \pi)$ функция котангенс убывает. Упорядочим углы: $\frac{\pi}{9} < \frac{2\pi}{9} < \frac{\pi}{3}$. Следовательно, для их котангенсов порядок будет обратным: $\cot(\frac{\pi}{3}) < \cot(\frac{2\pi}{9}) < \cot(\frac{\pi}{9})$.
Объединяя все, получаем следующий порядок возрастания: $\cot(-\frac{2\pi}{9}) < \cot(-\frac{5\pi}{3}) < \cot(\frac{11\pi}{9}) < \cot(\frac{\pi}{9})$.
Ответ: $\cot(-\frac{2\pi}{9})$, $\cot(-\frac{5\pi}{3})$, $\cot(\frac{11\pi}{9})$, $\cot(\frac{\pi}{9})$.
Другие задания:
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