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Номер 711, страница 205 - 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. 27. Формулы тригонометрических функций двойного и половинного углов - номер 711, страница 205.
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" => 1108452 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "205" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-711" "field_display_title" => "711" "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 [ 0 => App\Models\Term {#1042 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ …10] #original: array:10 [ …10] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ 0 => App\Models\Term {#1044 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ 0 => App\Models\Term {#1046 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false …25 } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1047 #items: array:3 [ 0 => App\Models\Term {#1048 …30} 1 => App\Models\Term {#1049 …30} 2 => App\Models\Term {#1050 …30} ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ 0 => App\Models\Term {#1053 …30} ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 …30} ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:1 [ 0 => App\Models\Term {#1057 …30} ] #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 [ "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 {#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 [ "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 {#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" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "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 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "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 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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" => 1107669 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "27. Формулы тригонометрических функций двойного и половинного углов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "203" "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" => 1107669 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "27. Формулы тригонометрических функций двойного и половинного углов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1145} "field_page_start" => "203" "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" => 1277066 "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 [ 0 => App\Models\Term {#1092 …30} ] #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" => "1108452" "type" => "task" ] "text" => "<p><strong>711.</strong> Преобразуйте каждое из выражений, используя формулу двойного угла:</p><p><strong>a)</strong> $sin 8\alpha$, $cos 6\alpha$, $tg 4\alpha$;</p><p><strong>б)</strong> $sin 5\alpha$, $cos 3\alpha$, $ctg \alpha$;</p><p><strong>в)</strong> $sin \frac{\alpha}{2}$, $cos \frac{\alpha}{3}$, $tg \frac{\alpha}{4}$;</p><p><strong>г)</strong> $sin \frac{\alpha}{5}$, $cos \frac{2\alpha}{7}$, $ctg \frac{3\alpha}{4}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "711-1.jpg" "alt" => null "width" => "1549" "height" => 447 "path" => "/media/algebra_09/soltan-u19/0-1/711-1.webp?ts=1752834710" ] ] ] #original: array:7 [ "id" => 1277066 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1086} "task" => array:2 [ "refs" => "1108452" "type" => "task" ] "text" => "<p><strong>711.