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Номер 773, страница 218 - 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. 29. Преобразование произведения тригонометрических функций в сумму и разность - номер 773, страница 218.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108516 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-773" "field_display_title" => "773" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …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 {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …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 {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …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" => "" 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"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 {#1068 #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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"field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …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 {#1074 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 …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 {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …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 {#1101 #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 {#1072} "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 {#1070} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1101} ] #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 {#1102 #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" => 1107671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "29. Преобразование произведения тригонометрических функций в сумму и разность" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "216" "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 {#1104 #items: array:1 [ 0 => App\Models\Book {#1105 #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 {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …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 {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …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 {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …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 {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …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 {#1133 #items: array:1 [ 0 => App\Models\Branch {#1134 #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 {#1135 …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 {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137 …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 {#1135 …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 {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137 …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" => 1107671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "29. Преобразование произведения тригонометрических функций в сумму и разность" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103} "field_page_start" => "216" "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 {#1104} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1133} ] #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 {#1148 #items: array:3 [ 0 => App\Models\Element {#1158 #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" => 1277128 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1167 #items: array:1 [ 0 => App\Models\Edition {#1159 #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 {#1161 …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 {#1162 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1164 …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 {#1161 …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 {#1162 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1164 …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" => "1108516" "type" => "task" ] "text" => "<p><strong>773.