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Номер 811, страница 224 - 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. 30. Упражнения на повторение раздела «Тригонометрия» - номер 811, страница 224.
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" => 1108557 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "224" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-811" "field_display_title" => "811" "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 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "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 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "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. 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" => 1107672 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "30. Упражнения на повторение раздела «Тригонометрия»" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "221" "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: array:1 [ 0 => 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" => 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 {#1095 …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 {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125 …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 {#1095 …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 {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125 …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" => 1107672 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "30. Упражнения на повторение раздела «Тригонометрия»" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "221" "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 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:3 [ 0 => App\Models\Element {#1127 #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" => 1277166 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Edition {#1129 #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 {#1130 …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 {#1132 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1134 …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 {#1130 …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 {#1132 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1134 …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" => "1108557" "type" => "task" ] "text" => "<p><strong>811.</strong> Докажите тождество:</p><p><strong>а)</strong></p><p>$4\sin^2 2\alpha \cdot \cos 2\alpha = \cos 2\alpha - \cos 6\alpha;$</p><p><strong>б)</strong></p><p>$8\sin^3 \alpha \cdot \cos \alpha = 2\sin 2\alpha - \sin 4\alpha;$</p><p><strong>в)</strong></p><p>$\frac{\sin^2 3\alpha}{\sin^2 \alpha} - \frac{\cos^2 3\alpha}{\cos^2 \alpha} = 8\cos 2\alpha;$</p><p><strong>г)</strong></p><p>$\frac{\sin 2\alpha}{1 + \cos 2\alpha} \cdot \frac{\cos \alpha}{1 + \cos \alpha} = \text{tg}\frac{\alpha}{2}.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "811-1.jpg" "alt" => null "width" => "931" "height" => 510 "path" => "/media/algebra_09/soltan-u19/0-1/811-1.webp?ts=1752834807" ] ] ] #original: array:7 [ "id" => 1277166 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128} "task" => array:2 [ "refs" => "1108557" "type" => "task" ] "text" => "<p><strong>811.</strong> Докажите тождество:</p><p><strong>а)</strong></p><p>$4\sin^2 2\alpha \cdot \cos 2\alpha = \cos 2\alpha - \cos 6\alpha;$</p><p><strong>б)</strong></p><p>$8\sin^3 \alpha \cdot \cos \alpha = 2\sin 2\alpha - \sin 4\alpha;$</p><p><strong>в)</strong></p><p>$\frac{\sin^2 3\alpha}{\sin^2 \alpha} - \frac{\cos^2 3\alpha}{\cos^2 \alpha} = 8\cos 2\alpha;$</p><p><strong>г)</strong></p><p>$\frac{\sin 2\alpha}{1 + \cos 2\alpha} \cdot \frac{\cos \alpha}{1 + \cos \alpha} = \text{tg}\frac{\alpha}{2}.