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Номер 734, страница 209 - 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 734, страница 209.
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" => 1108475 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-734" "field_display_title" => "734" "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 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1035 #items: array:1 [ …1] …1 } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 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 …1} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. 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Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #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 {#1064 #items: array:1 [ 0 => App\Models\Book {#1065 #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 {#1066 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 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 {#1066 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093 #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 {#1063} "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 {#1064} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093} ] #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 {#1094 #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 {#1095 #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 {#1096 #items: array:1 [ 0 => App\Models\Book {#1097 #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 {#1098 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1109 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1120 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 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 {#1098 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1109 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1120 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125 #items: array:1 [ 0 => App\Models\Branch {#1126 #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 {#1127 …2} "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 {#1128 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1157 …2} ] #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 {#1127 …2} "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 {#1128 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1157 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 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 {#1095} "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 {#1096} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125} ] #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 {#1158 #items: array:3 [ 0 => App\Models\Element {#1159 #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" => 1277089 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1160 #items: array:1 [ 0 => App\Models\Edition {#1161 #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 {#1162 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1164 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1166 …2} "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 {#1162 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1164 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1166 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108475" "type" => "task" ] "text" => "<p><strong>734.</strong> Докажите тождество:</p><p><strong>a)</strong></p><p>$\operatorname{tg} 3\alpha = \frac{3\operatorname{tg}\alpha - \operatorname{tg}^3\alpha}{1 - 3\operatorname{tg}^2\alpha}$;</p><p><strong>б)</strong></p><p>$\frac{\sin 3\alpha}{\sin \alpha} = 2 + \frac{\cos 3\alpha}{\cos \alpha}$;</p><p><strong>в)</strong></p><p>$\frac{\operatorname{tg}^2 2\alpha - \operatorname{tg}^2 \alpha}{1 - \operatorname{tg}^2 2\alpha \operatorname{tg}^2 \alpha} = \operatorname{tg} 3\alpha \cdot \operatorname{tg} \alpha.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "734-1.jpg" "alt" => null "width" => "1459" "height" => 407 "path" => "/media/algebra_09/soltan-u19/0-1/734-1.webp?ts=1752834734" ] ] ] #original: array:7 [ "id" => 1277089 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1160} "task" => array:2 [ "refs" => "1108475" "type" => "task" ] "text" => "<p><strong>734.</strong> Докажите тождество:</p><p><strong>a)</strong></p><p>$\operatorname{tg} 3\alpha = \frac{3\operatorname{tg}\alpha - \operatorname{tg}^3\alpha}{1 - 3\operatorname{tg}^2\alpha}$;</p><p><strong>б)</strong></p><p>$\frac{\sin 3\alpha}{\sin \alpha} = 2 + \frac{\cos 3\alpha}{\cos \alpha}$;</p><p><strong>в)</strong></p><p>$\frac{\operatorname{tg}^2 2\alpha - \operatorname{tg}^2 \alpha}{1 - \operatorname{tg}^2 2\alpha \operatorname{tg}^2 \alpha} = \operatorname{tg} 3\alpha \cdot \operatorname{tg} \alpha.