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Номер 715, страница 206 - 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 715, страница 206.
App\Models\Task {#1024 // 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" => 1108456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "206" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-715" "field_display_title" => "715" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #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 {#333 #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 {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #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 {#1032 #items: array:1 [ 0 => App\Models\Term {#1034 #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 {#1035 #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" => 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 {#1037 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1039 #items: array:3 [ 0 => App\Models\Term {#1040 #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 {#1041 #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 {#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: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 {#1043 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ 0 => App\Models\Term {#1045 #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 {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 #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 {#1048 #items: array:1 [ 0 => App\Models\Term {#1049 #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 {#1050 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 #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 {#1056 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 #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 {#1032} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037} "field_author" => Illuminate\Database\Eloquent\Collection {#1039} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044} "field_country" => Illuminate\Database\Eloquent\Collection {#1046} "field_city" => Illuminate\Database\Eloquent\Collection {#1048} "field_series" => Illuminate\Database\Eloquent\Collection {#1050} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055} "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 {#1056} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059} "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 {#1060 #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 {#333} "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 {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #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 {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #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 {#1063 #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 {#1064 #items: array:1 [ 0 => App\Models\Book {#1030} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1065 #items: array:1 [ 0 => App\Models\Branch {#1066 #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 {#1067 #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 {#1068 #items: array:1 [ 0 => App\Models\Book {#1069 …30} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 #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 {#1067} "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 {#1068} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 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 {#1063} "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 {#1064} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1065} ] #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 {#1098 #items: array:3 [ 0 => App\Models\Element {#1099 #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" => 1277070 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1101 #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 {#1102 #items: array:1 [ 0 => App\Models\Term {#1103 …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 {#1104 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 #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 {#1102} "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 {#1104} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106} "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" => "1108456" "type" => "task" ] "text" => "<p><strong>715.