</strong> Преобразуйте каждое из выражений, используя формулу двойного угла:</p><p><strong>a)</strong> $sin 8\alpha$, $cos 6\alpha$, $tg 4\alpha$;</p><p><strong>б)</strong> $sin 5\alpha$, $cos 3\alpha$, $ctg \alpha$;</p><p><strong>в)</strong> $sin \frac{\alpha}{2}$, $cos \frac{\alpha}{3}$, $tg \frac{\alpha}{4}$;</p><p><strong>г)</strong> $sin \frac{\alpha}{5}$, $cos \frac{2\alpha}{7}$, $ctg \frac{3\alpha}{4}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "711-1.jpg" "alt" => null "width" => "1549" "height" => 447 "path" => "/media/algebra_09/soltan-u19/0-1/711-1.webp?ts=1752834710" ] ] ] #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" => 1278357 "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 [ 0 => App\Models\Term {#1084 …30} ] #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" => "1108452" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "711-1.jpg" "alt" => null "width" => "895" "height" => 2265 "path" => "/media/algebra_09/soltan-u19/1-1/711-1.webp?ts=1752831506" ] ] ] #original: array:6 [ "id" => 1278357 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1076} "task" => array:2 [ "refs" => "1108452" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "711-1.jpg" "alt" => null "width" => "895" "height" => 2265 "path" => "/media/algebra_09/soltan-u19/1-1/711-1.webp?ts=1752831506" ] ] ] #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" => 1554312 "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 [ 0 => App\Models\Term {#1077 …30} ] #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" => "1108452" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Для преобразования $\sin 8\alpha$ используем формулу синуса двойного угла $\sin(2x) = 2 \sin x \cos x$. Полагая $8\alpha = 2 \cdot 4\alpha$, получаем $x = 4\alpha$:<br>$\sin 8\alpha = 2 \sin 4\alpha \cos 4\alpha$.<br>Ответ: $2 \sin 4\alpha \cos 4\alpha$.</p><p>Для преобразования $\cos 6\alpha$ используем формулу косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$. Полагая $6\alpha = 2 \cdot 3\alpha$, получаем $x = 3\alpha$:<br>$\cos 6\alpha = \cos^2 3\alpha - \sin^2 3\alpha$.<br>Ответ: $\cos^2 3\alpha - \sin^2 3\alpha$.</p><p>Для преобразования $\tan 4\alpha$ используем формулу тангенса двойного угла $\tan(2x) = \frac{2 \tan x}{1 - \tan^2 x}$. Полагая $4\alpha = 2 \cdot 2\alpha$, получаем $x = 2\alpha$:<br>$\tan 4\alpha = \frac{2 \tan 2\alpha}{1 - \tan^2 2\alpha}$.<br>Ответ: $\frac{2 \tan 2\alpha}{1 - \tan^2 2\alpha}$.</p><p><strong>б)</strong></p><p>Для преобразования $\sin 5\alpha$ используем формулу синуса двойного угла $\sin(2x) = 2 \sin x \cos x$. Полагая $5\alpha = 2 \cdot \frac{5\alpha}{2}$, получаем $x = \frac{5\alpha}{2}$:<br>$\sin 5\alpha = 2 \sin \frac{5\alpha}{2} \cos \frac{5\alpha}{2}$.<br>Ответ: $2 \sin \frac{5\alpha}{2} \cos \frac{5\alpha}{2}$.</p><p>Для преобразования $\cos 3\alpha$ используем формулу косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$. Полагая $3\alpha = 2 \cdot \frac{3\alpha}{2}$, получаем $x = \frac{3\alpha}{2}$:<br>$\cos 3\alpha = \cos^2 \frac{3\alpha}{2} - \sin^2 \frac{3\alpha}{2}$.<br>Ответ: $\cos^2 \frac{3\alpha}{2} - \sin^2 \frac{3\alpha}{2}$.</p><p>Для преобразования $\cot \alpha$ используем формулу котангенса двойного угла $\cot(2x) = \frac{\cot^2 x - 1}{2 \cot x}$. Полагая $\alpha = 2 \cdot \frac{\alpha}{2}$, получаем $x = \frac{\alpha}{2}$:<br>$\cot \alpha = \frac{\cot^2 \frac{\alpha}{2} - 1}{2 \cot \frac{\alpha}{2}}$.<br>Ответ: $\frac{\cot^2 \frac{\alpha}{2} - 1}{2 \cot \frac{\alpha}{2}}$.