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "734-1.jpg" "alt" => null "width" => "1459" "height" => 407 "path" => "/media/algebra_09/soltan-u19/0-1/734-1.webp?ts=1752834734" ] ] ] #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 {#1167 #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" => 1278380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1168 #items: array:1 [ 0 => App\Models\Edition {#1169 #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 {#1170 …2} "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 {#1172 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1174 …2} "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 {#1170 …2} "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 {#1172 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108475" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "734-1.jpg" "alt" => null "width" => "1182" "height" => 1473 "path" => "/media/algebra_09/soltan-u19/1-1/734-1.webp?ts=1752831534" ] 1 => array:5 [ "name" => "734-2.jpg" "alt" => null "width" => "1185" "height" => 1473 "path" => "/media/algebra_09/soltan-u19/1-1/734-2.webp?ts=1752831534" ] ] ] #original: array:6 [ "id" => 1278380 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1168} "task" => array:2 [ "refs" => "1108475" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "734-1.jpg" "alt" => null "width" => "1182" "height" => 1473 "path" => "/media/algebra_09/soltan-u19/1-1/734-1.webp?ts=1752831534" ] 1 => array:5 [ "name" => "734-2.jpg" "alt" => null "width" => "1185" "height" => 1473 "path" => "/media/algebra_09/soltan-u19/1-1/734-2.webp?ts=1752831534" ] ] ] #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 {#1175 #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" => 1554335 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1176 #items: array:1 [ 0 => App\Models\Edition {#1177 #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 {#1178 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1182 …2} "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 {#1178 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108475" "type" => "task" ] "text" => "<p><strong>а)</strong> Докажем тождество, преобразовав левую часть. Используем формулу тангенса суммы и тангенса двойного угла.</p><p>$\tg 3\alpha = \tg (2\alpha + \alpha) = \frac{\tg 2\alpha + \tg \alpha}{1 - \tg 2\alpha \cdot \tg \alpha}$<br>Подставим формулу тангенса двойного угла $\tg 2\alpha = \frac{2\tg \alpha}{1 - \tg^2 \alpha}$:<br>$\tg 3\alpha = \frac{\frac{2\tg \alpha}{1 - \tg^2 \alpha} + \tg \alpha}{1 - \frac{2\tg \alpha}{1 - \tg^2 \alpha} \cdot \tg \alpha} = \frac{\frac{2\tg \alpha + \tg \alpha(1 - \tg^2 \alpha)}{1 - \tg^2 \alpha}}{\frac{1 - \tg^2 \alpha - 2\tg^2 \alpha}{1 - \tg^2 \alpha}}$<br>Упростим числитель и знаменатель дроби:<br>$\tg 3\alpha = \frac{\frac{2\tg \alpha + \tg \alpha - \tg^3 \alpha}{1 - \tg^2 \alpha}}{\frac{1 - 3\tg^2 \alpha}{1 - \tg^2 \alpha}} = \frac{\frac{3\tg \alpha - \tg^3 \alpha}{1 - \tg^2 \alpha}}{\frac{1 - 3\tg^2 \alpha}{1 - \tg^2 \alpha}}$<br>Сократим общий множитель $(1 - \tg^2 \alpha)$ в числителе и знаменателе:<br>$\tg 3\alpha = \frac{3\tg \alpha - \tg^3 \alpha}{1 - 3\tg^2 \alpha}$<br>Левая часть равна правой, тождество доказано.<br>Ответ: $\tg 3\alpha = \frac{3\tg \alpha - \tg^3 \alpha}{1 - 3\tg^2 \alpha}$</p><p><strong>б)</strong> Преобразуем исходное равенство. Область допустимых значений: $\sin \alpha \neq 0$ и $\cos \alpha \neq 0$.<br>Перенесем слагаемое из правой части в левую:<br>$\frac{\sin 3\alpha}{\sin \alpha} - \frac{\cos 3\alpha}{\cos \alpha} = 2$<br>Приведем дроби в левой части к общему знаменателю:<br>$\frac{\sin 3\alpha \cos \alpha - \cos 3\alpha \sin \alpha}{\sin \alpha \cos \alpha} = 2$<br>В числителе дроби используем формулу синуса разности $\sin(x-y) = \sin x \cos y - \cos x \sin y$:<br>$\sin 3\alpha \cos \alpha - \cos 3\alpha \sin \alpha = \sin(3\alpha - \alpha) = \sin 2\alpha$<br>В знаменателе дроби используем формулу синуса двойного угла $\sin 2\alpha = 2\sin \alpha \cos \alpha$, откуда $\sin \alpha \cos \alpha = \frac{1}{2}\sin 2\alpha$.<br>Подставим полученные выражения в левую часть:<br>$\frac{\sin 2\alpha}{\frac{1}{2}\sin 2\alpha} = 2$<br>Сократив $\sin 2\alpha$ (что возможно, так как по ОДЗ $\sin \alpha \neq 0$ и $\cos \alpha \neq 0$, следовательно $\sin 2\alpha \neq 0$), получаем:<br>$2 = 2$<br>Получили верное равенство, тождество доказано.<br>Ответ: $\frac{\sin 3\alpha}{\sin \alpha} = 2 + \frac{\cos 3\alpha}{\cos \alpha}$</p><p><strong>в)</strong> Докажем тождество, преобразовав левую часть. Используем формулу разности квадратов $a^2-b^2=(a-b)(a+b)$.<br>$\frac{\tg^2 2\alpha - \tg^2 \alpha}{1 - \tg^2 2\alpha \tg^2 \alpha} = \frac{(\tg 2\alpha - \tg \alpha)(\tg 2\alpha + \tg \alpha)}{(1 - \tg 2\alpha \tg \alpha)(1 + \tg 2\alpha \tg \alpha)}$<br>Сгруппируем множители:<br>$= \left(\frac{\tg 2\alpha + \tg \alpha}{1 - \tg 2\alpha \tg \alpha}\right) \cdot \left(\frac{\tg 2\alpha - \tg \alpha}{1 + \tg 2\alpha \tg \alpha}\right)$<br>Первый множитель является формулой тангенса суммы $\tg(x+y) = \frac{\tg x + \tg y}{1 - \tg x \tg y}$:<br>$\frac{\tg 2\alpha + \tg \alpha}{1 - \tg 2\alpha \tg \alpha} = \tg(2\alpha + \alpha) = \tg 3\alpha$<br>Второй множитель является формулой тангенса разности $\tg(x-y) = \frac{\tg x - \tg y}{1 + \tg x \tg y}$:<br>$\frac{\tg 2\alpha - \tg \alpha}{1 + \tg 2\alpha \tg \alpha} = \tg(2\alpha - \alpha) = \tg \alpha$<br>Таким образом, левая часть равна:<br>$\tg 3\alpha \cdot \tg \alpha$<br>Левая часть равна правой, тождество доказано.