</strong> Найдите значение выражения:</p><p><strong>a)</strong> $ \cos 8\alpha - \cos 6\alpha - 2\cos 5\alpha \cdot \cos 3\alpha, \text{ если } \cos \alpha = -\frac{1}{\sqrt{3}}; $</p><p><strong>б)</strong> $ \sin 2\alpha \cdot \cos 5\alpha - \sin \alpha \cdot \cos 6\alpha, \text{ если } \sin \alpha = -\frac{1}{\sqrt{3}}; $</p><p><strong>в)</strong> $ \cos 7\alpha \cdot \cos 4\alpha - \cos 8\alpha \cdot \cos 3\alpha, \text{ если } \cos \alpha = \frac{\sqrt{3}}{3}; $</p><p><strong>г)</strong> $ \sin 2\alpha \cdot \cos 5\alpha - \sin \alpha \cdot \cos 6\alpha, \text{ если } \sin \alpha = \frac{\sqrt{5}}{4}. $</p>" "img" => array:1 [ 0 => array:5 [ "name" => "773-1.jpg" "alt" => null "width" => "1341" "height" => 666 "path" => "/media/algebra_09/soltan-u19/0-1/773-1.webp?ts=1752834761" ] ] ] #original: array:7 [ "id" => 1277128 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1167} "task" => array:2 [ "refs" => "1108516" "type" => "task" ] "text" => "<p><strong>773.</strong> Найдите значение выражения:</p><p><strong>a)</strong> $ \cos 8\alpha - \cos 6\alpha - 2\cos 5\alpha \cdot \cos 3\alpha, \text{ если } \cos \alpha = -\frac{1}{\sqrt{3}}; $</p><p><strong>б)</strong> $ \sin 2\alpha \cdot \cos 5\alpha - \sin \alpha \cdot \cos 6\alpha, \text{ если } \sin \alpha = -\frac{1}{\sqrt{3}}; $</p><p><strong>в)</strong> $ \cos 7\alpha \cdot \cos 4\alpha - \cos 8\alpha \cdot \cos 3\alpha, \text{ если } \cos \alpha = \frac{\sqrt{3}}{3}; $</p><p><strong>г)</strong> $ \sin 2\alpha \cdot \cos 5\alpha - \sin \alpha \cdot \cos 6\alpha, \text{ если } \sin \alpha = \frac{\sqrt{5}}{4}. $</p>" "img" => array:1 [ 0 => array:5 [ "name" => "773-1.jpg" "alt" => null "width" => "1341" "height" => 666 "path" => "/media/algebra_09/soltan-u19/0-1/773-1.webp?ts=1752834761" ] ] ] #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 {#1165 #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" => 1278419 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:1 [ 0 => App\Models\Edition {#1166 #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 {#1169 …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 {#1170 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172 …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 {#1169 …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 {#1170 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172 …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" => "1108516" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "773-1.jpg" "alt" => null "width" => "1361" "height" => 833 "path" => "/media/algebra_09/soltan-u19/1-1/773-1.webp?ts=1752831566" ] 1 => array:5 [ "name" => "773-2.jpg" "alt" => null "width" => "1418" "height" => 2414 "path" => "/media/algebra_09/soltan-u19/1-1/773-2.webp?ts=1752831566" ] 2 => array:5 [ "name" => "773-3.jpg" "alt" => null "width" => "1358" "height" => 1447 "path" => "/media/algebra_09/soltan-u19/1-1/773-3.webp?ts=1752831566" ] ] ] #original: array:6 [ "id" => 1278419 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175} "task" => array:2 [ "refs" => "1108516" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "773-1.jpg" "alt" => null "width" => "1361" "height" => 833 "path" => "/media/algebra_09/soltan-u19/1-1/773-1.webp?ts=1752831566" ] 1 => array:5 [ "name" => "773-2.jpg" "alt" => null "width" => "1418" "height" => 2414 "path" => "/media/algebra_09/soltan-u19/1-1/773-2.webp?ts=1752831566" ] 2 => array:5 [ "name" => "773-3.jpg" "alt" => null "width" => "1358" "height" => 1447 "path" => "/media/algebra_09/soltan-u19/1-1/773-3.webp?ts=1752831566" ] ] ] #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 {#1173 #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" => 1554374 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1183 #items: array:1 [ 0 => App\Models\Edition {#1174 #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 {#1177 …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 {#1178 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1180 …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 {#1177 …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 {#1178 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1180 …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" => "1108516" "type" => "task" ] "text" => "<p><strong>а)</strong> Для нахождения значения выражения $\cos 8\alpha - \cos 6\alpha - 2\cos 5\alpha \cdot \cos 3\alpha$ преобразуем его, используя тригонометрические формулы.<br>Применим формулу произведения косинусов для последнего слагаемого: $2\cos x \cos y = \cos(x+y) + \cos(x-y)$.<br>$2\cos 5\alpha \cdot \cos 3\alpha = \cos(5\alpha + 3\alpha) + \cos(5\alpha - 3\alpha) = \cos 8\alpha + \cos 2\alpha$.