</p><p><strong>в)</strong></p><p>Для преобразования $\sin \frac{\alpha}{2}$ используем формулу синуса двойного угла $\sin(2x) = 2 \sin x \cos x$. Полагая $\frac{\alpha}{2} = 2 \cdot \frac{\alpha}{4}$, получаем $x = \frac{\alpha}{4}$:<br>$\sin \frac{\alpha}{2} = 2 \sin \frac{\alpha}{4} \cos \frac{\alpha}{4}$.<br>Ответ: $2 \sin \frac{\alpha}{4} \cos \frac{\alpha}{4}$.</p><p>Для преобразования $\cos \frac{\alpha}{3}$ используем формулу косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$. Полагая $\frac{\alpha}{3} = 2 \cdot \frac{\alpha}{6}$, получаем $x = \frac{\alpha}{6}$:<br>$\cos \frac{\alpha}{3} = \cos^2 \frac{\alpha}{6} - \sin^2 \frac{\alpha}{6}$.<br>Ответ: $\cos^2 \frac{\alpha}{6} - \sin^2 \frac{\alpha}{6}$.</p><p>Для преобразования $\tan \frac{\alpha}{4}$ используем формулу тангенса двойного угла $\tan(2x) = \frac{2 \tan x}{1 - \tan^2 x}$. Полагая $\frac{\alpha}{4} = 2 \cdot \frac{\alpha}{8}$, получаем $x = \frac{\alpha}{8}$:<br>$\tan \frac{\alpha}{4} = \frac{2 \tan \frac{\alpha}{8}}{1 - \tan^2 \frac{\alpha}{8}}$.<br>Ответ: $\frac{2 \tan \frac{\alpha}{8}}{1 - \tan^2 \frac{\alpha}{8}}$.</p><p><strong>г)</strong></p><p>Для преобразования $\sin \frac{\alpha}{5}$ используем формулу синуса двойного угла $\sin(2x) = 2 \sin x \cos x$. Полагая $\frac{\alpha}{5} = 2 \cdot \frac{\alpha}{10}$, получаем $x = \frac{\alpha}{10}$:<br>$\sin \frac{\alpha}{5} = 2 \sin \frac{\alpha}{10} \cos \frac{\alpha}{10}$.<br>Ответ: $2 \sin \frac{\alpha}{10} \cos \frac{\alpha}{10}$.</p><p>Для преобразования $\cos \frac{2\alpha}{7}$ используем формулу косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$. Полагая $\frac{2\alpha}{7} = 2 \cdot \frac{\alpha}{7}$, получаем $x = \frac{\alpha}{7}$:<br>$\cos \frac{2\alpha}{7} = \cos^2 \frac{\alpha}{7} - \sin^2 \frac{\alpha}{7}$.<br>Ответ: $\cos^2 \frac{\alpha}{7} - \sin^2 \frac{\alpha}{7}$.</p><p>Для преобразования $\cot \frac{3\alpha}{4}$ используем формулу котангенса двойного угла $\cot(2x) = \frac{\cot^2 x - 1}{2 \cot x}$. Полагая $\frac{3\alpha}{4} = 2 \cdot \frac{3\alpha}{8}$, получаем $x = \frac{3\alpha}{8}$:<br>$\cot \frac{3\alpha}{4} = \frac{\cot^2 \frac{3\alpha}{8} - 1}{2 \cot \frac{3\alpha}{8}}$.<br>Ответ: $\frac{\cot^2 \frac{3\alpha}{8} - 1}{2 \cot \frac{3\alpha}{8}}$.</p>" ] #original: array:6 [ "id" => 1554312 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "1108452" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Для преобразования $\sin 8\alpha$ используем формулу синуса двойного угла $\sin(2x) = 2 \sin x \cos x$. Полагая $8\alpha = 2 \cdot 4\alpha$, получаем $x = 4\alpha$:<br>$\sin 8\alpha = 2 \sin 4\alpha \cos 4\alpha$.<br>Ответ: $2 \sin 4\alpha \cos 4\alpha$.</p><p>Для преобразования $\cos 6\alpha$ используем формулу косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$. Полагая $6\alpha = 2 \cdot 3\alpha$, получаем $x = 3\alpha$:<br>$\cos 6\alpha = \cos^2 3\alpha - \sin^2 3\alpha$.<br>Ответ: $\cos^2 3\alpha - \sin^2 3\alpha$.</p><p>Для преобразования $\tan 4\alpha$ используем формулу тангенса двойного угла $\tan(2x) = \frac{2 \tan x}{1 - \tan^2 x}$. Полагая $4\alpha = 2 \cdot 2\alpha$, получаем $x = 2\alpha$:<br>$\tan 4\alpha = \frac{2 \tan 2\alpha}{1 - \tan^2 2\alpha}$.<br>Ответ: $\frac{2 \tan 2\alpha}{1 - \tan^2 2\alpha}$.</p><p><strong>б)</strong></p><p>Для преобразования $\sin 5\alpha$ используем формулу синуса двойного угла $\sin(2x) = 2 \sin x \cos x$. Полагая $5\alpha = 2 \cdot \frac{5\alpha}{2}$, получаем $x = \frac{5\alpha}{2}$:<br>$\sin 5\alpha = 2 \sin \frac{5\alpha}{2} \cos \frac{5\alpha}{2}$.