<br>Ответ: $\frac{\tg^2 2\alpha - \tg^2 \alpha}{1 - \tg^2 2\alpha \tg^2 \alpha} = \tg 3\alpha \cdot \tg \alpha$</p>" ] #original: array:6 [ "id" => 1554335 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1176} "task" => array:2 [ "refs" => "1108475" "type" => "task" ] "text" => "<p><strong>а)</strong> Докажем тождество, преобразовав левую часть. Используем формулу тангенса суммы и тангенса двойного угла.</p><p>$\tg 3\alpha = \tg (2\alpha + \alpha) = \frac{\tg 2\alpha + \tg \alpha}{1 - \tg 2\alpha \cdot \tg \alpha}$<br>Подставим формулу тангенса двойного угла $\tg 2\alpha = \frac{2\tg \alpha}{1 - \tg^2 \alpha}$:<br>$\tg 3\alpha = \frac{\frac{2\tg \alpha}{1 - \tg^2 \alpha} + \tg \alpha}{1 - \frac{2\tg \alpha}{1 - \tg^2 \alpha} \cdot \tg \alpha} = \frac{\frac{2\tg \alpha + \tg \alpha(1 - \tg^2 \alpha)}{1 - \tg^2 \alpha}}{\frac{1 - \tg^2 \alpha - 2\tg^2 \alpha}{1 - \tg^2 \alpha}}$<br>Упростим числитель и знаменатель дроби:<br>$\tg 3\alpha = \frac{\frac{2\tg \alpha + \tg \alpha - \tg^3 \alpha}{1 - \tg^2 \alpha}}{\frac{1 - 3\tg^2 \alpha}{1 - \tg^2 \alpha}} = \frac{\frac{3\tg \alpha - \tg^3 \alpha}{1 - \tg^2 \alpha}}{\frac{1 - 3\tg^2 \alpha}{1 - \tg^2 \alpha}}$<br>Сократим общий множитель $(1 - \tg^2 \alpha)$ в числителе и знаменателе:<br>$\tg 3\alpha = \frac{3\tg \alpha - \tg^3 \alpha}{1 - 3\tg^2 \alpha}$<br>Левая часть равна правой, тождество доказано.<br>Ответ: $\tg 3\alpha = \frac{3\tg \alpha - \tg^3 \alpha}{1 - 3\tg^2 \alpha}$</p><p><strong>б)</strong> Преобразуем исходное равенство. Область допустимых значений: $\sin \alpha \neq 0$ и $\cos \alpha \neq 0$.<br>Перенесем слагаемое из правой части в левую:<br>$\frac{\sin 3\alpha}{\sin \alpha} - \frac{\cos 3\alpha}{\cos \alpha} = 2$<br>Приведем дроби в левой части к общему знаменателю:<br>$\frac{\sin 3\alpha \cos \alpha - \cos 3\alpha \sin \alpha}{\sin \alpha \cos \alpha} = 2$<br>В числителе дроби используем формулу синуса разности $\sin(x-y) = \sin x \cos y - \cos x \sin y$:<br>$\sin 3\alpha \cos \alpha - \cos 3\alpha \sin \alpha = \sin(3\alpha - \alpha) = \sin 2\alpha$<br>В знаменателе дроби используем формулу синуса двойного угла $\sin 2\alpha = 2\sin \alpha \cos \alpha$, откуда $\sin \alpha \cos \alpha = \frac{1}{2}\sin 2\alpha$.<br>Подставим полученные выражения в левую часть:<br>$\frac{\sin 2\alpha}{\frac{1}{2}\sin 2\alpha} = 2$<br>Сократив $\sin 2\alpha$ (что возможно, так как по ОДЗ $\sin \alpha \neq 0$ и $\cos \alpha \neq 0$, следовательно $\sin 2\alpha \neq 0$), получаем:<br>$2 = 2$<br>Получили верное равенство, тождество доказано.<br>Ответ: $\frac{\sin 3\alpha}{\sin \alpha} = 2 + \frac{\cos 3\alpha}{\cos \alpha}$</p><p><strong>в)</strong> Докажем тождество, преобразовав левую часть. Используем формулу разности квадратов $a^2-b^2=(a-b)(a+b)$.<br>$\frac{\tg^2 2\alpha - \tg^2 \alpha}{1 - \tg^2 2\alpha \tg^2 \alpha} = \frac{(\tg 2\alpha - \tg \alpha)(\tg 2\alpha + \tg \alpha)}{(1 - \tg 2\alpha \tg \alpha)(1 + \tg 2\alpha \tg \alpha)}$<br>Сгруппируем множители:<br>$= \left(\frac{\tg 2\alpha + \tg \alpha}{1 - \tg 2\alpha \tg \alpha}\right) \cdot \left(\frac{\tg 2\alpha - \tg \alpha}{1 + \tg 2\alpha \tg \alpha}\right)$<br>Первый множитель является формулой тангенса суммы $\tg(x+y) = \frac{\tg x + \tg y}{1 - \tg x \tg y}$:<br>$\frac{\tg 2\alpha + \tg \alpha}{1 - \tg 2\alpha \tg \alpha} = \tg(2\alpha + \alpha) = \tg 3\alpha$<br>Второй множитель является формулой тангенса разности $\tg(x-y) = \frac{\tg x - \tg y}{1 + \tg x \tg y}$:<br>$\frac{\tg 2\alpha - \tg \alpha}{1 + \tg 2\alpha \tg \alpha} = \tg(2\alpha - \alpha) = \tg \alpha$<br>Таким образом, левая часть равна:<br>$\tg 3\alpha \cdot \tg \alpha$<br>Левая часть равна правой, тождество доказано.<br>Ответ: $\frac{\tg^2 2\alpha - \tg^2 \alpha}{1 - \tg^2 2\alpha \tg^2 \alpha} = \tg 3\alpha \cdot \tg \alpha$</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" => "1108476" "type" => "task" ] "previous" => array:2 [ "refs" => "1108474" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1183 #items: array:1 [ 0 => App\Models\Book {#1184 #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 {#1185 #items: array:1 [ 0 => App\Models\Term {#1186 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1187 #items: array:1 [ 0 => App\Models\Term {#1188 #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 [ …6] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1189 #items: array:1 [ 0 => App\Models\Term {#1190 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:3 [ 0 => App\Models\Term {#1192 #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 {#1193 #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 {#1194 #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 {#1195 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1196 #items: array:1 [ 0 => App\Models\Term {#1197 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Term {#1199 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1200 #items: array:1 [ 0 => App\Models\Term {#1201 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1202 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1203 