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "811-1.jpg" "alt" => null "width" => "931" "height" => 510 "path" => "/media/algebra_09/soltan-u19/0-1/811-1.webp?ts=1752834807" ] ] ] #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 {#1135 #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" => 1278457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136 #items: array:1 [ 0 => App\Models\Edition {#1137 #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 {#1138 …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 {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …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 {#1138 …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 {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …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" => "1108557" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "811-1.jpg" "alt" => null "width" => "1445" "height" => 2410 "path" => "/media/algebra_09/soltan-u19/1-1/811-1.webp?ts=1752831631" ] 1 => array:5 [ "name" => "811-2.jpg" "alt" => null "width" => "1258" "height" => 706 "path" => "/media/algebra_09/soltan-u19/1-1/811-2.webp?ts=1752831631" ] ] ] #original: array:6 [ "id" => 1278457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136} "task" => array:2 [ "refs" => "1108557" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "811-1.jpg" "alt" => null "width" => "1445" "height" => 2410 "path" => "/media/algebra_09/soltan-u19/1-1/811-1.webp?ts=1752831631" ] 1 => array:5 [ "name" => "811-2.jpg" "alt" => null "width" => "1258" "height" => 706 "path" => "/media/algebra_09/soltan-u19/1-1/811-2.webp?ts=1752831631" ] ] ] #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 {#1143 #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" => 1554412 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:1 [ 0 => App\Models\Edition {#1145 #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 {#1146 …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 {#1148 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1150 …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 {#1146 …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 {#1148 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1150 …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" => "1108557" "type" => "task" ] "text" => "<p><strong>а)</strong> Для доказательства тождества $4\sin^2{2\alpha} \cdot \cos 2\alpha = \cos 2\alpha - \cos 6\alpha$ преобразуем его правую часть. Воспользуемся формулой разности косинусов: $\cos x - \cos y = -2\sin\frac{x+y}{2}\sin\frac{x-y}{2}$.</p><p>$\cos 2\alpha - \cos 6\alpha = -2\sin\frac{2\alpha+6\alpha}{2}\sin\frac{2\alpha-6\alpha}{2} = -2\sin(4\alpha)\sin(-2\alpha)$.</p><p>Поскольку синус является нечетной функцией, $\sin(-2\alpha) = -\sin(2\alpha)$. Тогда выражение принимает вид:</p><p>$2\sin(4\alpha)\sin(2\alpha)$.</p><p>Применим формулу синуса двойного угла $\sin(2x) = 2\sin x \cos x$ для члена $\sin(4\alpha)$:</p><p>$2 \cdot (2\sin(2\alpha)\cos(2\alpha)) \cdot \sin(2\alpha) = 4\sin^2(2\alpha)\cos(2\alpha)$.</p><p>Мы получили выражение, стоящее в левой части исходного тождества. Таким образом, тождество доказано.</p><p><strong>Ответ:</strong> Тождество $4\sin^2{2\alpha} \cdot \cos 2\alpha = \cos 2\alpha - \cos 6\alpha$ доказано.</p><p><strong>б)</strong> Для доказательства тождества $8\sin^3\alpha \cdot \cos \alpha = 2\sin 2\alpha - \sin 4\alpha$ преобразуем его левую часть.</p><p>$8\sin^3\alpha \cdot \cos \alpha = 4\sin^2\alpha \cdot (2\sin\alpha\cos\alpha)$.</p><p>Используем формулу синуса двойного угла $2\sin\alpha\cos\alpha = \sin(2\alpha)$:</p><p>$4\sin^2\alpha \sin(2\alpha)$.</p><p>Теперь применим формулу понижения степени $\sin^2\alpha = \frac{1-\cos(2\alpha)}{2}$:</p><p>$4 \cdot \frac{1-\cos(2\alpha)}{2} \cdot \sin(2\alpha) = 2(1-\cos(2\alpha))\sin(2\alpha) = 2\sin(2\alpha) - 2\sin(2\alpha)\cos(2\alpha)$.</p><p>Снова используем формулу синуса двойного угла для выражения $2\sin(2\alpha)\cos(2\alpha)$, которое равно $\sin(4\alpha)$:</p><p>$2\sin(2\alpha) - \sin(4\alpha)$.</p><p>Мы получили выражение, стоящее в правой части, следовательно, тождество доказано.</p><p><strong>Ответ:</strong> Тождество $8\sin^3\alpha \cdot \cos \alpha = 2\sin 2\alpha - \sin 4\alpha$ доказано.</p><p><strong>в)</strong> Для доказательства тождества $\frac{\sin^2{3\alpha}}{\sin^2\alpha} - \frac{\cos^2{3\alpha}}{\cos^2\alpha} = 8\cos 2\alpha$ преобразуем его левую часть. Приведем дроби к общему знаменателю:</p><p>$\frac{\sin^2(3\alpha)\cos^2\alpha - \cos^2(3\alpha)\sin^2\alpha}{\sin^2\alpha\cos^2\alpha}$.</p><p>Числитель является разностью квадратов, которую можно разложить на множители:</p><p>$(\sin(3\alpha)\cos\alpha - \cos(3\alpha)\sin\alpha)(\sin(3\alpha)\cos\alpha + \cos(3\alpha)\sin\alpha)$.