</strong> Вычислите:</p><p><strong>а)</strong> sin $2\alpha$, если cos $\alpha = -\frac{5}{13}$ и $\frac{\pi}{2} < \alpha < \pi$;</p><p><strong>г)</strong> cos $2\alpha$, если tg $\alpha = 3$;</p><p><strong>б)</strong> cos $2\alpha$, если sin $\alpha = -0,8$ и $\frac{3\pi}{2} < \alpha < 2\pi$;</p><p><strong>д)</strong> sin $4\alpha$, если tg $\alpha = 2$;</p><p><strong>в)</strong> tg $2\alpha$, если cos $\alpha = -0,6$ и $\pi < \alpha < \frac{3\pi}{2}$;</p><p><strong>е)</strong> cos $4\alpha$, если sin $\alpha = \frac{1}{3}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "715-1.jpg" "alt" => null "width" => "1557" "height" => 529 "path" => "/media/algebra_09/soltan-u19/0-1/715-1.webp?ts=1752834710" ] ] ] #original: array:7 [ "id" => 1277070 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1108456" "type" => "task" ] "text" => "<p><strong>715.</strong> Вычислите:</p><p><strong>а)</strong> sin $2\alpha$, если cos $\alpha = -\frac{5}{13}$ и $\frac{\pi}{2} < \alpha < \pi$;</p><p><strong>г)</strong> cos $2\alpha$, если tg $\alpha = 3$;</p><p><strong>б)</strong> cos $2\alpha$, если sin $\alpha = -0,8$ и $\frac{3\pi}{2} < \alpha < 2\pi$;</p><p><strong>д)</strong> sin $4\alpha$, если tg $\alpha = 2$;</p><p><strong>в)</strong> tg $2\alpha$, если cos $\alpha = -0,6$ и $\pi < \alpha < \frac{3\pi}{2}$;</p><p><strong>е)</strong> cos $4\alpha$, если sin $\alpha = \frac{1}{3}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "715-1.jpg" "alt" => null "width" => "1557" "height" => 529 "path" => "/media/algebra_09/soltan-u19/0-1/715-1.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 {#1107 #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" => 1278361 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1109 #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 {#1110 #items: array:1 [ 0 => App\Models\Term {#1111 …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 {#1112 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 #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 {#1110} "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 {#1112} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114} "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" => "1108456" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "715-1.jpg" "alt" => null "width" => "1261" "height" => 1000 "path" => "/media/algebra_09/soltan-u19/1-1/715-1.webp?ts=1752831506" ] 1 => array:5 [ "name" => "715-2.jpg" "alt" => null "width" => "1340" "height" => 1959 "path" => "/media/algebra_09/soltan-u19/1-1/715-2.webp?ts=1752831506" ] ] ] #original: array:6 [ "id" => 1278361 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1108456" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "715-1.jpg" "alt" => null "width" => "1261" "height" => 1000 "path" => "/media/algebra_09/soltan-u19/1-1/715-1.webp?ts=1752831506" ] 1 => array:5 [ "name" => "715-2.jpg" "alt" => null "width" => "1340" "height" => 1959 "path" => "/media/algebra_09/soltan-u19/1-1/715-2.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 {#1115 #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" => 1554316 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1117 #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 {#1118 #items: array:1 [ 0 => App\Models\Term {#1119 …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 {#1120 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 #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 {#1118} "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 {#1120} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122} "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" => "1108456" "type" => "task" ] "text" => "<p><strong>а)</strong> Для вычисления $sin(2\alpha)$ воспользуемся формулой синуса двойного угла: $sin(2\alpha) = 2sin(\alpha)cos(\alpha)$.<br>Нам дан $cos(\alpha) = -\frac{5}{13}$. Необходимо найти $sin(\alpha)$. Используем основное тригонометрическое тождество: $sin^2(\alpha) + cos^2(\alpha) = 1$.<br>$sin^2(\alpha) = 1 - cos^2(\alpha) = 1 - \left(-\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{169 - 25}{169} = \frac{144}{169}$.<br>$sin(\alpha) = \pm\sqrt{\frac{144}{169}} = \pm\frac{12}{13}$.<br>По условию, угол $\alpha$ принадлежит второй четверти ($\frac{\pi}{2} < \alpha < \pi$), где $sin(\alpha)$ имеет положительное значение. Значит, $sin(\alpha) = \frac{12}{13}$.<br>Теперь подставляем значения в формулу для $sin(2\alpha)$:<br>$sin(2\alpha) = 2 \cdot \frac{12}{13} \cdot \left(-\frac{5}{13}\right) = -\frac{120}{169}$.<br>Ответ: $-\frac{120}{169}$.</p><p><strong>б)</strong> Для вычисления $cos(2\alpha)$ используем формулу, связывающую его с $sin(\alpha)$: $cos(2\alpha) = 1 - 2sin^2(\alpha)$.<br>Нам дан $sin(\alpha) = -0,8 = -\frac{4}{5}$.<br>Подставляем это значение в формулу:<br>$cos(2\alpha) = 1 - 2 \cdot (-0,8)^2 = 1 - 2 \cdot 0,64 = 1 - 1,28 = -0,28$.<br>В дробях: $cos(2\alpha) = 1 - 2 \cdot \left(-\frac{4}{5}\right)^2 = 1 - 2 \cdot \frac{16}{25} = 1 - \frac{32}{25} = \frac{25 - 32}{25} = -\frac{7}{25}$.