<br>Ответ: $2 \sin \frac{5\alpha}{2} \cos \frac{5\alpha}{2}$.</p><p>Для преобразования $\cos 3\alpha$ используем формулу косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$. Полагая $3\alpha = 2 \cdot \frac{3\alpha}{2}$, получаем $x = \frac{3\alpha}{2}$:<br>$\cos 3\alpha = \cos^2 \frac{3\alpha}{2} - \sin^2 \frac{3\alpha}{2}$.<br>Ответ: $\cos^2 \frac{3\alpha}{2} - \sin^2 \frac{3\alpha}{2}$.</p><p>Для преобразования $\cot \alpha$ используем формулу котангенса двойного угла $\cot(2x) = \frac{\cot^2 x - 1}{2 \cot x}$. Полагая $\alpha = 2 \cdot \frac{\alpha}{2}$, получаем $x = \frac{\alpha}{2}$:<br>$\cot \alpha = \frac{\cot^2 \frac{\alpha}{2} - 1}{2 \cot \frac{\alpha}{2}}$.<br>Ответ: $\frac{\cot^2 \frac{\alpha}{2} - 1}{2 \cot \frac{\alpha}{2}}$.</p><p><strong>в)</strong></p><p>Для преобразования $\sin \frac{\alpha}{2}$ используем формулу синуса двойного угла $\sin(2x) = 2 \sin x \cos x$. Полагая $\frac{\alpha}{2} = 2 \cdot \frac{\alpha}{4}$, получаем $x = \frac{\alpha}{4}$:<br>$\sin \frac{\alpha}{2} = 2 \sin \frac{\alpha}{4} \cos \frac{\alpha}{4}$.<br>Ответ: $2 \sin \frac{\alpha}{4} \cos \frac{\alpha}{4}$.</p><p>Для преобразования $\cos \frac{\alpha}{3}$ используем формулу косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$. Полагая $\frac{\alpha}{3} = 2 \cdot \frac{\alpha}{6}$, получаем $x = \frac{\alpha}{6}$:<br>$\cos \frac{\alpha}{3} = \cos^2 \frac{\alpha}{6} - \sin^2 \frac{\alpha}{6}$.<br>Ответ: $\cos^2 \frac{\alpha}{6} - \sin^2 \frac{\alpha}{6}$.</p><p>Для преобразования $\tan \frac{\alpha}{4}$ используем формулу тангенса двойного угла $\tan(2x) = \frac{2 \tan x}{1 - \tan^2 x}$. Полагая $\frac{\alpha}{4} = 2 \cdot \frac{\alpha}{8}$, получаем $x = \frac{\alpha}{8}$:<br>$\tan \frac{\alpha}{4} = \frac{2 \tan \frac{\alpha}{8}}{1 - \tan^2 \frac{\alpha}{8}}$.<br>Ответ: $\frac{2 \tan \frac{\alpha}{8}}{1 - \tan^2 \frac{\alpha}{8}}$.</p><p><strong>г)</strong></p><p>Для преобразования $\sin \frac{\alpha}{5}$ используем формулу синуса двойного угла $\sin(2x) = 2 \sin x \cos x$. Полагая $\frac{\alpha}{5} = 2 \cdot \frac{\alpha}{10}$, получаем $x = \frac{\alpha}{10}$:<br>$\sin \frac{\alpha}{5} = 2 \sin \frac{\alpha}{10} \cos \frac{\alpha}{10}$.<br>Ответ: $2 \sin \frac{\alpha}{10} \cos \frac{\alpha}{10}$.</p><p>Для преобразования $\cos \frac{2\alpha}{7}$ используем формулу косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$. Полагая $\frac{2\alpha}{7} = 2 \cdot \frac{\alpha}{7}$, получаем $x = \frac{\alpha}{7}$:<br>$\cos \frac{2\alpha}{7} = \cos^2 \frac{\alpha}{7} - \sin^2 \frac{\alpha}{7}$.<br>Ответ: $\cos^2 \frac{\alpha}{7} - \sin^2 \frac{\alpha}{7}$.</p><p>Для преобразования $\cot \frac{3\alpha}{4}$ используем формулу котангенса двойного угла $\cot(2x) = \frac{\cot^2 x - 1}{2 \cot x}$. Полагая $\frac{3\alpha}{4} = 2 \cdot \frac{3\alpha}{8}$, получаем $x = \frac{3\alpha}{8}$:<br>$\cot \frac{3\alpha}{4} = \frac{\cot^2 \frac{3\alpha}{8} - 1}{2 \cot \frac{3\alpha}{8}}$.<br>Ответ: $\frac{\cot^2 \frac{3\alpha}{8} - 1}{2 \cot \frac{3\alpha}{8}}$.