</p><p>Используя формулы синуса суммы и разности углов, получаем:</p><p>$\sin(3\alpha-\alpha)\sin(3\alpha+\alpha) = \sin(2\alpha)\sin(4\alpha)$.</p><p>Знаменатель преобразуем с помощью формулы синуса двойного угла $\sin(2\alpha)=2\sin\alpha\cos\alpha$, из которой следует $\sin\alpha\cos\alpha = \frac{\sin(2\alpha)}{2}$:</p><p>$\sin^2\alpha\cos^2\alpha = (\sin\alpha\cos\alpha)^2 = \left(\frac{\sin(2\alpha)}{2}\right)^2 = \frac{\sin^2(2\alpha)}{4}$.</p><p>Теперь подставим преобразованные числитель и знаменатель в дробь:</p><p>$\frac{\sin(2\alpha)\sin(4\alpha)}{\frac{\sin^2(2\alpha)}{4}} = \frac{4\sin(2\alpha)\sin(4\alpha)}{\sin^2(2\alpha)} = \frac{4\sin(4\alpha)}{\sin(2\alpha)}$.</p><p>Применим формулу синуса двойного угла $\sin(4\alpha) = 2\sin(2\alpha)\cos(2\alpha)$:</p><p>$\frac{4 \cdot 2\sin(2\alpha)\cos(2\alpha)}{\sin(2\alpha)} = 8\cos(2\alpha)$.</p><p>Левая часть тождества равна правой, что и требовалось доказать.</p><p><strong>Ответ:</strong> Тождество $\frac{\sin^2{3\alpha}}{\sin^2\alpha} - \frac{\cos^2{3\alpha}}{\cos^2\alpha} = 8\cos 2\alpha$ доказано.</p><p><strong>г)</strong> Для доказательства тождества $\frac{\sin 2\alpha}{1+\cos 2\alpha} \cdot \frac{\cos\alpha}{1+\cos\alpha} = \text{tg}\frac{\alpha}{2}$ преобразуем его левую часть. Рассмотрим первую дробь:</p><p>$\frac{\sin 2\alpha}{1+\cos 2\alpha}$.</p><p>Используем формулы двойного угла: $\sin(2\alpha) = 2\sin\alpha\cos\alpha$ и $1+\cos(2\alpha) = 2\cos^2\alpha$.</p><p>$\frac{2\sin\alpha\cos\alpha}{2\cos^2\alpha} = \frac{\sin\alpha}{\cos\alpha} = \text{tg}\alpha$.</p><p>Теперь левая часть тождества имеет вид:</p><p>$\text{tg}\alpha \cdot \frac{\cos\alpha}{1+\cos\alpha} = \frac{\sin\alpha}{\cos\alpha} \cdot \frac{\cos\alpha}{1+\cos\alpha} = \frac{\sin\alpha}{1+\cos\alpha}$.</p><p>Применим формулы половинного угла: $\sin\alpha = 2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}$ и $1+\cos\alpha = 2\cos^2\frac{\alpha}{2}$.</p><p>$\frac{2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}}{2\cos^2\frac{\alpha}{2}} = \frac{\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}} = \text{tg}\frac{\alpha}{2}$.</p><p>Левая часть равна правой, тождество доказано.</p><p><strong>Ответ:</strong> Тождество $\frac{\sin 2\alpha}{1+\cos 2\alpha} \cdot \frac{\cos\alpha}{1+\cos\alpha} = \text{tg}\frac{\alpha}{2}$ доказано.</p>" ] #original: array:6 [ "id" => 1554412 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144} "task" => array:2 [ "refs" => "1108557" "type" => "task" ] "text" => "<p><strong>а)</strong> Для доказательства тождества $4\sin^2{2\alpha} \cdot \cos 2\alpha = \cos 2\alpha - \cos 6\alpha$ преобразуем его правую часть. Воспользуемся формулой разности косинусов: $\cos x - \cos y = -2\sin\frac{x+y}{2}\sin\frac{x-y}{2}$.</p><p>$\cos 2\alpha - \cos 6\alpha = -2\sin\frac{2\alpha+6\alpha}{2}\sin\frac{2\alpha-6\alpha}{2} = -2\sin(4\alpha)\sin(-2\alpha)$.</p><p>Поскольку синус является нечетной функцией, $\sin(-2\alpha) = -\sin(2\alpha)$. Тогда выражение принимает вид:</p><p>$2\sin(4\alpha)\sin(2\alpha)$.</p><p>Применим формулу синуса двойного угла $\sin(2x) = 2\sin x \cos x$ для члена $\sin(4\alpha)$:</p><p>$2 \cdot (2\sin(2\alpha)\cos(2\alpha)) \cdot \sin(2\alpha) = 4\sin^2(2\alpha)\cos(2\alpha)$.</p><p>Мы получили выражение, стоящее в левой части исходного тождества. Таким образом, тождество доказано.</p><p><strong>Ответ:</strong> Тождество $4\sin^2{2\alpha} \cdot \cos 2\alpha = \cos 2\alpha - \cos 6\alpha$ доказано.</p><p><strong>б)</strong> Для доказательства тождества $8\sin^3\alpha \cdot \cos \alpha = 2\sin 2\alpha - \sin 4\alpha$ преобразуем его левую часть.</p><p>$8\sin^3\alpha \cdot \cos \alpha = 4\sin^2\alpha \cdot (2\sin\alpha\cos\alpha)$.</p><p>Используем формулу синуса двойного угла $2\sin\alpha\cos\alpha = \sin(2\alpha)$:</p><p>$4\sin^2\alpha \sin(2\alpha)$.</p><p>Теперь применим формулу понижения степени $\sin^2\alpha = \frac{1-\cos(2\alpha)}{2}$:</p><p>$4 \cdot \frac{1-\cos(2\alpha)}{2} \cdot \sin(2\alpha) = 2(1-\cos(2\alpha))\sin(2\alpha) = 2\sin(2\alpha) - 2\sin(2\alpha)\cos(2\alpha)$.</p><p>Снова используем формулу синуса двойного угла для выражения $2\sin(2\alpha)\cos(2\alpha)$, которое равно $\sin(4\alpha)$:</p><p>$2\sin(2\alpha) - \sin(4\alpha)$.