<br>Условие $\frac{3\pi}{2} < \alpha < 2\pi$ (четвертая четверть) согласуется со знаком $sin(\alpha) < 0$, но для вычисления $cos(2\alpha)$ по данной формуле оно не требовалось.<br>Ответ: $-0,28$.</p><p><strong>в)</strong> Для вычисления $tg(2\alpha)$ воспользуемся формулой тангенса двойного угла: $tg(2\alpha) = \frac{2tg(\alpha)}{1 - tg^2(\alpha)}$.<br>Сначала найдем $tg(\alpha)$. Нам дан $cos(\alpha) = -0,6 = -\frac{3}{5}$.<br>Найдем $sin(\alpha)$ из тождества $sin^2(\alpha) + cos^2(\alpha) = 1$:<br>$sin^2(\alpha) = 1 - (-0,6)^2 = 1 - 0,36 = 0,64$.<br>$sin(\alpha) = \pm\sqrt{0,64} = \pm0,8$.<br>По условию, угол $\alpha$ находится в третьей четверти ($\pi < \alpha < \frac{3\pi}{2}$), где и синус, и косинус отрицательны. Значит, $sin(\alpha) = -0,8 = -\frac{4}{5}$.<br>Теперь найдем $tg(\alpha) = \frac{sin(\alpha)}{cos(\alpha)} = \frac{-0,8}{-0,6} = \frac{8}{6} = \frac{4}{3}$.<br>Подставляем значение $tg(\alpha)$ в формулу для $tg(2\alpha)$:<br>$tg(2\alpha) = \frac{2 \cdot \frac{4}{3}}{1 - (\frac{4}{3})^2} = \frac{\frac{8}{3}}{1 - \frac{16}{9}} = \frac{\frac{8}{3}}{\frac{9 - 16}{9}} = \frac{\frac{8}{3}}{-\frac{7}{9}} = \frac{8}{3} \cdot \left(-\frac{9}{7}\right) = -\frac{24}{7}$.<br>Ответ: $-\frac{24}{7}$.</p><p><strong>г)</strong> Для вычисления $cos(2\alpha)$ при известном $tg(\alpha)$ удобно использовать формулу, выражающую косинус двойного угла через тангенс: $cos(2\alpha) = \frac{1 - tg^2(\alpha)}{1 + tg^2(\alpha)}$.<br>Нам дано $tg(\alpha) = 3$.<br>Подставляем это значение в формулу:<br>$cos(2\alpha) = \frac{1 - 3^2}{1 + 3^2} = \frac{1 - 9}{1 + 9} = \frac{-8}{10} = -\frac{4}{5}$.<br>Ответ: $-\frac{4}{5}$.</p><p><strong>д)</strong> Для вычисления $sin(4\alpha)$ используем формулу синуса двойного угла: $sin(4\alpha) = 2sin(2\alpha)cos(2\alpha)$.<br>Сначала найдем $sin(2\alpha)$ и $cos(2\alpha)$, используя формулы их выражения через $tg(\alpha) = 2$:<br>$sin(2\alpha) = \frac{2tg(\alpha)}{1 + tg^2(\alpha)} = \frac{2 \cdot 2}{1 + 2^2} = \frac{4}{1 + 4} = \frac{4}{5}$.<br>$cos(2\alpha) = \frac{1 - tg^2(\alpha)}{1 + tg^2(\alpha)} = \frac{1 - 2^2}{1 + 2^2} = \frac{1 - 4}{1 + 4} = -\frac{3}{5}$.<br>Теперь вычисляем $sin(4\alpha)$:<br>$sin(4\alpha) = 2 \cdot sin(2\alpha) \cdot cos(2\alpha) = 2 \cdot \frac{4}{5} \cdot \left(-\frac{3}{5}\right) = -\frac{24}{25}$.<br>Ответ: $-\frac{24}{25}$.</p><p><strong>е)</strong> Для вычисления $cos(4\alpha)$ представим его как косинус двойного угла от $2\alpha$: $cos(4\alpha) = cos(2 \cdot 2\alpha)$.<br>Воспользуемся одной из формул косинуса двойного угла, например, $cos(2x) = 2cos^2(x) - 1$. В нашем случае $x = 2\alpha$, так что $cos(4\alpha) = 2cos^2(2\alpha) - 1$.<br>Сначала нужно найти $cos(2\alpha)$. Используем формулу $cos(2\alpha) = 1 - 2sin^2(\alpha)$, так как нам дан $sin(\alpha) = \frac{1}{3}$.<br>$cos(2\alpha) = 1 - 2 \cdot \left(\frac{1}{3}\right)^2 = 1 - 2 \cdot \frac{1}{9} = 1 - \frac{2}{9} = \frac{7}{9}$.<br>Теперь подставляем найденное значение $cos(2\alpha)$ в формулу для $cos(4\alpha)$:<br>$cos(4\alpha) = 2 \cdot \left(\frac{7}{9}\right)^2 - 1 = 2 \cdot \frac{49}{81} - 1 = \frac{98}{81} - \frac{81}{81} = \frac{17}{81}$.<br>Ответ: $\frac{17}{81}$.</p>" ] #original: array:6 [ "id" => 1554316 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1108456" "type" => "task" ] "text" => "<p><strong>а)</strong> Для вычисления $sin(2\alpha)$ воспользуемся формулой синуса двойного угла: $sin(2\alpha) = 2sin(\alpha)cos(\alpha)$.<br>Нам дан $cos(\alpha) = -\frac{5}{13}$. Необходимо найти $sin(\alpha)$. Используем основное тригонометрическое тождество: $sin^2(\alpha) + cos^2(\alpha) = 1$.<br>$sin^2(\alpha) = 1 - cos^2(\alpha) = 1 - \left(-\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{169 - 25}{169} = \frac{144}{169}$.<br>$sin(\alpha) = \pm\sqrt{\frac{144}{169}} = \pm\frac{12}{13}$.<br>По условию, угол $\alpha$ принадлежит второй четверти ($\frac{\pi}{2} < \alpha < \pi$), где $sin(\alpha)$ имеет положительное значение. Значит, $sin(\alpha) = \frac{12}{13}$.<br>Теперь подставляем значения в формулу для $sin(2\alpha)$:<br>$sin(2\alpha) = 2 \cdot \frac{12}{13} \cdot \left(-\frac{5}{13}\right) = -\frac{120}{169}$.<br>Ответ: $-\frac{120}{169}$.</p><p><strong>б)</strong> Для вычисления $cos(2\alpha)$ используем формулу, связывающую его с $sin(\alpha)$: $cos(2\alpha) = 1 - 2sin^2(\alpha)$.<br>Нам дан $sin(\alpha) = -0,8 = -\frac{4}{5}$.<br>Подставляем это значение в формулу:<br>$cos(2\alpha) = 1 - 2 \cdot (-0,8)^2 = 1 - 2 \cdot 0,64 = 1 - 1,28 = -0,28$.<br>В дробях: $cos(2\alpha) = 1 - 2 \cdot \left(-\frac{4}{5}\right)^2 = 1 - 2 \cdot \frac{16}{25} = 1 - \frac{32}{25} = \frac{25 - 32}{25} = -\frac{7}{25}$.