<br>Подставим полученное выражение в исходное:<br>$\cos 8\alpha - \cos 6\alpha - (\cos 8\alpha + \cos 2\alpha) = \cos 8\alpha - \cos 6\alpha - \cos 8\alpha - \cos 2\alpha = -(\cos 6\alpha + \cos 2\alpha)$.<br>Теперь необходимо найти значения $\cos 2\alpha$ и $\cos 6\alpha$, зная, что $\cos \alpha = -\frac{1}{\sqrt{3}}$.<br>Используем формулу косинуса двойного угла $\cos 2\alpha = 2\cos^2 \alpha - 1$:<br>$\cos 2\alpha = 2\left(-\frac{1}{\sqrt{3}}\right)^2 - 1 = 2 \cdot \frac{1}{3} - 1 = \frac{2}{3} - 1 = -\frac{1}{3}$.<br>Для вычисления $\cos 6\alpha$ применим формулу косинуса тройного угла к аргументу $2\alpha$: $\cos 3x = 4\cos^3 x - 3\cos x$.<br>$\cos 6\alpha = \cos(3 \cdot 2\alpha) = 4\cos^3(2\alpha) - 3\cos(2\alpha)$.<br>$\cos 6\alpha = 4\left(-\frac{1}{3}\right)^3 - 3\left(-\frac{1}{3}\right) = 4\left(-\frac{1}{27}\right) + 1 = -\frac{4}{27} + \frac{27}{27} = \frac{23}{27}$.<br>Теперь вычислим значение всего выражения:<br>$-(\cos 6\alpha + \cos 2\alpha) = -\left(\frac{23}{27} + \left(-\frac{1}{3}\right)\right) = -\left(\frac{23}{27} - \frac{9}{27}\right) = -\frac{14}{27}$.<br>Ответ: $-\frac{14}{27}$</p><p><strong>б)</strong> Для нахождения значения выражения $\sin 2\alpha \cdot \cos 5\alpha - \sin \alpha \cdot \cos 6\alpha$ преобразуем его, используя формулу произведения синуса на косинус $\sin x \cos y = \frac{1}{2}(\sin(x+y) + \sin(x-y))$.<br>$\sin 2\alpha \cos 5\alpha = \frac{1}{2}(\sin(2\alpha+5\alpha) + \sin(2\alpha-5\alpha)) = \frac{1}{2}(\sin 7\alpha + \sin(-3\alpha)) = \frac{1}{2}(\sin 7\alpha - \sin 3\alpha)$.<br>$\sin \alpha \cos 6\alpha = \frac{1}{2}(\sin(\alpha+6\alpha) + \sin(\alpha-6\alpha)) = \frac{1}{2}(\sin 7\alpha + \sin(-5\alpha)) = \frac{1}{2}(\sin 7\alpha - \sin 5\alpha)$.<br>Подставим в исходное выражение:<br>$\frac{1}{2}(\sin 7\alpha - \sin 3\alpha) - \frac{1}{2}(\sin 7\alpha - \sin 5\alpha) = \frac{1}{2}\sin 7\alpha - \frac{1}{2}\sin 3\alpha - \frac{1}{2}\sin 7\alpha + \frac{1}{2}\sin 5\alpha = \frac{1}{2}(\sin 5\alpha - \sin 3\alpha)$.<br>Теперь применим формулу разности синусов $\sin x - \sin y = 2\cos\frac{x+y}{2}\sin\frac{x-y}{2}$:<br>$\frac{1}{2}(2\cos\frac{5\alpha+3\alpha}{2}\sin\frac{5\alpha-3\alpha}{2}) = \cos 4\alpha \sin \alpha$.<br>Нам дано, что $\sin \alpha = -\frac{1}{\sqrt{3}}$. Найдем $\cos 4\alpha$.<br>Сначала найдем $\cos 2\alpha$ по формуле $\cos 2\alpha = 1 - 2\sin^2 \alpha$:<br>$\cos 2\alpha = 1 - 2\left(-\frac{1}{\sqrt{3}}\right)^2 = 1 - 2 \cdot \frac{1}{3} = \frac{1}{3}$.<br>Теперь найдем $\cos 4\alpha$ по формуле $\cos 4\alpha = 2\cos^2(2\alpha) - 1$:<br>$\cos 4\alpha = 2\left(\frac{1}{3}\right)^2 - 1 = 2 \cdot \frac{1}{9} - 1 = \frac{2}{9} - 1 = -\frac{7}{9}$.<br>Вычислим итоговое значение:<br>$\cos 4\alpha \sin \alpha = \left(-\frac{7}{9}\right) \cdot \left(-\frac{1}{\sqrt{3}}\right) = \frac{7}{9\sqrt{3}} = \frac{7\sqrt{3}}{27}$.<br>Ответ: $\frac{7\sqrt{3}}{27}$</p><p><strong>в)</strong> Для нахождения значения выражения $\cos 7\alpha \cdot \cos 4\alpha - \cos 8\alpha \cdot \cos 3\alpha$ преобразуем его, используя формулу произведения косинусов $\cos x \cos y = \frac{1}{2}(\cos(x+y) + \cos(x-y))$.<br>$\cos 7\alpha \cos 4\alpha = \frac{1}{2}(\cos(7\alpha+4\alpha) + \cos(7\alpha-4\alpha)) = \frac{1}{2}(\cos 11\alpha + \cos 3\alpha)$.<br>$\cos 8\alpha \cos 3\alpha = \frac{1}{2}(\cos(8\alpha+3\alpha) + \cos(8\alpha-3\alpha)) = \frac{1}{2}(\cos 11\alpha + \cos 5\alpha)$.<br>Подставим в исходное выражение:<br>$\frac{1}{2}(\cos 11\alpha + \cos 3\alpha) - \frac{1}{2}(\cos 11\alpha + \cos 5\alpha) = \frac{1}{2}\cos 11\alpha + \frac{1}{2}\cos 3\alpha - \frac{1}{2}\cos 11\alpha - \frac{1}{2}\cos 5\alpha = \frac{1}{2}(\cos 3\alpha - \cos 5\alpha)$.<br>Применим формулу разности косинусов $\cos x - \cos y = -2\sin\frac{x+y}{2}\sin\frac{x-y}{2}$:<br>$\frac{1}{2}(-2\sin\frac{3\alpha+5\alpha}{2}\sin\frac{3\alpha-5\alpha}{2}) = -\sin(4\alpha)\sin(-\alpha) = \sin 4\alpha \sin \alpha$.<br>Выражение упростилось до $\sin 4\alpha \sin \alpha$. Нам дано $\cos \alpha = \frac{\sqrt{3}}{3}$.<br>Чтобы найти значение, выразим $\sin 4\alpha \sin \alpha$ через $\cos \alpha$.<br>$\sin 4\alpha = 2\sin 2\alpha \cos 2\alpha = 2(2\sin\alpha\cos\alpha)(2\cos^2\alpha-1)$.<br>Тогда все выражение равно $(\sin\alpha) \cdot (4\sin\alpha\cos\alpha(2\cos^2\alpha-1)) = 4\sin^2\alpha\cos\alpha(2\cos^2\alpha-1)$.