</p><p>Мы получили выражение, стоящее в правой части, следовательно, тождество доказано.</p><p><strong>Ответ:</strong> Тождество $8\sin^3\alpha \cdot \cos \alpha = 2\sin 2\alpha - \sin 4\alpha$ доказано.</p><p><strong>в)</strong> Для доказательства тождества $\frac{\sin^2{3\alpha}}{\sin^2\alpha} - \frac{\cos^2{3\alpha}}{\cos^2\alpha} = 8\cos 2\alpha$ преобразуем его левую часть. Приведем дроби к общему знаменателю:</p><p>$\frac{\sin^2(3\alpha)\cos^2\alpha - \cos^2(3\alpha)\sin^2\alpha}{\sin^2\alpha\cos^2\alpha}$.</p><p>Числитель является разностью квадратов, которую можно разложить на множители:</p><p>$(\sin(3\alpha)\cos\alpha - \cos(3\alpha)\sin\alpha)(\sin(3\alpha)\cos\alpha + \cos(3\alpha)\sin\alpha)$.</p><p>Используя формулы синуса суммы и разности углов, получаем:</p><p>$\sin(3\alpha-\alpha)\sin(3\alpha+\alpha) = \sin(2\alpha)\sin(4\alpha)$.</p><p>Знаменатель преобразуем с помощью формулы синуса двойного угла $\sin(2\alpha)=2\sin\alpha\cos\alpha$, из которой следует $\sin\alpha\cos\alpha = \frac{\sin(2\alpha)}{2}$:</p><p>$\sin^2\alpha\cos^2\alpha = (\sin\alpha\cos\alpha)^2 = \left(\frac{\sin(2\alpha)}{2}\right)^2 = \frac{\sin^2(2\alpha)}{4}$.</p><p>Теперь подставим преобразованные числитель и знаменатель в дробь:</p><p>$\frac{\sin(2\alpha)\sin(4\alpha)}{\frac{\sin^2(2\alpha)}{4}} = \frac{4\sin(2\alpha)\sin(4\alpha)}{\sin^2(2\alpha)} = \frac{4\sin(4\alpha)}{\sin(2\alpha)}$.</p><p>Применим формулу синуса двойного угла $\sin(4\alpha) = 2\sin(2\alpha)\cos(2\alpha)$:</p><p>$\frac{4 \cdot 2\sin(2\alpha)\cos(2\alpha)}{\sin(2\alpha)} = 8\cos(2\alpha)$.</p><p>Левая часть тождества равна правой, что и требовалось доказать.</p><p><strong>Ответ:</strong> Тождество $\frac{\sin^2{3\alpha}}{\sin^2\alpha} - \frac{\cos^2{3\alpha}}{\cos^2\alpha} = 8\cos 2\alpha$ доказано.</p><p><strong>г)</strong> Для доказательства тождества $\frac{\sin 2\alpha}{1+\cos 2\alpha} \cdot \frac{\cos\alpha}{1+\cos\alpha} = \text{tg}\frac{\alpha}{2}$ преобразуем его левую часть. Рассмотрим первую дробь:</p><p>$\frac{\sin 2\alpha}{1+\cos 2\alpha}$.</p><p>Используем формулы двойного угла: $\sin(2\alpha) = 2\sin\alpha\cos\alpha$ и $1+\cos(2\alpha) = 2\cos^2\alpha$.</p><p>$\frac{2\sin\alpha\cos\alpha}{2\cos^2\alpha} = \frac{\sin\alpha}{\cos\alpha} = \text{tg}\alpha$.</p><p>Теперь левая часть тождества имеет вид:</p><p>$\text{tg}\alpha \cdot \frac{\cos\alpha}{1+\cos\alpha} = \frac{\sin\alpha}{\cos\alpha} \cdot \frac{\cos\alpha}{1+\cos\alpha} = \frac{\sin\alpha}{1+\cos\alpha}$.</p><p>Применим формулы половинного угла: $\sin\alpha = 2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}$ и $1+\cos\alpha = 2\cos^2\frac{\alpha}{2}$.</p><p>$\frac{2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}}{2\cos^2\frac{\alpha}{2}} = \frac{\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}} = \text{tg}\frac{\alpha}{2}$.</p><p>Левая часть равна правой, тождество доказано.</p><p><strong>Ответ:</strong> Тождество $\frac{\sin 2\alpha}{1+\cos 2\alpha} \cdot \frac{\cos\alpha}{1+\cos\alpha} = \text{tg}\frac{\alpha}{2}$ доказано.</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" => "1108558" "type" => "task" ] "previous" => array:2 [ "refs" => "1108556" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1151 #items: array:1 [ 0 => App\Models\Book {#1152 #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" => 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#dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1474 #items: array:1 [ 0 => App\Models\BookPage {#1183 #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" => 1097966 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "224" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-224" "field_display_title" => "224" "field_folder" => "folder1" "field_image_name" => "224" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1187 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1186 #items: array:1 [ 0 => App\Models\Book {#1188 #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 {#1189 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1202 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1204 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1209 