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Номер 739, страница 212 - 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. 28. Преобразование суммы и разности тригонометрических функций в произведение - номер 739, страница 212.
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Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1035 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1050 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037} "field_author" => Illuminate\Database\Eloquent\Collection {#1039} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044} "field_country" => Illuminate\Database\Eloquent\Collection {#1046} "field_city" => Illuminate\Database\Eloquent\Collection {#1048} "field_series" => Illuminate\Database\Eloquent\Collection {#1050} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107670 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "28. Преобразование суммы и разности тригонометрических функций в произведение" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "210" "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 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "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} "field_class" => Illuminate\Database\Eloquent\Collection {#1068} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070} "field_author" => Illuminate\Database\Eloquent\Collection {#1072} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077} "field_country" => Illuminate\Database\Eloquent\Collection {#1079} "field_city" => Illuminate\Database\Eloquent\Collection {#1081} "field_series" => Illuminate\Database\Eloquent\Collection {#1083} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086} "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. 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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 {#1097 …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" => 1107670 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "28. Преобразование суммы и разности тригонометрических функций в произведение" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "210" "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 {#1098 #items: array:3 [ 0 => App\Models\Element {#1099 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1277094 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1101 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 …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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …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 {#1102 …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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …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" => "1108481" "type" => "task" ] "text" => "<p><strong>739.</strong> Верно ли равенство:</p><p><strong>а)</strong> $\sin 10^{\circ} + \sin 50^{\circ} = \cos 20^{\circ};$</p><p><strong>б)</strong> $\sin 80^{\circ} - \sin 20^{\circ} = \sin 40^{\circ};$</p><p><strong>в)</strong> $\cos \frac{7\pi}{24} + \cos \frac{\pi}{24} = \cos \frac{\pi}{8};$</p><p><strong>г)</strong> $\cos \frac{7\pi}{18} - \sin \frac{4\pi}{9} = -\sin \frac{2\pi}{9};$</p><p><strong>д)</strong> $\sin 40^{\circ} + \sin 20^{\circ} - 2\cos 10^{\circ} = -\cos 10^{\circ};$</p><p><strong>е)</strong> $\operatorname{tg} 43^{\circ} + \operatorname{tg} 17^{\circ} = \frac{\sqrt{3}}{2 \sin 47^{\circ} \cdot \cos 17^{\circ}}?$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "739-1.jpg" "alt" => null "width" => "1017" "height" => 673 "path" => "/media/algebra_09/soltan-u19/0-1/739-1.webp?ts=1752834734" ] ] ] #original: array:7 [ "id" => 1277094 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1108481" "type" => "task" ] "text" => "<p><strong>739.</strong> Верно ли равенство:</p><p><strong>а)</strong> $\sin 10^{\circ} + \sin 50^{\circ} = \cos 20^{\circ};$</p><p><strong>б)</strong> $\sin 80^{\circ} - \sin 20^{\circ} = \sin 40^{\circ};$</p><p><strong>в)</strong> $\cos \frac{7\pi}{24} + \cos \frac{\pi}{24} = \cos \frac{\pi}{8};$</p><p><strong>г)</strong> $\cos \frac{7\pi}{18} - \sin \frac{4\pi}{9} = -\sin \frac{2\pi}{9};$</p><p><strong>д)</strong> $\sin 40^{\circ} + \sin 20^{\circ} - 2\cos 10^{\circ} = -\cos 10^{\circ};$</p><p><strong>е)</strong> $\operatorname{tg} 43^{\circ} + \operatorname{tg} 17^{\circ} = \frac{\sqrt{3}}{2 \sin 47^{\circ} \cdot \cos 17^{\circ}}?