<br>Условие $\frac{3\pi}{2} < \alpha < 2\pi$ (четвертая четверть) согласуется со знаком $sin(\alpha) < 0$, но для вычисления $cos(2\alpha)$ по данной формуле оно не требовалось.<br>Ответ: $-0,28$.</p><p><strong>в)</strong> Для вычисления $tg(2\alpha)$ воспользуемся формулой тангенса двойного угла: $tg(2\alpha) = \frac{2tg(\alpha)}{1 - tg^2(\alpha)}$.<br>Сначала найдем $tg(\alpha)$. Нам дан $cos(\alpha) = -0,6 = -\frac{3}{5}$.<br>Найдем $sin(\alpha)$ из тождества $sin^2(\alpha) + cos^2(\alpha) = 1$:<br>$sin^2(\alpha) = 1 - (-0,6)^2 = 1 - 0,36 = 0,64$.<br>$sin(\alpha) = \pm\sqrt{0,64} = \pm0,8$.<br>По условию, угол $\alpha$ находится в третьей четверти ($\pi < \alpha < \frac{3\pi}{2}$), где и синус, и косинус отрицательны. Значит, $sin(\alpha) = -0,8 = -\frac{4}{5}$.<br>Теперь найдем $tg(\alpha) = \frac{sin(\alpha)}{cos(\alpha)} = \frac{-0,8}{-0,6} = \frac{8}{6} = \frac{4}{3}$.<br>Подставляем значение $tg(\alpha)$ в формулу для $tg(2\alpha)$:<br>$tg(2\alpha) = \frac{2 \cdot \frac{4}{3}}{1 - (\frac{4}{3})^2} = \frac{\frac{8}{3}}{1 - \frac{16}{9}} = \frac{\frac{8}{3}}{\frac{9 - 16}{9}} = \frac{\frac{8}{3}}{-\frac{7}{9}} = \frac{8}{3} \cdot \left(-\frac{9}{7}\right) = -\frac{24}{7}$.<br>Ответ: $-\frac{24}{7}$.</p><p><strong>г)</strong> Для вычисления $cos(2\alpha)$ при известном $tg(\alpha)$ удобно использовать формулу, выражающую косинус двойного угла через тангенс: $cos(2\alpha) = \frac{1 - tg^2(\alpha)}{1 + tg^2(\alpha)}$.<br>Нам дано $tg(\alpha) = 3$.<br>Подставляем это значение в формулу:<br>$cos(2\alpha) = \frac{1 - 3^2}{1 + 3^2} = \frac{1 - 9}{1 + 9} = \frac{-8}{10} = -\frac{4}{5}$.<br>Ответ: $-\frac{4}{5}$.</p><p><strong>д)</strong> Для вычисления $sin(4\alpha)$ используем формулу синуса двойного угла: $sin(4\alpha) = 2sin(2\alpha)cos(2\alpha)$.<br>Сначала найдем $sin(2\alpha)$ и $cos(2\alpha)$, используя формулы их выражения через $tg(\alpha) = 2$:<br>$sin(2\alpha) = \frac{2tg(\alpha)}{1 + tg^2(\alpha)} = \frac{2 \cdot 2}{1 + 2^2} = \frac{4}{1 + 4} = \frac{4}{5}$.<br>$cos(2\alpha) = \frac{1 - tg^2(\alpha)}{1 + tg^2(\alpha)} = \frac{1 - 2^2}{1 + 2^2} = \frac{1 - 4}{1 + 4} = -\frac{3}{5}$.<br>Теперь вычисляем $sin(4\alpha)$:<br>$sin(4\alpha) = 2 \cdot sin(2\alpha) \cdot cos(2\alpha) = 2 \cdot \frac{4}{5} \cdot \left(-\frac{3}{5}\right) = -\frac{24}{25}$.<br>Ответ: $-\frac{24}{25}$.</p><p><strong>е)</strong> Для вычисления $cos(4\alpha)$ представим его как косинус двойного угла от $2\alpha$: $cos(4\alpha) = cos(2 \cdot 2\alpha)$.<br>Воспользуемся одной из формул косинуса двойного угла, например, $cos(2x) = 2cos^2(x) - 1$. В нашем случае $x = 2\alpha$, так что $cos(4\alpha) = 2cos^2(2\alpha) - 1$.<br>Сначала нужно найти $cos(2\alpha)$. Используем формулу $cos(2\alpha) = 1 - 2sin^2(\alpha)$, так как нам дан $sin(\alpha) = \frac{1}{3}$.<br>$cos(2\alpha) = 1 - 2 \cdot \left(\frac{1}{3}\right)^2 = 1 - 2 \cdot \frac{1}{9} = 1 - \frac{2}{9} = \frac{7}{9}$.<br>Теперь подставляем найденное значение $cos(2\alpha)$ в формулу для $cos(4\alpha)$:<br>$cos(4\alpha) = 2 \cdot \left(\frac{7}{9}\right)^2 - 1 = 2 \cdot \frac{49}{81} - 1 = \frac{98}{81} - \frac{81}{81} = \frac{17}{81}$.<br>Ответ: $\frac{17}{81}$.</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" => "1108457" "type" => "task" ] "previous" => array:2 [ "refs" => "1108455" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => 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array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" …1 ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1156} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1157} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1158} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1159} "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 } "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 {#1160 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1161 #items: array:1 [ 0 => App\Models\Element {#1162 #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" => 1261099 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1163 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1097948" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1261099 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1163} "book_page" => array:2 [ "refs" => "1097948" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #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" => "1097949" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097947" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:6 [ 0 => App\Models\Task {#1171 #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" => 1108456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "206" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-715" "field_display_title" => "715" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1172 #items: array:1 [ …1] #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 {#1174 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1208 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1245 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108457" "type" => "task" ] "previous" => array:2 [ "refs" => "1108455" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1270 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097948" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108456 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "206" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-715" "field_display_title" => "715" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1172} "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 {#1174} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1175} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1208} "content" => Illuminate\Database\Eloquent\Collection {#1245} "next" => array:2 [ "refs" => "1108457" "type" => "task" ] "previous" => array:2 [ "refs" => "1108455" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1270} "page" => array:2 [ "refs" => "1097948" "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 => "*" ] } 1 => App\Models\Task {#1271 #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" => 1108457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "206" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-716" "field_display_title" => "716" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1272 #items: array:1 [ …1] #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 {#1274 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1275 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1308 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1345 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108458" "type" => "task" ] "previous" => array:2 [ "refs" => "1108456" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1370 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097948" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "206" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-716" "field_display_title" => "716" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1272} "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 {#1274} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1275} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1308} "content" => Illuminate\Database\Eloquent\Collection {#1345} "next" => array:2 [ "refs" => "1108458" "type" => "task" ] "previous" => array:2 [ "refs" => "1108456" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1370} "page" => array:2 [ "refs" => "1097948" "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" => 1108458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "206" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-717" "field_display_title" => "717" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1372 #items: array:1 [ …1] #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 {#1374 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1375 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1408 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1445 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108459" "type" => "task" ] "previous" => array:2 [ "refs" => "1108457" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1470 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097948" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "206" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-717" "field_display_title" => "717" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1372} "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 {#1374} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1375} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1408} "content" => Illuminate\Database\Eloquent\Collection {#1445} "next" => array:2 [ "refs" => "1108459" "type" => "task" ] "previous" => array:2 [ "refs" => "1108457" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1470} "page" => array:2 [ "refs" => "1097948" "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 => "*" ] } 3 => App\Models\Task {#1471 #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" => 1108459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "206" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-718" "field_display_title" => "718" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1472 #items: array:1 [ …1] #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 {#1474 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1475 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1508 