<br>Найдем $\sin^2\alpha$ из основного тригонометрического тождества $\sin^2\alpha = 1 - \cos^2\alpha$:<br>$\sin^2\alpha = 1 - \left(\frac{\sqrt{3}}{3}\right)^2 = 1 - \frac{3}{9} = 1 - \frac{1}{3} = \frac{2}{3}$.<br>Подставим известные значения:<br>$4\left(\frac{2}{3}\right)\left(\frac{\sqrt{3}}{3}\right)\left(2\left(\frac{\sqrt{3}}{3}\right)^2-1\right) = \frac{8\sqrt{3}}{9}\left(2\cdot\frac{1}{3}-1\right) = \frac{8\sqrt{3}}{9}\left(\frac{2}{3}-1\right) = \frac{8\sqrt{3}}{9}\left(-\frac{1}{3}\right) = -\frac{8\sqrt{3}}{27}$.<br>Ответ: $-\frac{8\sqrt{3}}{27}$</p><p><strong>г)</strong> Исходное выражение $\sin 2\alpha \cdot \cos 5\alpha - \sin \alpha \cdot \cos 6\alpha$ полностью совпадает с выражением из пункта б).<br>Как было показано в решении пункта б), данное выражение можно упростить до вида $\cos 4\alpha \sin \alpha$.<br>По условию $\sin \alpha = \frac{\sqrt{5}}{4}$. Нам осталось найти $\cos 4\alpha$.<br>Найдем $\cos 2\alpha$ по формуле $\cos 2\alpha = 1 - 2\sin^2 \alpha$:<br>$\cos 2\alpha = 1 - 2\left(\frac{\sqrt{5}}{4}\right)^2 = 1 - 2 \cdot \frac{5}{16} = 1 - \frac{10}{16} = 1 - \frac{5}{8} = \frac{3}{8}$.<br>Теперь найдем $\cos 4\alpha$ по формуле $\cos 4\alpha = 2\cos^2(2\alpha) - 1$:<br>$\cos 4\alpha = 2\left(\frac{3}{8}\right)^2 - 1 = 2 \cdot \frac{9}{64} - 1 = \frac{18}{64} - 1 = \frac{9}{32} - \frac{32}{32} = -\frac{23}{32}$.<br>Вычислим итоговое значение $\cos 4\alpha \sin \alpha$:<br>$\left(-\frac{23}{32}\right) \cdot \left(\frac{\sqrt{5}}{4}\right) = -\frac{23\sqrt{5}}{128}$.<br>Ответ: $-\frac{23\sqrt{5}}{128}$</p>" ] #original: array:6 [ "id" => 1554374 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1183} "task" => array:2 [ "refs" => "1108516" "type" => "task" ] "text" => "<p><strong>а)</strong> Для нахождения значения выражения $\cos 8\alpha - \cos 6\alpha - 2\cos 5\alpha \cdot \cos 3\alpha$ преобразуем его, используя тригонометрические формулы.<br>Применим формулу произведения косинусов для последнего слагаемого: $2\cos x \cos y = \cos(x+y) + \cos(x-y)$.<br>$2\cos 5\alpha \cdot \cos 3\alpha = \cos(5\alpha + 3\alpha) + \cos(5\alpha - 3\alpha) = \cos 8\alpha + \cos 2\alpha$.<br>Подставим полученное выражение в исходное:<br>$\cos 8\alpha - \cos 6\alpha - (\cos 8\alpha + \cos 2\alpha) = \cos 8\alpha - \cos 6\alpha - \cos 8\alpha - \cos 2\alpha = -(\cos 6\alpha + \cos 2\alpha)$.<br>Теперь необходимо найти значения $\cos 2\alpha$ и $\cos 6\alpha$, зная, что $\cos \alpha = -\frac{1}{\sqrt{3}}$.<br>Используем формулу косинуса двойного угла $\cos 2\alpha = 2\cos^2 \alpha - 1$:<br>$\cos 2\alpha = 2\left(-\frac{1}{\sqrt{3}}\right)^2 - 1 = 2 \cdot \frac{1}{3} - 1 = \frac{2}{3} - 1 = -\frac{1}{3}$.<br>Для вычисления $\cos 6\alpha$ применим формулу косинуса тройного угла к аргументу $2\alpha$: $\cos 3x = 4\cos^3 x - 3\cos x$.<br>$\cos 6\alpha = \cos(3 \cdot 2\alpha) = 4\cos^3(2\alpha) - 3\cos(2\alpha)$.<br>$\cos 6\alpha = 4\left(-\frac{1}{3}\right)^3 - 3\left(-\frac{1}{3}\right) = 4\left(-\frac{1}{27}\right) + 1 = -\frac{4}{27} + \frac{27}{27} = \frac{23}{27}$.<br>Теперь вычислим значение всего выражения:<br>$-(\cos 6\alpha + \cos 2\alpha) = -\left(\frac{23}{27} + \left(-\frac{1}{3}\right)\right) = -\left(\frac{23}{27} - \frac{9}{27}\right) = -\frac{14}{27}$.<br>Ответ: $-\frac{14}{27}$</p><p><strong>б)</strong> Для нахождения значения выражения $\sin 2\alpha \cdot \cos 5\alpha - \sin \alpha \cdot \cos 6\alpha$ преобразуем его, используя формулу произведения синуса на косинус $\sin x \cos y = \frac{1}{2}(\sin(x+y) + \sin(x-y))$.<br>$\sin 2\alpha \cos 5\alpha = \frac{1}{2}(\sin(2\alpha+5\alpha) + \sin(2\alpha-5\alpha)) = \frac{1}{2}(\sin 7\alpha + \sin(-3\alpha)) = \frac{1}{2}(\sin 7\alpha - \sin 3\alpha)$.<br>$\sin \alpha \cos 6\alpha = \frac{1}{2}(\sin(\alpha+6\alpha) + \sin(\alpha-6\alpha)) = \frac{1}{2}(\sin 7\alpha + \sin(-5\alpha)) = \frac{1}{2}(\sin 7\alpha - \sin 5\alpha)$.<br>Подставим в исходное выражение:<br>$\frac{1}{2}(\sin 7\alpha - \sin 3\alpha) - \frac{1}{2}(\sin 7\alpha - \sin 5\alpha) = \frac{1}{2}\sin 7\alpha - \frac{1}{2}\sin 3\alpha - \frac{1}{2}\sin 7\alpha + \frac{1}{2}\sin 5\alpha = \frac{1}{2}(\sin 5\alpha - \sin 3\alpha)$.