$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "739-1.jpg" "alt" => null "width" => "1017" "height" => 673 "path" => "/media/algebra_09/soltan-u19/0-1/739-1.webp?ts=1752834734" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1107 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1278385 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1109 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …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 {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …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 {#1110 …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 {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …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" => "1108481" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "739-1.jpg" "alt" => null "width" => "1003" "height" => 530 "path" => "/media/algebra_09/soltan-u19/1-1/739-1.webp?ts=1752831534" ] ] ] #original: array:6 [ "id" => 1278385 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1108481" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "739-1.jpg" "alt" => null "width" => "1003" "height" => 530 "path" => "/media/algebra_09/soltan-u19/1-1/739-1.webp?ts=1752831534" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1115 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1554340 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1117 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1118 …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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …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 {#1118 …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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …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" => "1108481" "type" => "task" ] "text" => "<p><strong>а)</strong> Для проверки равенства преобразуем его левую часть, используя формулу суммы синусов $ \sin \alpha + \sin \beta = 2 \sin\frac{\alpha+\beta}{2} \cos\frac{\alpha-\beta}{2} $.</p><p>$ \sin 10° + \sin 50° = 2 \sin\frac{10°+50°}{2} \cos\frac{50°-10°}{2} = 2 \sin 30° \cos 20° $.</p><p>Поскольку $ \sin 30° = \frac{1}{2} $, выражение упрощается до $ 2 \cdot \frac{1}{2} \cdot \cos 20° = \cos 20° $.</p><p>Левая часть равна правой, следовательно, равенство верно.</p><p><strong>Ответ:</strong> Верно.</p><p><strong>б)</strong> Преобразуем левую часть равенства, используя формулу разности синусов $ \sin \alpha - \sin \beta = 2 \sin\frac{\alpha-\beta}{2} \cos\frac{\alpha+\beta}{2} $.</p><p>$ \sin 80° - \sin 20° = 2 \sin\frac{80°-20°}{2} \cos\frac{80°+20°}{2} = 2 \sin 30° \cos 50° $.</p><p>Так как $ \sin 30° = \frac{1}{2} $, получаем $ 2 \cdot \frac{1}{2} \cdot \cos 50° = \cos 50° $.</p><p>Используем формулу приведения $ \cos \alpha = \sin(90°-\alpha) $. Тогда $ \cos 50° = \sin(90°-50°) = \sin 40° $.</p><p>Левая часть равна правой, следовательно, равенство верно.</p><p><strong>Ответ:</strong> Верно.</p><p><strong>в)</strong> Преобразуем левую часть равенства, используя формулу суммы косинусов $ \cos \alpha + \cos \beta = 2 \cos\frac{\alpha+\beta}{2} \cos\frac{\alpha-\beta}{2} $.</p><p>$ \cos \frac{7\pi}{24} + \cos \frac{\pi}{24} = 2 \cos\frac{\frac{7\pi}{24}+\frac{\pi}{24}}{2} \cos\frac{\frac{7\pi}{24}-\frac{\pi}{24}}{2} = 2 \cos\frac{\frac{8\pi}{24}}{2} \cos\frac{\frac{6\pi}{24}}{2} = 2 \cos\frac{\pi}{6} \cos\frac{\pi}{8} $.</p><p>Поскольку $ \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2} $, получаем $ 2 \cdot \frac{\sqrt{3}}{2} \cdot \cos\frac{\pi}{8} = \sqrt{3} \cos\frac{\pi}{8} $.</p><p>Правая часть равенства равна $ \cos\frac{\pi}{8} $. Так как $ \sqrt{3} \neq 1 $ и $ \cos\frac{\pi}{8} \neq 0 $, то $ \sqrt{3} \cos\frac{\pi}{8} \neq \cos\frac{\pi}{8} $.</p><p>Левая часть не равна правой, следовательно, равенство неверно.</p><p><strong>Ответ:</strong> Неверно.