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1204 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1205 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1206 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1207 #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 {#1208 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1209 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1210 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1211 #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 {#1185} "field_class" => Illuminate\Database\Eloquent\Collection {#1187} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1189} "field_author" => Illuminate\Database\Eloquent\Collection {#1191} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1195} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1196} "field_country" => Illuminate\Database\Eloquent\Collection {#1198} "field_city" => Illuminate\Database\Eloquent\Collection {#1200} "field_series" => Illuminate\Database\Eloquent\Collection {#1202} "field_umk" => Illuminate\Database\Eloquent\Collection {#1203} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1204} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1205} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1206} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1207} "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 {#1208} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1209} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1210} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1211} "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 } "page" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:1 [ 0 => App\Models\BookPage {#1225 #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" => 1097951 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-209" "field_display_title" => "209" "field_folder" => "folder1" "field_image_name" => "209" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1227 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1258 #items: array:1 [ 0 => App\Models\Book {#1228 #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 {#1230 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1231 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1233 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1235 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1239 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1240 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1242 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1244 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1246 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1247 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1248 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1249 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1250 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1251 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1252 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1253 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1254 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1255 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 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 {#1230 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1231 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1233 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1235 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1239 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1240 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1242 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1244 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1246 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1247 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1248 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1249 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1250 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1251 …2} "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 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1252 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1253 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1254 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1255 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "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 {#1256 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1267 #items: array:1 [ 0 => App\Models\Element {#1266 #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" => 1261102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1276 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1276 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1097952" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097950" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1802 #items: array:4 [ 0 => App\Models\Task {#1812 #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" => 1108475 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-734" "field_display_title" => "734" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1816 