<br>Теперь применим формулу разности синусов $\sin x - \sin y = 2\cos\frac{x+y}{2}\sin\frac{x-y}{2}$:<br>$\frac{1}{2}(2\cos\frac{5\alpha+3\alpha}{2}\sin\frac{5\alpha-3\alpha}{2}) = \cos 4\alpha \sin \alpha$.<br>Нам дано, что $\sin \alpha = -\frac{1}{\sqrt{3}}$. Найдем $\cos 4\alpha$.<br>Сначала найдем $\cos 2\alpha$ по формуле $\cos 2\alpha = 1 - 2\sin^2 \alpha$:<br>$\cos 2\alpha = 1 - 2\left(-\frac{1}{\sqrt{3}}\right)^2 = 1 - 2 \cdot \frac{1}{3} = \frac{1}{3}$.<br>Теперь найдем $\cos 4\alpha$ по формуле $\cos 4\alpha = 2\cos^2(2\alpha) - 1$:<br>$\cos 4\alpha = 2\left(\frac{1}{3}\right)^2 - 1 = 2 \cdot \frac{1}{9} - 1 = \frac{2}{9} - 1 = -\frac{7}{9}$.<br>Вычислим итоговое значение:<br>$\cos 4\alpha \sin \alpha = \left(-\frac{7}{9}\right) \cdot \left(-\frac{1}{\sqrt{3}}\right) = \frac{7}{9\sqrt{3}} = \frac{7\sqrt{3}}{27}$.<br>Ответ: $\frac{7\sqrt{3}}{27}$</p><p><strong>в)</strong> Для нахождения значения выражения $\cos 7\alpha \cdot \cos 4\alpha - \cos 8\alpha \cdot \cos 3\alpha$ преобразуем его, используя формулу произведения косинусов $\cos x \cos y = \frac{1}{2}(\cos(x+y) + \cos(x-y))$.<br>$\cos 7\alpha \cos 4\alpha = \frac{1}{2}(\cos(7\alpha+4\alpha) + \cos(7\alpha-4\alpha)) = \frac{1}{2}(\cos 11\alpha + \cos 3\alpha)$.<br>$\cos 8\alpha \cos 3\alpha = \frac{1}{2}(\cos(8\alpha+3\alpha) + \cos(8\alpha-3\alpha)) = \frac{1}{2}(\cos 11\alpha + \cos 5\alpha)$.<br>Подставим в исходное выражение:<br>$\frac{1}{2}(\cos 11\alpha + \cos 3\alpha) - \frac{1}{2}(\cos 11\alpha + \cos 5\alpha) = \frac{1}{2}\cos 11\alpha + \frac{1}{2}\cos 3\alpha - \frac{1}{2}\cos 11\alpha - \frac{1}{2}\cos 5\alpha = \frac{1}{2}(\cos 3\alpha - \cos 5\alpha)$.<br>Применим формулу разности косинусов $\cos x - \cos y = -2\sin\frac{x+y}{2}\sin\frac{x-y}{2}$:<br>$\frac{1}{2}(-2\sin\frac{3\alpha+5\alpha}{2}\sin\frac{3\alpha-5\alpha}{2}) = -\sin(4\alpha)\sin(-\alpha) = \sin 4\alpha \sin \alpha$.<br>Выражение упростилось до $\sin 4\alpha \sin \alpha$. Нам дано $\cos \alpha = \frac{\sqrt{3}}{3}$.<br>Чтобы найти значение, выразим $\sin 4\alpha \sin \alpha$ через $\cos \alpha$.<br>$\sin 4\alpha = 2\sin 2\alpha \cos 2\alpha = 2(2\sin\alpha\cos\alpha)(2\cos^2\alpha-1)$.<br>Тогда все выражение равно $(\sin\alpha) \cdot (4\sin\alpha\cos\alpha(2\cos^2\alpha-1)) = 4\sin^2\alpha\cos\alpha(2\cos^2\alpha-1)$.<br>Найдем $\sin^2\alpha$ из основного тригонометрического тождества $\sin^2\alpha = 1 - \cos^2\alpha$:<br>$\sin^2\alpha = 1 - \left(\frac{\sqrt{3}}{3}\right)^2 = 1 - \frac{3}{9} = 1 - \frac{1}{3} = \frac{2}{3}$.<br>Подставим известные значения:<br>$4\left(\frac{2}{3}\right)\left(\frac{\sqrt{3}}{3}\right)\left(2\left(\frac{\sqrt{3}}{3}\right)^2-1\right) = \frac{8\sqrt{3}}{9}\left(2\cdot\frac{1}{3}-1\right) = \frac{8\sqrt{3}}{9}\left(\frac{2}{3}-1\right) = \frac{8\sqrt{3}}{9}\left(-\frac{1}{3}\right) = -\frac{8\sqrt{3}}{27}$.<br>Ответ: $-\frac{8\sqrt{3}}{27}$</p><p><strong>г)</strong> Исходное выражение $\sin 2\alpha \cdot \cos 5\alpha - \sin \alpha \cdot \cos 6\alpha$ полностью совпадает с выражением из пункта б).<br>Как было показано в решении пункта б), данное выражение можно упростить до вида $\cos 4\alpha \sin \alpha$.<br>По условию $\sin \alpha = \frac{\sqrt{5}}{4}$. Нам осталось найти $\cos 4\alpha$.<br>Найдем $\cos 2\alpha$ по формуле $\cos 2\alpha = 1 - 2\sin^2 \alpha$:<br>$\cos 2\alpha = 1 - 2\left(\frac{\sqrt{5}}{4}\right)^2 = 1 - 2 \cdot \frac{5}{16} = 1 - \frac{10}{16} = 1 - \frac{5}{8} = \frac{3}{8}$.<br>Теперь найдем $\cos 4\alpha$ по формуле $\cos 4\alpha = 2\cos^2(2\alpha) - 1$:<br>$\cos 4\alpha = 2\left(\frac{3}{8}\right)^2 - 1 = 2 \cdot \frac{9}{64} - 1 = \frac{18}{64} - 1 = \frac{9}{32} - \frac{32}{32} = -\frac{23}{32}$.<br>Вычислим итоговое значение $\cos 4\alpha \sin \alpha$:<br>$\left(-\frac{23}{32}\right) \cdot \left(\frac{\sqrt{5}}{4}\right) = -\frac{23\sqrt{5}}{128}$.