</p><p><strong>г)</strong> Для преобразования левой части используем формулу приведения $ \cos \alpha = \sin(\frac{\pi}{2}-\alpha) $.</p><p>$ \cos\frac{7\pi}{18} = \sin(\frac{\pi}{2}-\frac{7\pi}{18}) = \sin(\frac{9\pi-7\pi}{18}) = \sin\frac{2\pi}{18} = \sin\frac{\pi}{9} $.</p><p>Тогда левая часть примет вид: $ \sin\frac{\pi}{9} - \sin\frac{4\pi}{9} $.</p><p>Применим формулу разности синусов $ \sin \alpha - \sin \beta = 2 \sin\frac{\alpha-\beta}{2} \cos\frac{\alpha+\beta}{2} $.</p><p>$ \sin\frac{\pi}{9} - \sin\frac{4\pi}{9} = 2 \sin\frac{\frac{\pi}{9}-\frac{4\pi}{9}}{2} \cos\frac{\frac{\pi}{9}+\frac{4\pi}{9}}{2} = 2 \sin(-\frac{\pi}{6}) \cos\frac{5\pi}{18} $.</p><p>Используя нечетность синуса и значение $ \sin\frac{\pi}{6} = \frac{1}{2} $, получаем $ -2 \sin\frac{\pi}{6} \cos\frac{5\pi}{18} = -2 \cdot \frac{1}{2} \cos\frac{5\pi}{18} = -\cos\frac{5\pi}{18} $.</p><p>Правая часть равна $ -\sin\frac{2\pi}{9} $. Равенство верно, если $ \cos\frac{5\pi}{18} = \sin\frac{2\pi}{9} $.</p><p>Так как $ \frac{5\pi}{18} + \frac{2\pi}{9} = \frac{5\pi}{18} + \frac{4\pi}{18} = \frac{9\pi}{18} = \frac{\pi}{2} $, то по формуле приведения $ \cos \alpha = \sin(\frac{\pi}{2}-\alpha) $ это соотношение верно.</p><p>Следовательно, исходное равенство верно.</p><p><strong>Ответ:</strong> Верно.</p><p><strong>д)</strong> Преобразуем левую часть равенства. Применим к первым двум слагаемым формулу суммы синусов.</p><p>$ \sin 40° + \sin 20° = 2 \sin\frac{40°+20°}{2} \cos\frac{40°-20°}{2} = 2 \sin 30° \cos 10° = 2 \cdot \frac{1}{2} \cos 10° = \cos 10° $.</p><p>Подставим полученный результат в левую часть исходного равенства:</p><p>$ (\sin 40° + \sin 20°) - 2\cos 10° = \cos 10° - 2\cos 10° = -\cos 10° $.</p><p>Левая часть равна правой, следовательно, равенство верно.</p><p><strong>Ответ:</strong> Верно.</p><p><strong>е)</strong> Преобразуем левую часть, используя формулу суммы тангенсов $ \text{tg } \alpha + \text{tg } \beta = \frac{\sin(\alpha+\beta)}{\cos\alpha\cos\beta} $.</p><p>$ \text{tg } 43° + \text{tg } 17° = \frac{\sin(43°+17°)}{\cos 43° \cos 17°} = \frac{\sin 60°}{\cos 43° \cos 17°} $.</p><p>Поскольку $ \sin 60° = \frac{\sqrt{3}}{2} $, левая часть равна $ \frac{\frac{\sqrt{3}}{2}}{\cos 43° \cos 17°} = \frac{\sqrt{3}}{2 \cos 43° \cos 17°} $.</p><p>Правая часть равенства равна $ \frac{\sqrt{3}}{2 \sin 47° \cos 17°} $.</p><p>Равенство будет верным, если $ \cos 43° = \sin 47° $.</p><p>По формуле приведения $ \sin \alpha = \cos(90°-\alpha) $, имеем $ \sin 47° = \cos(90°-47°) = \cos 43° $.</p><p>Знаменатели дробей равны, значит, левая часть равна правой, и равенство верно.</p><p><strong>Ответ:</strong> Верно.</p>" ] #original: array:6 [ "id" => 1554340 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1108481" "type" => "task" ] "text" => "<p><strong>а)</strong> Для проверки равенства преобразуем его левую часть, используя формулу суммы синусов $ \sin \alpha + \sin \beta = 2 \sin\frac{\alpha+\beta}{2} \cos\frac{\alpha-\beta}{2} $.</p><p>$ \sin 10° + \sin 50° = 2 \sin\frac{10°+50°}{2} \cos\frac{50°-10°}{2} = 2 \sin 30° \cos 20° $.</p><p>Поскольку $ \sin 30° = \frac{1}{2} $, выражение упрощается до $ 2 \cdot \frac{1}{2} \cdot \cos 20° = \cos 20° $.</p><p>Левая часть равна правой, следовательно, равенство верно.</p><p><strong>Ответ:</strong> Верно.</p><p><strong>б)</strong> Преобразуем левую часть равенства, используя формулу разности синусов $ \sin \alpha - \sin \beta = 2 \sin\frac{\alpha-\beta}{2} \cos\frac{\alpha+\beta}{2} $.</p><p>$ \sin 80° - \sin 20° = 2 \sin\frac{80°-20°}{2} \cos\frac{80°+20°}{2} = 2 \sin 30° \cos 50° $.</p><p>Так как $ \sin 30° = \frac{1}{2} $, получаем $ 2 \cdot \frac{1}{2} \cdot \cos 50° = \cos 50° $.</p><p>Используем формулу приведения $ \cos \alpha = \sin(90°-\alpha) $. Тогда $ \cos 50° = \sin(90°-50°) = \sin 40° $.</p><p>Левая часть равна правой, следовательно, равенство верно.</p><p><strong>Ответ:</strong> Верно.</p><p><strong>в)</strong> Преобразуем левую часть равенства, используя формулу суммы косинусов $ \cos \alpha + \cos \beta = 2 \cos\frac{\alpha+\beta}{2} \cos\frac{\alpha-\beta}{2} $.