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1815 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1850 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1945 …2} "content" => Illuminate\Database\Eloquent\Collection {#1971 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1972 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108475 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-734" "field_display_title" => "734" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1816 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1815 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1850 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1945 …2} "content" => Illuminate\Database\Eloquent\Collection {#1971 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1972 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1970 #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" => 1108476 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-735" "field_display_title" => "735" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1973 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1974 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1975 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1976 …2} "content" => Illuminate\Database\Eloquent\Collection {#2002 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2003 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108476 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-735" "field_display_title" => "735" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1973 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1974 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1975 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1976 …2} "content" => Illuminate\Database\Eloquent\Collection {#2002 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2003 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#2001 #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" => 1108477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-736" "field_display_title" => "736" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2004 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#2005 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2006 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2007 …2} "content" => Illuminate\Database\Eloquent\Collection {#2033 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2034 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-736" "field_display_title" => "736" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2004 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#2005 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2006 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2007 …2} "content" => Illuminate\Database\Eloquent\Collection {#2033 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2034 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#2032 #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" => 1108478 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-737" "field_display_title" => "737" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2035 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#2036 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2037 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2038 …2} "content" => Illuminate\Database\Eloquent\Collection {#2064 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => 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№734 (с. 209)
Условие. №734 (с. 209)
скриншот условия
734. Докажите тождество:
a)
$\operatorname{tg} 3\alpha = \frac{3\operatorname{tg}\alpha - \operatorname{tg}^3\alpha}{1 - 3\operatorname{tg}^2\alpha}$;
б)
$\frac{\sin 3\alpha}{\sin \alpha} = 2 + \frac{\cos 3\alpha}{\cos \alpha}$;
в)
$\frac{\operatorname{tg}^2 2\alpha - \operatorname{tg}^2 \alpha}{1 - \operatorname{tg}^2 2\alpha \operatorname{tg}^2 \alpha} = \operatorname{tg} 3\alpha \cdot \operatorname{tg} \alpha.$
Решение. №734 (с. 209)
Решение 2 (rus). №734 (с. 209)
а) Докажем тождество, преобразовав левую часть. Используем формулу тангенса суммы и тангенса двойного угла.
$\tg 3\alpha = \tg (2\alpha + \alpha) = \frac{\tg 2\alpha + \tg \alpha}{1 - \tg 2\alpha \cdot \tg \alpha}$
Подставим формулу тангенса двойного угла $\tg 2\alpha = \frac{2\tg \alpha}{1 - \tg^2 \alpha}$:
$\tg 3\alpha = \frac{\frac{2\tg \alpha}{1 - \tg^2 \alpha} + \tg \alpha}{1 - \frac{2\tg \alpha}{1 - \tg^2 \alpha} \cdot \tg \alpha} = \frac{\frac{2\tg \alpha + \tg \alpha(1 - \tg^2 \alpha)}{1 - \tg^2 \alpha}}{\frac{1 - \tg^2 \alpha - 2\tg^2 \alpha}{1 - \tg^2 \alpha}}$
Упростим числитель и знаменатель дроби:
$\tg 3\alpha = \frac{\frac{2\tg \alpha + \tg \alpha - \tg^3 \alpha}{1 - \tg^2 \alpha}}{\frac{1 - 3\tg^2 \alpha}{1 - \tg^2 \alpha}} = \frac{\frac{3\tg \alpha - \tg^3 \alpha}{1 - \tg^2 \alpha}}{\frac{1 - 3\tg^2 \alpha}{1 - \tg^2 \alpha}}$
Сократим общий множитель $(1 - \tg^2 \alpha)$ в числителе и знаменателе:
$\tg 3\alpha = \frac{3\tg \alpha - \tg^3 \alpha}{1 - 3\tg^2 \alpha}$
Левая часть равна правой, тождество доказано.