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1545 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108460" "type" => "task" ] "previous" => array:2 [ "refs" => "1108458" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1570 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097948" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "206" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-718" "field_display_title" => "718" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1472} "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 {#1474} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1475} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1508} "content" => Illuminate\Database\Eloquent\Collection {#1545} "next" => array:2 [ "refs" => "1108460" "type" => "task" ] "previous" => array:2 [ "refs" => "1108458" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1570} "page" => array:2 [ "refs" => "1097948" "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 => "*" ] } 4 => App\Models\Task {#1571 #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" => 1108460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "206" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-719" "field_display_title" => "719" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1572 #items: array:1 [ …1] #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 {#1574 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1575 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1608 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1645 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108461" "type" => "task" ] "previous" => array:2 [ "refs" => "1108459" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1670 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097948" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "206" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-719" "field_display_title" => "719" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1572} "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 {#1574} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1575} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1608} "content" => Illuminate\Database\Eloquent\Collection {#1645} "next" => array:2 [ "refs" => "1108461" "type" => "task" ] "previous" => array:2 [ "refs" => "1108459" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1670} "page" => array:2 [ "refs" => "1097948" "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 => "*" ] } 5 => App\Models\Task {#1671 #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" => 1108461 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "206" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-720" "field_display_title" => "720" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1672 #items: array:1 [ …1] #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 {#1674 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1675 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1708 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1745 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108462" "type" => "task" ] "previous" => array:2 [ "refs" => "1108460" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1770 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097948" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108461 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "206" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-720" "field_display_title" => "720" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1672} "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 {#1674} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1675} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1708} "content" => Illuminate\Database\Eloquent\Collection {#1745} "next" => array:2 [ "refs" => "1108462" "type" => "task" ] "previous" => array:2 [ "refs" => "1108460" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1770} "page" => array:2 [ "refs" => "1097948" "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 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1097948 "created_at" => "2026-04-10 13:58:26" "updated_at" 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№715 (с. 206)
Условие. №715 (с. 206)
скриншот условия
715. Вычислите:
а) sin $2\alpha$, если cos $\alpha = -\frac{5}{13}$ и $\frac{\pi}{2} < \alpha < \pi$;
г) cos $2\alpha$, если tg $\alpha = 3$;
б) cos $2\alpha$, если sin $\alpha = -0,8$ и $\frac{3\pi}{2} < \alpha < 2\pi$;
д) sin $4\alpha$, если tg $\alpha = 2$;
в) tg $2\alpha$, если cos $\alpha = -0,6$ и $\pi < \alpha < \frac{3\pi}{2}$;
е) cos $4\alpha$, если sin $\alpha = \frac{1}{3}$.