</p><p>$ \cos \frac{7\pi}{24} + \cos \frac{\pi}{24} = 2 \cos\frac{\frac{7\pi}{24}+\frac{\pi}{24}}{2} \cos\frac{\frac{7\pi}{24}-\frac{\pi}{24}}{2} = 2 \cos\frac{\frac{8\pi}{24}}{2} \cos\frac{\frac{6\pi}{24}}{2} = 2 \cos\frac{\pi}{6} \cos\frac{\pi}{8} $.</p><p>Поскольку $ \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2} $, получаем $ 2 \cdot \frac{\sqrt{3}}{2} \cdot \cos\frac{\pi}{8} = \sqrt{3} \cos\frac{\pi}{8} $.</p><p>Правая часть равенства равна $ \cos\frac{\pi}{8} $. Так как $ \sqrt{3} \neq 1 $ и $ \cos\frac{\pi}{8} \neq 0 $, то $ \sqrt{3} \cos\frac{\pi}{8} \neq \cos\frac{\pi}{8} $.</p><p>Левая часть не равна правой, следовательно, равенство неверно.</p><p><strong>Ответ:</strong> Неверно.</p><p><strong>г)</strong> Для преобразования левой части используем формулу приведения $ \cos \alpha = \sin(\frac{\pi}{2}-\alpha) $.</p><p>$ \cos\frac{7\pi}{18} = \sin(\frac{\pi}{2}-\frac{7\pi}{18}) = \sin(\frac{9\pi-7\pi}{18}) = \sin\frac{2\pi}{18} = \sin\frac{\pi}{9} $.</p><p>Тогда левая часть примет вид: $ \sin\frac{\pi}{9} - \sin\frac{4\pi}{9} $.</p><p>Применим формулу разности синусов $ \sin \alpha - \sin \beta = 2 \sin\frac{\alpha-\beta}{2} \cos\frac{\alpha+\beta}{2} $.</p><p>$ \sin\frac{\pi}{9} - \sin\frac{4\pi}{9} = 2 \sin\frac{\frac{\pi}{9}-\frac{4\pi}{9}}{2} \cos\frac{\frac{\pi}{9}+\frac{4\pi}{9}}{2} = 2 \sin(-\frac{\pi}{6}) \cos\frac{5\pi}{18} $.</p><p>Используя нечетность синуса и значение $ \sin\frac{\pi}{6} = \frac{1}{2} $, получаем $ -2 \sin\frac{\pi}{6} \cos\frac{5\pi}{18} = -2 \cdot \frac{1}{2} \cos\frac{5\pi}{18} = -\cos\frac{5\pi}{18} $.</p><p>Правая часть равна $ -\sin\frac{2\pi}{9} $. Равенство верно, если $ \cos\frac{5\pi}{18} = \sin\frac{2\pi}{9} $.</p><p>Так как $ \frac{5\pi}{18} + \frac{2\pi}{9} = \frac{5\pi}{18} + \frac{4\pi}{18} = \frac{9\pi}{18} = \frac{\pi}{2} $, то по формуле приведения $ \cos \alpha = \sin(\frac{\pi}{2}-\alpha) $ это соотношение верно.</p><p>Следовательно, исходное равенство верно.</p><p><strong>Ответ:</strong> Верно.</p><p><strong>д)</strong> Преобразуем левую часть равенства. Применим к первым двум слагаемым формулу суммы синусов.</p><p>$ \sin 40° + \sin 20° = 2 \sin\frac{40°+20°}{2} \cos\frac{40°-20°}{2} = 2 \sin 30° \cos 10° = 2 \cdot \frac{1}{2} \cos 10° = \cos 10° $.</p><p>Подставим полученный результат в левую часть исходного равенства:</p><p>$ (\sin 40° + \sin 20°) - 2\cos 10° = \cos 10° - 2\cos 10° = -\cos 10° $.</p><p>Левая часть равна правой, следовательно, равенство верно.</p><p><strong>Ответ:</strong> Верно.</p><p><strong>е)</strong> Преобразуем левую часть, используя формулу суммы тангенсов $ \text{tg } \alpha + \text{tg } \beta = \frac{\sin(\alpha+\beta)}{\cos\alpha\cos\beta} $.</p><p>$ \text{tg } 43° + \text{tg } 17° = \frac{\sin(43°+17°)}{\cos 43° \cos 17°} = \frac{\sin 60°}{\cos 43° \cos 17°} $.</p><p>Поскольку $ \sin 60° = \frac{\sqrt{3}}{2} $, левая часть равна $ \frac{\frac{\sqrt{3}}{2}}{\cos 43° \cos 17°} = \frac{\sqrt{3}}{2 \cos 43° \cos 17°} $.</p><p>Правая часть равенства равна $ \frac{\sqrt{3}}{2 \sin 47° \cos 17°} $.</p><p>Равенство будет верным, если $ \cos 43° = \sin 47° $.</p><p>По формуле приведения $ \sin \alpha = \cos(90°-\alpha) $, имеем $ \sin 47° = \cos(90°-47°) = \cos 43° $.</p><p>Знаменатели дробей равны, значит, левая часть равна правой, и равенство верно.</p><p><strong>Ответ:</strong> Верно.</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" => "1108482" "type" => "task" ] "previous" => array:2 [ "refs" => "1108480" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Book {#1124 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true 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Illuminate\Database\Eloquent\Collection {#1138} "field_city" => Illuminate\Database\Eloquent\Collection {#1140} "field_series" => Illuminate\Database\Eloquent\Collection {#1142} "field_umk" => Illuminate\Database\Eloquent\Collection {#1143} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1144} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1145} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1147} "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 {#1148} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151} "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 {#1360 #items: array:1 [ 0 => App\Models\BookPage {#1155 #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" => 1097954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "212" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-212" "field_display_title" => "212" "field_folder" => "folder1" "field_image_name" => "212" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1159 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Book {#1160 #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 {#1161 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1183 …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 {#1184 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1187 …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 {#1161 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1183 …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 {#1184 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1187 …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 {#1188 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1189 #items: array:1 [ 0 => App\Models\Element {#1190 #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" => 1261105 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261105 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191 …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" => "1097955" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097953" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:6 [ 0 => App\Models\Task {#1199 #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" => 1108480 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "212" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-738" "field_display_title" => "738" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200 …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 {#1202 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236 …2} "content" => Illuminate\Database\Eloquent\Collection {#1273 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1292 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108480 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "212" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-738" "field_display_title" => "738" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200 …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 {#1202 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236 …2} "content" => Illuminate\Database\Eloquent\Collection {#1273 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1292 …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 {#1293 #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" => 1108481 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "212" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-739" "field_display_title" => "739" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1294 …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 {#1295 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1296 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1297 …2} "content" => Illuminate\Database\Eloquent\Collection {#1298 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1305 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108481 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "212" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-739" "field_display_title" => "739" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1294 …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 {#1295 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1296 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1297 …2} "content" => Illuminate\Database\Eloquent\Collection {#1298 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1305 …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 {#1306 #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" => 1108482 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "212" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-740" "field_display_title" => "740" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1307 …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 {#1308 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1309 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1310 …2} "content" => Illuminate\Database\Eloquent\Collection {#1311 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1318 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108482 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "212" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-740" "field_display_title" => "740" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1307 …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 {#1308 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1309 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1310 …2} "content" => Illuminate\Database\Eloquent\Collection {#1311 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1318 …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 {#1319 #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" => 1108483 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "212" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-741" "field_display_title" => "741" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1320 …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 {#1321 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1322 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1323 …2} "content" => Illuminate\Database\Eloquent\Collection {#1324 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1331 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108483 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "212" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-741" "field_display_title" => "741" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1320 …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 {#1321 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1322 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1323 …2} "content" => 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№739 (с. 212)
Условие. №739 (с. 212)
скриншот условия
739. Верно ли равенство:
а) $\sin 10^{\circ} + \sin 50^{\circ} = \cos 20^{\circ};$
б) $\sin 80^{\circ} - \sin 20^{\circ} = \sin 40^{\circ};$
в) $\cos \frac{7\pi}{24} + \cos \frac{\pi}{24} = \cos \frac{\pi}{8};$
г) $\cos \frac{7\pi}{18} - \sin \frac{4\pi}{9} = -\sin \frac{2\pi}{9};$
д) $\sin 40^{\circ} + \sin 20^{\circ} - 2\cos 10^{\circ} = -\cos 10^{\circ};$
е) $\operatorname{tg} 43^{\circ} + \operatorname{tg} 17^{\circ} = \frac{\sqrt{3}}{2 \sin 47^{\circ} \cdot \cos 17^{\circ}}?$
Решение. №739 (с. 212)
Решение 2 (rus). №739 (с. 212)
а) Для проверки равенства преобразуем его левую часть, используя формулу суммы синусов $ \sin \alpha + \sin \beta = 2 \sin\frac{\alpha+\beta}{2} \cos\frac{\alpha-\beta}{2} $.
$ \sin 10° + \sin 50° = 2 \sin\frac{10°+50°}{2} \cos\frac{50°-10°}{2} = 2 \sin 30° \cos 20° $.
Поскольку $ \sin 30° = \frac{1}{2} $, выражение упрощается до $ 2 \cdot \frac{1}{2} \cdot \cos 20° = \cos 20° $.
Левая часть равна правой, следовательно, равенство верно.
Ответ: Верно.
б) Преобразуем левую часть равенства, используя формулу разности синусов $ \sin \alpha - \sin \beta = 2 \sin\frac{\alpha-\beta}{2} \cos\frac{\alpha+\beta}{2} $.
$ \sin 80° - \sin 20° = 2 \sin\frac{80°-20°}{2} \cos\frac{80°+20°}{2} = 2 \sin 30° \cos 50° $.
Так как $ \sin 30° = \frac{1}{2} $, получаем $ 2 \cdot \frac{1}{2} \cdot \cos 50° = \cos 50° $.
Используем формулу приведения $ \cos \alpha = \sin(90°-\alpha) $. Тогда $ \cos 50° = \sin(90°-50°) = \sin 40° $.
Левая часть равна правой, следовательно, равенство верно.
Ответ: Верно.
в) Преобразуем левую часть равенства, используя формулу суммы косинусов $ \cos \alpha + \cos \beta = 2 \cos\frac{\alpha+\beta}{2} \cos\frac{\alpha-\beta}{2} $.
$ \cos \frac{7\pi}{24} + \cos \frac{\pi}{24} = 2 \cos\frac{\frac{7\pi}{24}+\frac{\pi}{24}}{2} \cos\frac{\frac{7\pi}{24}-\frac{\pi}{24}}{2} = 2 \cos\frac{\frac{8\pi}{24}}{2} \cos\frac{\frac{6\pi}{24}}{2} = 2 \cos\frac{\pi}{6} \cos\frac{\pi}{8} $.
Поскольку $ \cos\frac{\pi}{6} = \frac{\sqrt{3}}{2} $, получаем $ 2 \cdot \frac{\sqrt{3}}{2} \cdot \cos\frac{\pi}{8} = \sqrt{3} \cos\frac{\pi}{8} $.