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1211 …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 {#1212 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1215 …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 {#1189 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1202 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1204 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1209 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1211 …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 {#1212 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1215 …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 {#1216 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ 0 => App\Models\Element {#1218 #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" => 1261117 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261117 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219 …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" => "1097967" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097965" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:8 [ 0 => App\Models\Task {#1227 #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" => 1108551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "224" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-805" "field_display_title" => "805" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228 …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 {#1230 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1231 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1264 …2} "content" => Illuminate\Database\Eloquent\Collection {#1361 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1380 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108551 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "224" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-805" "field_display_title" => "805" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228 …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 {#1230 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1231 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1264 …2} "content" => Illuminate\Database\Eloquent\Collection {#1361 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1380 …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 {#1381 #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" => 1108552 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "224" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-806" "field_display_title" => "806" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1382 …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 {#1383 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1384 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1385 …2} "content" => Illuminate\Database\Eloquent\Collection {#1386 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1393 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108552 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "224" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-806" "field_display_title" => "806" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1382 …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 {#1383 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1384 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1385 …2} "content" => Illuminate\Database\Eloquent\Collection {#1386 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1393 …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 {#1394 #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" => 1108553 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "224" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-807" "field_display_title" => "807" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1395 …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 {#1396 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1397 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1398 …2} "content" => Illuminate\Database\Eloquent\Collection {#1399 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1406 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108553 