<br>Ответ: $-\frac{23\sqrt{5}}{128}$</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" 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App\Models\Term {#1146 #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 {#1145 #items: array:1 [ 0 => App\Models\Term {#1144 #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 {#1142 #items: array:3 [ 0 => App\Models\Term {#1143 #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 {#1149 #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 {#1147 #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 {#1154 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:1 [ 0 => App\Models\Term {#1152 #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 {#1150 #items: array:1 [ 0 => App\Models\Term {#1153 #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 {#1151 #items: array:1 [ 0 => App\Models\Term {#1181 #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 {#1182 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1184 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1185 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1186 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1187 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1188 #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 {#1189 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1190 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1191 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192 #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 {#1141} "field_class" => Illuminate\Database\Eloquent\Collection {#1139} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1145} "field_author" => Illuminate\Database\Eloquent\Collection {#1142} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1154} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1156} "field_country" => Illuminate\Database\Eloquent\Collection {#1150} "field_city" => Illuminate\Database\Eloquent\Collection {#1151} "field_series" => Illuminate\Database\Eloquent\Collection {#1182} "field_umk" => Illuminate\Database\Eloquent\Collection {#1184} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1185} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1186} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1187} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1188} "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 {#1189} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1190} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1191} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192} "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 {#1357 #items: array:1 [ 0 => App\Models\BookPage {#1032 #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" => 1097960 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "218" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-218" "field_display_title" => "218" "field_folder" => "folder1" "field_image_name" => "218" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1023 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1026 #items: array:1 [ 0 => App\Models\Book {#1193 #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 {#1194 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1197 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1201 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1205 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1216 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1217 …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 {#1218 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1220 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1221 …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 {#1194 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1197 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1201 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1205 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1216 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1217 …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" 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№773 (с. 218)
Условие. №773 (с. 218)
скриншот условия
773. Найдите значение выражения:
a) $ \cos 8\alpha - \cos 6\alpha - 2\cos 5\alpha \cdot \cos 3\alpha, \text{ если } \cos \alpha = -\frac{1}{\sqrt{3}}; $
б) $ \sin 2\alpha \cdot \cos 5\alpha - \sin \alpha \cdot \cos 6\alpha, \text{ если } \sin \alpha = -\frac{1}{\sqrt{3}}; $
в) $ \cos 7\alpha \cdot \cos 4\alpha - \cos 8\alpha \cdot \cos 3\alpha, \text{ если } \cos \alpha = \frac{\sqrt{3}}{3}; $
г) $ \sin 2\alpha \cdot \cos 5\alpha - \sin \alpha \cdot \cos 6\alpha, \text{ если } \sin \alpha = \frac{\sqrt{5}}{4}. $
Решение. №773 (с. 218)
Решение 2 (rus). №773 (с. 218)
а) Для нахождения значения выражения $\cos 8\alpha - \cos 6\alpha - 2\cos 5\alpha \cdot \cos 3\alpha$ преобразуем его, используя тригонометрические формулы.