</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" => "1108453" "type" => "task" ] "previous" => array:2 [ "refs" => "1108451" "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" => 1097947 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "205" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-205" "field_display_title" => "205" "field_folder" => "folder1" "field_image_name" => "205" "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" => 1261098 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155 #items: array:1 [ 0 => App\Models\Edition {#1093} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1097947" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "205-1.jpg" "alt" => null "width" => "2009" "height" => 2972 "path" => "/media/algebra_09/soltan-u19/0-folder1/205-1.webp?ts=1752141221" ] ] ] #original: array:6 [ "id" => 1261098 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" 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"1097947" "type" => "book_page" ] ] #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 {#1371 #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" => 1108453 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "205" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-712" "field_display_title" => "712" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1384 #items: array:1 [ 0 => App\Models\Term {#1036} ] #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 {#1385 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1386 #items: array:1 [ 0 => App\Models\Branch {#1112} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1387 #items: array:2 [ 0 => App\Models\Branch {#1112} 1 => App\Models\Branch {#1143} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1390 #items: array:3 [ 0 => App\Models\Element {#1399 …30} 1 => App\Models\Element {#1401 …30} 2 => App\Models\Element {#1403 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108454" "type" => "task" ] "previous" => array:2 [ "refs" => "1108452" "type" => 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№711 (с. 205)
Условие. №711 (с. 205)
скриншот условия
711. Преобразуйте каждое из выражений, используя формулу двойного угла:
a) $sin 8\alpha$, $cos 6\alpha$, $tg 4\alpha$;
б) $sin 5\alpha$, $cos 3\alpha$, $ctg \alpha$;
в) $sin \frac{\alpha}{2}$, $cos \frac{\alpha}{3}$, $tg \frac{\alpha}{4}$;
г) $sin \frac{\alpha}{5}$, $cos \frac{2\alpha}{7}$, $ctg \frac{3\alpha}{4}$.
Решение. №711 (с. 205)
Решение 2 (rus). №711 (с. 205)
а)
Для преобразования $\sin 8\alpha$ используем формулу синуса двойного угла $\sin(2x) = 2 \sin x \cos x$. Полагая $8\alpha = 2 \cdot 4\alpha$, получаем $x = 4\alpha$:
$\sin 8\alpha = 2 \sin 4\alpha \cos 4\alpha$.
Ответ: $2 \sin 4\alpha \cos 4\alpha$.
Для преобразования $\cos 6\alpha$ используем формулу косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$. Полагая $6\alpha = 2 \cdot 3\alpha$, получаем $x = 3\alpha$:
$\cos 6\alpha = \cos^2 3\alpha - \sin^2 3\alpha$.
Ответ: $\cos^2 3\alpha - \sin^2 3\alpha$.
Для преобразования $\tan 4\alpha$ используем формулу тангенса двойного угла $\tan(2x) = \frac{2 \tan x}{1 - \tan^2 x}$. Полагая $4\alpha = 2 \cdot 2\alpha$, получаем $x = 2\alpha$:
$\tan 4\alpha = \frac{2 \tan 2\alpha}{1 - \tan^2 2\alpha}$.
Ответ: $\frac{2 \tan 2\alpha}{1 - \tan^2 2\alpha}$.
б)
Для преобразования $\sin 5\alpha$ используем формулу синуса двойного угла $\sin(2x) = 2 \sin x \cos x$. Полагая $5\alpha = 2 \cdot \frac{5\alpha}{2}$, получаем $x = \frac{5\alpha}{2}$:
$\sin 5\alpha = 2 \sin \frac{5\alpha}{2} \cos \frac{5\alpha}{2}$.
Ответ: $2 \sin \frac{5\alpha}{2} \cos \frac{5\alpha}{2}$.
Для преобразования $\cos 3\alpha$ используем формулу косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$. Полагая $3\alpha = 2 \cdot \frac{3\alpha}{2}$, получаем $x = \frac{3\alpha}{2}$:
$\cos 3\alpha = \cos^2 \frac{3\alpha}{2} - \sin^2 \frac{3\alpha}{2}$.