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "224" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-807" "field_display_title" => "807" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1395 …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 {#1396 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1397 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1398 …2} "content" => Illuminate\Database\Eloquent\Collection {#1399 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1406 …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 {#1407 #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" => 1108554 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "224" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-808" "field_display_title" => "808" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1408 …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 {#1409 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1410 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1411 …2} "content" => Illuminate\Database\Eloquent\Collection {#1412 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1419 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108554 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "224" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-808" "field_display_title" => "808" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1408 …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 {#1409 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1410 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1411 …2} "content" => Illuminate\Database\Eloquent\Collection {#1412 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1419 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1420 #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" => 1108555 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "224" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-809" "field_display_title" => "809" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1421 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null 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№811 (с. 224)
Условие. №811 (с. 224)
скриншот условия
811. Докажите тождество:
а)
$4\sin^2 2\alpha \cdot \cos 2\alpha = \cos 2\alpha - \cos 6\alpha;$
б)
$8\sin^3 \alpha \cdot \cos \alpha = 2\sin 2\alpha - \sin 4\alpha;$
в)
$\frac{\sin^2 3\alpha}{\sin^2 \alpha} - \frac{\cos^2 3\alpha}{\cos^2 \alpha} = 8\cos 2\alpha;$
г)
$\frac{\sin 2\alpha}{1 + \cos 2\alpha} \cdot \frac{\cos \alpha}{1 + \cos \alpha} = \text{tg}\frac{\alpha}{2}.$
Решение. №811 (с. 224)
Решение 2 (rus). №811 (с. 224)
а) Для доказательства тождества $4\sin^2{2\alpha} \cdot \cos 2\alpha = \cos 2\alpha - \cos 6\alpha$ преобразуем его правую часть. Воспользуемся формулой разности косинусов: $\cos x - \cos y = -2\sin\frac{x+y}{2}\sin\frac{x-y}{2}$.
$\cos 2\alpha - \cos 6\alpha = -2\sin\frac{2\alpha+6\alpha}{2}\sin\frac{2\alpha-6\alpha}{2} = -2\sin(4\alpha)\sin(-2\alpha)$.
Поскольку синус является нечетной функцией, $\sin(-2\alpha) = -\sin(2\alpha)$. Тогда выражение принимает вид:
$2\sin(4\alpha)\sin(2\alpha)$.
Применим формулу синуса двойного угла $\sin(2x) = 2\sin x \cos x$ для члена $\sin(4\alpha)$:
$2 \cdot (2\sin(2\alpha)\cos(2\alpha)) \cdot \sin(2\alpha) = 4\sin^2(2\alpha)\cos(2\alpha)$.
Мы получили выражение, стоящее в левой части исходного тождества. Таким образом, тождество доказано.
Ответ: Тождество $4\sin^2{2\alpha} \cdot \cos 2\alpha = \cos 2\alpha - \cos 6\alpha$ доказано.
б) Для доказательства тождества $8\sin^3\alpha \cdot \cos \alpha = 2\sin 2\alpha - \sin 4\alpha$ преобразуем его левую часть.
$8\sin^3\alpha \cdot \cos \alpha = 4\sin^2\alpha \cdot (2\sin\alpha\cos\alpha)$.
Используем формулу синуса двойного угла $2\sin\alpha\cos\alpha = \sin(2\alpha)$:
$4\sin^2\alpha \sin(2\alpha)$.
Теперь применим формулу понижения степени $\sin^2\alpha = \frac{1-\cos(2\alpha)}{2}$:
$4 \cdot \frac{1-\cos(2\alpha)}{2} \cdot \sin(2\alpha) = 2(1-\cos(2\alpha))\sin(2\alpha) = 2\sin(2\alpha) - 2\sin(2\alpha)\cos(2\alpha)$.
Снова используем формулу синуса двойного угла для выражения $2\sin(2\alpha)\cos(2\alpha)$, которое равно $\sin(4\alpha)$:
$2\sin(2\alpha) - \sin(4\alpha)$.
Мы получили выражение, стоящее в правой части, следовательно, тождество доказано.