Применим формулу произведения косинусов для последнего слагаемого: $2\cos x \cos y = \cos(x+y) + \cos(x-y)$.
$2\cos 5\alpha \cdot \cos 3\alpha = \cos(5\alpha + 3\alpha) + \cos(5\alpha - 3\alpha) = \cos 8\alpha + \cos 2\alpha$.
Подставим полученное выражение в исходное:
$\cos 8\alpha - \cos 6\alpha - (\cos 8\alpha + \cos 2\alpha) = \cos 8\alpha - \cos 6\alpha - \cos 8\alpha - \cos 2\alpha = -(\cos 6\alpha + \cos 2\alpha)$.
Теперь необходимо найти значения $\cos 2\alpha$ и $\cos 6\alpha$, зная, что $\cos \alpha = -\frac{1}{\sqrt{3}}$.
Используем формулу косинуса двойного угла $\cos 2\alpha = 2\cos^2 \alpha - 1$:
$\cos 2\alpha = 2\left(-\frac{1}{\sqrt{3}}\right)^2 - 1 = 2 \cdot \frac{1}{3} - 1 = \frac{2}{3} - 1 = -\frac{1}{3}$.
Для вычисления $\cos 6\alpha$ применим формулу косинуса тройного угла к аргументу $2\alpha$: $\cos 3x = 4\cos^3 x - 3\cos x$.
$\cos 6\alpha = \cos(3 \cdot 2\alpha) = 4\cos^3(2\alpha) - 3\cos(2\alpha)$.
$\cos 6\alpha = 4\left(-\frac{1}{3}\right)^3 - 3\left(-\frac{1}{3}\right) = 4\left(-\frac{1}{27}\right) + 1 = -\frac{4}{27} + \frac{27}{27} = \frac{23}{27}$.
Теперь вычислим значение всего выражения:
$-(\cos 6\alpha + \cos 2\alpha) = -\left(\frac{23}{27} + \left(-\frac{1}{3}\right)\right) = -\left(\frac{23}{27} - \frac{9}{27}\right) = -\frac{14}{27}$.
Ответ: $-\frac{14}{27}$
б) Для нахождения значения выражения $\sin 2\alpha \cdot \cos 5\alpha - \sin \alpha \cdot \cos 6\alpha$ преобразуем его, используя формулу произведения синуса на косинус $\sin x \cos y = \frac{1}{2}(\sin(x+y) + \sin(x-y))$.
$\sin 2\alpha \cos 5\alpha = \frac{1}{2}(\sin(2\alpha+5\alpha) + \sin(2\alpha-5\alpha)) = \frac{1}{2}(\sin 7\alpha + \sin(-3\alpha)) = \frac{1}{2}(\sin 7\alpha - \sin 3\alpha)$.
$\sin \alpha \cos 6\alpha = \frac{1}{2}(\sin(\alpha+6\alpha) + \sin(\alpha-6\alpha)) = \frac{1}{2}(\sin 7\alpha + \sin(-5\alpha)) = \frac{1}{2}(\sin 7\alpha - \sin 5\alpha)$.
Подставим в исходное выражение:
$\frac{1}{2}(\sin 7\alpha - \sin 3\alpha) - \frac{1}{2}(\sin 7\alpha - \sin 5\alpha) = \frac{1}{2}\sin 7\alpha - \frac{1}{2}\sin 3\alpha - \frac{1}{2}\sin 7\alpha + \frac{1}{2}\sin 5\alpha = \frac{1}{2}(\sin 5\alpha - \sin 3\alpha)$.
Теперь применим формулу разности синусов $\sin x - \sin y = 2\cos\frac{x+y}{2}\sin\frac{x-y}{2}$:
$\frac{1}{2}(2\cos\frac{5\alpha+3\alpha}{2}\sin\frac{5\alpha-3\alpha}{2}) = \cos 4\alpha \sin \alpha$.
Нам дано, что $\sin \alpha = -\frac{1}{\sqrt{3}}$. Найдем $\cos 4\alpha$.
Сначала найдем $\cos 2\alpha$ по формуле $\cos 2\alpha = 1 - 2\sin^2 \alpha$:
$\cos 2\alpha = 1 - 2\left(-\frac{1}{\sqrt{3}}\right)^2 = 1 - 2 \cdot \frac{1}{3} = \frac{1}{3}$.
Теперь найдем $\cos 4\alpha$ по формуле $\cos 4\alpha = 2\cos^2(2\alpha) - 1$:
$\cos 4\alpha = 2\left(\frac{1}{3}\right)^2 - 1 = 2 \cdot \frac{1}{9} - 1 = \frac{2}{9} - 1 = -\frac{7}{9}$.
Вычислим итоговое значение:
$\cos 4\alpha \sin \alpha = \left(-\frac{7}{9}\right) \cdot \left(-\frac{1}{\sqrt{3}}\right) = \frac{7}{9\sqrt{3}} = \frac{7\sqrt{3}}{27}$.