Ответ: $\tg 3\alpha = \frac{3\tg \alpha - \tg^3 \alpha}{1 - 3\tg^2 \alpha}$
б) Преобразуем исходное равенство. Область допустимых значений: $\sin \alpha \neq 0$ и $\cos \alpha \neq 0$.
Перенесем слагаемое из правой части в левую:
$\frac{\sin 3\alpha}{\sin \alpha} - \frac{\cos 3\alpha}{\cos \alpha} = 2$
Приведем дроби в левой части к общему знаменателю:
$\frac{\sin 3\alpha \cos \alpha - \cos 3\alpha \sin \alpha}{\sin \alpha \cos \alpha} = 2$
В числителе дроби используем формулу синуса разности $\sin(x-y) = \sin x \cos y - \cos x \sin y$:
$\sin 3\alpha \cos \alpha - \cos 3\alpha \sin \alpha = \sin(3\alpha - \alpha) = \sin 2\alpha$
В знаменателе дроби используем формулу синуса двойного угла $\sin 2\alpha = 2\sin \alpha \cos \alpha$, откуда $\sin \alpha \cos \alpha = \frac{1}{2}\sin 2\alpha$.
Подставим полученные выражения в левую часть:
$\frac{\sin 2\alpha}{\frac{1}{2}\sin 2\alpha} = 2$
Сократив $\sin 2\alpha$ (что возможно, так как по ОДЗ $\sin \alpha \neq 0$ и $\cos \alpha \neq 0$, следовательно $\sin 2\alpha \neq 0$), получаем:
$2 = 2$
Получили верное равенство, тождество доказано.
Ответ: $\frac{\sin 3\alpha}{\sin \alpha} = 2 + \frac{\cos 3\alpha}{\cos \alpha}$
в) Докажем тождество, преобразовав левую часть. Используем формулу разности квадратов $a^2-b^2=(a-b)(a+b)$.
$\frac{\tg^2 2\alpha - \tg^2 \alpha}{1 - \tg^2 2\alpha \tg^2 \alpha} = \frac{(\tg 2\alpha - \tg \alpha)(\tg 2\alpha + \tg \alpha)}{(1 - \tg 2\alpha \tg \alpha)(1 + \tg 2\alpha \tg \alpha)}$
Сгруппируем множители:
$= \left(\frac{\tg 2\alpha + \tg \alpha}{1 - \tg 2\alpha \tg \alpha}\right) \cdot \left(\frac{\tg 2\alpha - \tg \alpha}{1 + \tg 2\alpha \tg \alpha}\right)$
Первый множитель является формулой тангенса суммы $\tg(x+y) = \frac{\tg x + \tg y}{1 - \tg x \tg y}$:
$\frac{\tg 2\alpha + \tg \alpha}{1 - \tg 2\alpha \tg \alpha} = \tg(2\alpha + \alpha) = \tg 3\alpha$
Второй множитель является формулой тангенса разности $\tg(x-y) = \frac{\tg x - \tg y}{1 + \tg x \tg y}$:
$\frac{\tg 2\alpha - \tg \alpha}{1 + \tg 2\alpha \tg \alpha} = \tg(2\alpha - \alpha) = \tg \alpha$
Таким образом, левая часть равна:
$\tg 3\alpha \cdot \tg \alpha$
Левая часть равна правой, тождество доказано.
Ответ: $\frac{\tg^2 2\alpha - \tg^2 \alpha}{1 - \tg^2 2\alpha \tg^2 \alpha} = \tg 3\alpha \cdot \tg \alpha$
Другие задания:
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