Решение. №715 (с. 206)
Решение 2 (rus). №715 (с. 206)
а) Для вычисления $sin(2\alpha)$ воспользуемся формулой синуса двойного угла: $sin(2\alpha) = 2sin(\alpha)cos(\alpha)$.
Нам дан $cos(\alpha) = -\frac{5}{13}$. Необходимо найти $sin(\alpha)$. Используем основное тригонометрическое тождество: $sin^2(\alpha) + cos^2(\alpha) = 1$.
$sin^2(\alpha) = 1 - cos^2(\alpha) = 1 - \left(-\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{169 - 25}{169} = \frac{144}{169}$.
$sin(\alpha) = \pm\sqrt{\frac{144}{169}} = \pm\frac{12}{13}$.
По условию, угол $\alpha$ принадлежит второй четверти ($\frac{\pi}{2} < \alpha < \pi$), где $sin(\alpha)$ имеет положительное значение. Значит, $sin(\alpha) = \frac{12}{13}$.
Теперь подставляем значения в формулу для $sin(2\alpha)$:
$sin(2\alpha) = 2 \cdot \frac{12}{13} \cdot \left(-\frac{5}{13}\right) = -\frac{120}{169}$.
Ответ: $-\frac{120}{169}$.
б) Для вычисления $cos(2\alpha)$ используем формулу, связывающую его с $sin(\alpha)$: $cos(2\alpha) = 1 - 2sin^2(\alpha)$.
Нам дан $sin(\alpha) = -0,8 = -\frac{4}{5}$.
Подставляем это значение в формулу:
$cos(2\alpha) = 1 - 2 \cdot (-0,8)^2 = 1 - 2 \cdot 0,64 = 1 - 1,28 = -0,28$.
В дробях: $cos(2\alpha) = 1 - 2 \cdot \left(-\frac{4}{5}\right)^2 = 1 - 2 \cdot \frac{16}{25} = 1 - \frac{32}{25} = \frac{25 - 32}{25} = -\frac{7}{25}$.
Условие $\frac{3\pi}{2} < \alpha < 2\pi$ (четвертая четверть) согласуется со знаком $sin(\alpha) < 0$, но для вычисления $cos(2\alpha)$ по данной формуле оно не требовалось.
Ответ: $-0,28$.
в) Для вычисления $tg(2\alpha)$ воспользуемся формулой тангенса двойного угла: $tg(2\alpha) = \frac{2tg(\alpha)}{1 - tg^2(\alpha)}$.
Сначала найдем $tg(\alpha)$. Нам дан $cos(\alpha) = -0,6 = -\frac{3}{5}$.
Найдем $sin(\alpha)$ из тождества $sin^2(\alpha) + cos^2(\alpha) = 1$:
$sin^2(\alpha) = 1 - (-0,6)^2 = 1 - 0,36 = 0,64$.
$sin(\alpha) = \pm\sqrt{0,64} = \pm0,8$.