Ответ: Тождество $8\sin^3\alpha \cdot \cos \alpha = 2\sin 2\alpha - \sin 4\alpha$ доказано.
в) Для доказательства тождества $\frac{\sin^2{3\alpha}}{\sin^2\alpha} - \frac{\cos^2{3\alpha}}{\cos^2\alpha} = 8\cos 2\alpha$ преобразуем его левую часть. Приведем дроби к общему знаменателю:
$\frac{\sin^2(3\alpha)\cos^2\alpha - \cos^2(3\alpha)\sin^2\alpha}{\sin^2\alpha\cos^2\alpha}$.
Числитель является разностью квадратов, которую можно разложить на множители:
$(\sin(3\alpha)\cos\alpha - \cos(3\alpha)\sin\alpha)(\sin(3\alpha)\cos\alpha + \cos(3\alpha)\sin\alpha)$.
Используя формулы синуса суммы и разности углов, получаем:
$\sin(3\alpha-\alpha)\sin(3\alpha+\alpha) = \sin(2\alpha)\sin(4\alpha)$.
Знаменатель преобразуем с помощью формулы синуса двойного угла $\sin(2\alpha)=2\sin\alpha\cos\alpha$, из которой следует $\sin\alpha\cos\alpha = \frac{\sin(2\alpha)}{2}$:
$\sin^2\alpha\cos^2\alpha = (\sin\alpha\cos\alpha)^2 = \left(\frac{\sin(2\alpha)}{2}\right)^2 = \frac{\sin^2(2\alpha)}{4}$.
Теперь подставим преобразованные числитель и знаменатель в дробь:
$\frac{\sin(2\alpha)\sin(4\alpha)}{\frac{\sin^2(2\alpha)}{4}} = \frac{4\sin(2\alpha)\sin(4\alpha)}{\sin^2(2\alpha)} = \frac{4\sin(4\alpha)}{\sin(2\alpha)}$.
Применим формулу синуса двойного угла $\sin(4\alpha) = 2\sin(2\alpha)\cos(2\alpha)$:
$\frac{4 \cdot 2\sin(2\alpha)\cos(2\alpha)}{\sin(2\alpha)} = 8\cos(2\alpha)$.
Левая часть тождества равна правой, что и требовалось доказать.
Ответ: Тождество $\frac{\sin^2{3\alpha}}{\sin^2\alpha} - \frac{\cos^2{3\alpha}}{\cos^2\alpha} = 8\cos 2\alpha$ доказано.
г) Для доказательства тождества $\frac{\sin 2\alpha}{1+\cos 2\alpha} \cdot \frac{\cos\alpha}{1+\cos\alpha} = \text{tg}\frac{\alpha}{2}$ преобразуем его левую часть. Рассмотрим первую дробь:
$\frac{\sin 2\alpha}{1+\cos 2\alpha}$.
Используем формулы двойного угла: $\sin(2\alpha) = 2\sin\alpha\cos\alpha$ и $1+\cos(2\alpha) = 2\cos^2\alpha$.
$\frac{2\sin\alpha\cos\alpha}{2\cos^2\alpha} = \frac{\sin\alpha}{\cos\alpha} = \text{tg}\alpha$.
Теперь левая часть тождества имеет вид:
$\text{tg}\alpha \cdot \frac{\cos\alpha}{1+\cos\alpha} = \frac{\sin\alpha}{\cos\alpha} \cdot \frac{\cos\alpha}{1+\cos\alpha} = \frac{\sin\alpha}{1+\cos\alpha}$.
Применим формулы половинного угла: $\sin\alpha = 2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}$ и $1+\cos\alpha = 2\cos^2\frac{\alpha}{2}$.
$\frac{2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}}{2\cos^2\frac{\alpha}{2}} = \frac{\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}} = \text{tg}\frac{\alpha}{2}$.
Левая часть равна правой, тождество доказано.
Ответ: Тождество $\frac{\sin 2\alpha}{1+\cos 2\alpha} \cdot \frac{\cos\alpha}{1+\cos\alpha} = \text{tg}\frac{\alpha}{2}$ доказано.
Другие задания:
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