Правая часть равенства равна $ \cos\frac{\pi}{8} $. Так как $ \sqrt{3} \neq 1 $ и $ \cos\frac{\pi}{8} \neq 0 $, то $ \sqrt{3} \cos\frac{\pi}{8} \neq \cos\frac{\pi}{8} $.
Левая часть не равна правой, следовательно, равенство неверно.
Ответ: Неверно.
г) Для преобразования левой части используем формулу приведения $ \cos \alpha = \sin(\frac{\pi}{2}-\alpha) $.
$ \cos\frac{7\pi}{18} = \sin(\frac{\pi}{2}-\frac{7\pi}{18}) = \sin(\frac{9\pi-7\pi}{18}) = \sin\frac{2\pi}{18} = \sin\frac{\pi}{9} $.
Тогда левая часть примет вид: $ \sin\frac{\pi}{9} - \sin\frac{4\pi}{9} $.
Применим формулу разности синусов $ \sin \alpha - \sin \beta = 2 \sin\frac{\alpha-\beta}{2} \cos\frac{\alpha+\beta}{2} $.
$ \sin\frac{\pi}{9} - \sin\frac{4\pi}{9} = 2 \sin\frac{\frac{\pi}{9}-\frac{4\pi}{9}}{2} \cos\frac{\frac{\pi}{9}+\frac{4\pi}{9}}{2} = 2 \sin(-\frac{\pi}{6}) \cos\frac{5\pi}{18} $.
Используя нечетность синуса и значение $ \sin\frac{\pi}{6} = \frac{1}{2} $, получаем $ -2 \sin\frac{\pi}{6} \cos\frac{5\pi}{18} = -2 \cdot \frac{1}{2} \cos\frac{5\pi}{18} = -\cos\frac{5\pi}{18} $.
Правая часть равна $ -\sin\frac{2\pi}{9} $. Равенство верно, если $ \cos\frac{5\pi}{18} = \sin\frac{2\pi}{9} $.
Так как $ \frac{5\pi}{18} + \frac{2\pi}{9} = \frac{5\pi}{18} + \frac{4\pi}{18} = \frac{9\pi}{18} = \frac{\pi}{2} $, то по формуле приведения $ \cos \alpha = \sin(\frac{\pi}{2}-\alpha) $ это соотношение верно.
Следовательно, исходное равенство верно.
Ответ: Верно.
д) Преобразуем левую часть равенства. Применим к первым двум слагаемым формулу суммы синусов.
$ \sin 40° + \sin 20° = 2 \sin\frac{40°+20°}{2} \cos\frac{40°-20°}{2} = 2 \sin 30° \cos 10° = 2 \cdot \frac{1}{2} \cos 10° = \cos 10° $.
Подставим полученный результат в левую часть исходного равенства:
$ (\sin 40° + \sin 20°) - 2\cos 10° = \cos 10° - 2\cos 10° = -\cos 10° $.
Левая часть равна правой, следовательно, равенство верно.
Ответ: Верно.
е) Преобразуем левую часть, используя формулу суммы тангенсов $ \text{tg } \alpha + \text{tg } \beta = \frac{\sin(\alpha+\beta)}{\cos\alpha\cos\beta} $.
$ \text{tg } 43° + \text{tg } 17° = \frac{\sin(43°+17°)}{\cos 43° \cos 17°} = \frac{\sin 60°}{\cos 43° \cos 17°} $.
Поскольку $ \sin 60° = \frac{\sqrt{3}}{2} $, левая часть равна $ \frac{\frac{\sqrt{3}}{2}}{\cos 43° \cos 17°} = \frac{\sqrt{3}}{2 \cos 43° \cos 17°} $.
Правая часть равенства равна $ \frac{\sqrt{3}}{2 \sin 47° \cos 17°} $.
Равенство будет верным, если $ \cos 43° = \sin 47° $.
По формуле приведения $ \sin \alpha = \cos(90°-\alpha) $, имеем $ \sin 47° = \cos(90°-47°) = \cos 43° $.
Знаменатели дробей равны, значит, левая часть равна правой, и равенство верно.
Ответ: Верно.
Другие задания:
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