По условию, угол $\alpha$ находится в третьей четверти ($\pi < \alpha < \frac{3\pi}{2}$), где и синус, и косинус отрицательны. Значит, $sin(\alpha) = -0,8 = -\frac{4}{5}$.
Теперь найдем $tg(\alpha) = \frac{sin(\alpha)}{cos(\alpha)} = \frac{-0,8}{-0,6} = \frac{8}{6} = \frac{4}{3}$.
Подставляем значение $tg(\alpha)$ в формулу для $tg(2\alpha)$:
$tg(2\alpha) = \frac{2 \cdot \frac{4}{3}}{1 - (\frac{4}{3})^2} = \frac{\frac{8}{3}}{1 - \frac{16}{9}} = \frac{\frac{8}{3}}{\frac{9 - 16}{9}} = \frac{\frac{8}{3}}{-\frac{7}{9}} = \frac{8}{3} \cdot \left(-\frac{9}{7}\right) = -\frac{24}{7}$.
Ответ: $-\frac{24}{7}$.
г) Для вычисления $cos(2\alpha)$ при известном $tg(\alpha)$ удобно использовать формулу, выражающую косинус двойного угла через тангенс: $cos(2\alpha) = \frac{1 - tg^2(\alpha)}{1 + tg^2(\alpha)}$.
Нам дано $tg(\alpha) = 3$.
Подставляем это значение в формулу:
$cos(2\alpha) = \frac{1 - 3^2}{1 + 3^2} = \frac{1 - 9}{1 + 9} = \frac{-8}{10} = -\frac{4}{5}$.
Ответ: $-\frac{4}{5}$.
д) Для вычисления $sin(4\alpha)$ используем формулу синуса двойного угла: $sin(4\alpha) = 2sin(2\alpha)cos(2\alpha)$.
Сначала найдем $sin(2\alpha)$ и $cos(2\alpha)$, используя формулы их выражения через $tg(\alpha) = 2$:
$sin(2\alpha) = \frac{2tg(\alpha)}{1 + tg^2(\alpha)} = \frac{2 \cdot 2}{1 + 2^2} = \frac{4}{1 + 4} = \frac{4}{5}$.
$cos(2\alpha) = \frac{1 - tg^2(\alpha)}{1 + tg^2(\alpha)} = \frac{1 - 2^2}{1 + 2^2} = \frac{1 - 4}{1 + 4} = -\frac{3}{5}$.
Теперь вычисляем $sin(4\alpha)$:
$sin(4\alpha) = 2 \cdot sin(2\alpha) \cdot cos(2\alpha) = 2 \cdot \frac{4}{5} \cdot \left(-\frac{3}{5}\right) = -\frac{24}{25}$.
Ответ: $-\frac{24}{25}$.
е) Для вычисления $cos(4\alpha)$ представим его как косинус двойного угла от $2\alpha$: $cos(4\alpha) = cos(2 \cdot 2\alpha)$.
Воспользуемся одной из формул косинуса двойного угла, например, $cos(2x) = 2cos^2(x) - 1$. В нашем случае $x = 2\alpha$, так что $cos(4\alpha) = 2cos^2(2\alpha) - 1$.
Сначала нужно найти $cos(2\alpha)$. Используем формулу $cos(2\alpha) = 1 - 2sin^2(\alpha)$, так как нам дан $sin(\alpha) = \frac{1}{3}$.
$cos(2\alpha) = 1 - 2 \cdot \left(\frac{1}{3}\right)^2 = 1 - 2 \cdot \frac{1}{9} = 1 - \frac{2}{9} = \frac{7}{9}$.
Теперь подставляем найденное значение $cos(2\alpha)$ в формулу для $cos(4\alpha)$:
$cos(4\alpha) = 2 \cdot \left(\frac{7}{9}\right)^2 - 1 = 2 \cdot \frac{49}{81} - 1 = \frac{98}{81} - \frac{81}{81} = \frac{17}{81}$.
Ответ: $\frac{17}{81}$.
Другие задания:
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