Ответ: $\frac{7\sqrt{3}}{27}$
в) Для нахождения значения выражения $\cos 7\alpha \cdot \cos 4\alpha - \cos 8\alpha \cdot \cos 3\alpha$ преобразуем его, используя формулу произведения косинусов $\cos x \cos y = \frac{1}{2}(\cos(x+y) + \cos(x-y))$.
$\cos 7\alpha \cos 4\alpha = \frac{1}{2}(\cos(7\alpha+4\alpha) + \cos(7\alpha-4\alpha)) = \frac{1}{2}(\cos 11\alpha + \cos 3\alpha)$.
$\cos 8\alpha \cos 3\alpha = \frac{1}{2}(\cos(8\alpha+3\alpha) + \cos(8\alpha-3\alpha)) = \frac{1}{2}(\cos 11\alpha + \cos 5\alpha)$.
Подставим в исходное выражение:
$\frac{1}{2}(\cos 11\alpha + \cos 3\alpha) - \frac{1}{2}(\cos 11\alpha + \cos 5\alpha) = \frac{1}{2}\cos 11\alpha + \frac{1}{2}\cos 3\alpha - \frac{1}{2}\cos 11\alpha - \frac{1}{2}\cos 5\alpha = \frac{1}{2}(\cos 3\alpha - \cos 5\alpha)$.
Применим формулу разности косинусов $\cos x - \cos y = -2\sin\frac{x+y}{2}\sin\frac{x-y}{2}$:
$\frac{1}{2}(-2\sin\frac{3\alpha+5\alpha}{2}\sin\frac{3\alpha-5\alpha}{2}) = -\sin(4\alpha)\sin(-\alpha) = \sin 4\alpha \sin \alpha$.
Выражение упростилось до $\sin 4\alpha \sin \alpha$. Нам дано $\cos \alpha = \frac{\sqrt{3}}{3}$.
Чтобы найти значение, выразим $\sin 4\alpha \sin \alpha$ через $\cos \alpha$.
$\sin 4\alpha = 2\sin 2\alpha \cos 2\alpha = 2(2\sin\alpha\cos\alpha)(2\cos^2\alpha-1)$.
Тогда все выражение равно $(\sin\alpha) \cdot (4\sin\alpha\cos\alpha(2\cos^2\alpha-1)) = 4\sin^2\alpha\cos\alpha(2\cos^2\alpha-1)$.
Найдем $\sin^2\alpha$ из основного тригонометрического тождества $\sin^2\alpha = 1 - \cos^2\alpha$:
$\sin^2\alpha = 1 - \left(\frac{\sqrt{3}}{3}\right)^2 = 1 - \frac{3}{9} = 1 - \frac{1}{3} = \frac{2}{3}$.
Подставим известные значения:
$4\left(\frac{2}{3}\right)\left(\frac{\sqrt{3}}{3}\right)\left(2\left(\frac{\sqrt{3}}{3}\right)^2-1\right) = \frac{8\sqrt{3}}{9}\left(2\cdot\frac{1}{3}-1\right) = \frac{8\sqrt{3}}{9}\left(\frac{2}{3}-1\right) = \frac{8\sqrt{3}}{9}\left(-\frac{1}{3}\right) = -\frac{8\sqrt{3}}{27}$.
Ответ: $-\frac{8\sqrt{3}}{27}$
г) Исходное выражение $\sin 2\alpha \cdot \cos 5\alpha - \sin \alpha \cdot \cos 6\alpha$ полностью совпадает с выражением из пункта б).
Как было показано в решении пункта б), данное выражение можно упростить до вида $\cos 4\alpha \sin \alpha$.
По условию $\sin \alpha = \frac{\sqrt{5}}{4}$. Нам осталось найти $\cos 4\alpha$.
Найдем $\cos 2\alpha$ по формуле $\cos 2\alpha = 1 - 2\sin^2 \alpha$:
$\cos 2\alpha = 1 - 2\left(\frac{\sqrt{5}}{4}\right)^2 = 1 - 2 \cdot \frac{5}{16} = 1 - \frac{10}{16} = 1 - \frac{5}{8} = \frac{3}{8}$.
Теперь найдем $\cos 4\alpha$ по формуле $\cos 4\alpha = 2\cos^2(2\alpha) - 1$:
$\cos 4\alpha = 2\left(\frac{3}{8}\right)^2 - 1 = 2 \cdot \frac{9}{64} - 1 = \frac{18}{64} - 1 = \frac{9}{32} - \frac{32}{32} = -\frac{23}{32}$.
Вычислим итоговое значение $\cos 4\alpha \sin \alpha$:
$\left(-\frac{23}{32}\right) \cdot \left(\frac{\sqrt{5}}{4}\right) = -\frac{23\sqrt{5}}{128}$.
Ответ: $-\frac{23\sqrt{5}}{128}$
Другие задания:
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