Ответ: $\cos^2 \frac{3\alpha}{2} - \sin^2 \frac{3\alpha}{2}$.
Для преобразования $\cot \alpha$ используем формулу котангенса двойного угла $\cot(2x) = \frac{\cot^2 x - 1}{2 \cot x}$. Полагая $\alpha = 2 \cdot \frac{\alpha}{2}$, получаем $x = \frac{\alpha}{2}$:
$\cot \alpha = \frac{\cot^2 \frac{\alpha}{2} - 1}{2 \cot \frac{\alpha}{2}}$.
Ответ: $\frac{\cot^2 \frac{\alpha}{2} - 1}{2 \cot \frac{\alpha}{2}}$.
в)
Для преобразования $\sin \frac{\alpha}{2}$ используем формулу синуса двойного угла $\sin(2x) = 2 \sin x \cos x$. Полагая $\frac{\alpha}{2} = 2 \cdot \frac{\alpha}{4}$, получаем $x = \frac{\alpha}{4}$:
$\sin \frac{\alpha}{2} = 2 \sin \frac{\alpha}{4} \cos \frac{\alpha}{4}$.
Ответ: $2 \sin \frac{\alpha}{4} \cos \frac{\alpha}{4}$.
Для преобразования $\cos \frac{\alpha}{3}$ используем формулу косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$. Полагая $\frac{\alpha}{3} = 2 \cdot \frac{\alpha}{6}$, получаем $x = \frac{\alpha}{6}$:
$\cos \frac{\alpha}{3} = \cos^2 \frac{\alpha}{6} - \sin^2 \frac{\alpha}{6}$.
Ответ: $\cos^2 \frac{\alpha}{6} - \sin^2 \frac{\alpha}{6}$.
Для преобразования $\tan \frac{\alpha}{4}$ используем формулу тангенса двойного угла $\tan(2x) = \frac{2 \tan x}{1 - \tan^2 x}$. Полагая $\frac{\alpha}{4} = 2 \cdot \frac{\alpha}{8}$, получаем $x = \frac{\alpha}{8}$:
$\tan \frac{\alpha}{4} = \frac{2 \tan \frac{\alpha}{8}}{1 - \tan^2 \frac{\alpha}{8}}$.
Ответ: $\frac{2 \tan \frac{\alpha}{8}}{1 - \tan^2 \frac{\alpha}{8}}$.
г)
Для преобразования $\sin \frac{\alpha}{5}$ используем формулу синуса двойного угла $\sin(2x) = 2 \sin x \cos x$. Полагая $\frac{\alpha}{5} = 2 \cdot \frac{\alpha}{10}$, получаем $x = \frac{\alpha}{10}$:
$\sin \frac{\alpha}{5} = 2 \sin \frac{\alpha}{10} \cos \frac{\alpha}{10}$.
Ответ: $2 \sin \frac{\alpha}{10} \cos \frac{\alpha}{10}$.
Для преобразования $\cos \frac{2\alpha}{7}$ используем формулу косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$. Полагая $\frac{2\alpha}{7} = 2 \cdot \frac{\alpha}{7}$, получаем $x = \frac{\alpha}{7}$:
$\cos \frac{2\alpha}{7} = \cos^2 \frac{\alpha}{7} - \sin^2 \frac{\alpha}{7}$.
Ответ: $\cos^2 \frac{\alpha}{7} - \sin^2 \frac{\alpha}{7}$.
Для преобразования $\cot \frac{3\alpha}{4}$ используем формулу котангенса двойного угла $\cot(2x) = \frac{\cot^2 x - 1}{2 \cot x}$. Полагая $\frac{3\alpha}{4} = 2 \cdot \frac{3\alpha}{8}$, получаем $x = \frac{3\alpha}{8}$:
$\cot \frac{3\alpha}{4} = \frac{\cot^2 \frac{3\alpha}{8} - 1}{2 \cot \frac{3\alpha}{8}}$.
Ответ: $\frac{\cot^2 \frac{3\alpha}{8} - 1}{2 \cot \frac{3\alpha}{8}}$.
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