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Номер 803, страница 223 - гдз по алгебре 9 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2019 - 2026
ISBN: 978-601-317-424-2
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 9 классе
IV. Тригонометрия. 30. Упражнения на повторение раздела «Тригонометрия» - номер 803, страница 223.
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" => 1108549 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "223" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-803" "field_display_title" => "803" "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" => "" "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] ] #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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Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072 #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 {#1070 #items: array:1 [ 0 => App\Models\Book {#1073 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false 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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] ] #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" => 1107672 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "30. Упражнения на повторение раздела «Тригонометрия»" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "221" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#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 {#1165 …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 {#1165 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107672 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "30. Упражнения на повторение раздела «Тригонометрия»" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103} "field_page_start" => "221" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#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 {#1169 #items: array:3 [ 0 => App\Models\Element {#1178 #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" => 1277158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1187 #items: array:1 [ 0 => App\Models\Edition {#1179 #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 {#1181 …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 {#1182 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1184 …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 {#1181 …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 {#1182 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1184 …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" => "1108549" "type" => "task" ] "text" => "<p><strong>803.</strong> Упростите выражение:</p><p><strong>а)</strong> $\sin\frac{\alpha}{4} \cdot \cos\frac{\alpha}{4} \cdot \cos\frac{\alpha}{2}$;</p><p><strong>б)</strong> $\cos^4\frac{\alpha}{2} - \sin^4\frac{\alpha}{2}$;</p><p><strong>в)</strong> $\cos^2\alpha - 4\sin^2\frac{\alpha}{2} \cdot \cos^2\frac{\alpha}{2}$;</p><p><strong>г)</strong> $1 - 8\sin^2\alpha \cdot \cos^2\alpha$;</p><p><strong>д)</strong> $(\cot\frac{\alpha}{2} - \tan\frac{\alpha}{2}) \cdot \sin\alpha$;</p><p><strong>е)</strong> $(2\cot\frac{\alpha}{2} \cdot \sin^2\frac{\alpha}{2}) : (\sin^2\frac{\alpha}{2} - \cos^2\frac{\alpha}{2})$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "803-1.jpg" "alt" => null "width" => "1534" "height" => 519 "path" => "/media/algebra_09/soltan-u19/0-1/803-1.webp?ts=1752834807" ] ] ] #original: array:7 [ "id" => 1277158 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1187} "task" => array:2 [ "refs" => "1108549" "type" => "task" ] "text" => "<p><strong>803.</strong> Упростите выражение:</p><p><strong>а)</strong> $\sin\frac{\alpha}{4} \cdot \cos\frac{\alpha}{4} \cdot \cos\frac{\alpha}{2}$;</p><p><strong>б)</strong> $\cos^4\frac{\alpha}{2} - \sin^4\frac{\alpha}{2}$;</p><p><strong>в)</strong> $\cos^2\alpha - 4\sin^2\frac{\alpha}{2} \cdot \cos^2\frac{\alpha}{2}$;</p><p><strong>г)</strong> $1 - 8\sin^2\alpha \cdot \cos^2\alpha$;</p><p><strong>д)</strong> $(\cot\frac{\alpha}{2} - \tan\frac{\alpha}{2}) \cdot \sin\alpha$;</p><p><strong>е)</strong> $(2\cot\frac{\alpha}{2} \cdot \sin^2\frac{\alpha}{2}) : (\sin^2\frac{\alpha}{2} - \cos^2\frac{\alpha}{2})$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "803-1.jpg" "alt" => null "width" => "1534" "height" => 519 "path" => "/media/algebra_09/soltan-u19/0-1/803-1.webp?ts=1752834807" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1185 #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" => 1278449 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Edition {#1186 #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 {#1189 …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 {#1190 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1192 …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 {#1189 …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 {#1190 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1192 …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" => "1108549" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "803-1.jpg" "alt" => null "width" => "1327" "height" => 2210 "path" => "/media/algebra_09/soltan-u19/1-1/803-1.webp?ts=1752831631" ] ] ] #original: array:6 [ "id" => 1278449 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1195} "task" => array:2 [ "refs" => "1108549" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "803-1.jpg" "alt" => null "width" => "1327" "height" => 2210 "path" => "/media/algebra_09/soltan-u19/1-1/803-1.webp?ts=1752831631" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1193 #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" => 1554404 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ 0 => App\Models\Edition {#1194 #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 {#1197 …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 {#1198 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1200 …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 {#1197 …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 {#1198 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1200 …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" => "1108549" "type" => "task" ] "text" => "<p><strong>а)</strong> $\sin\frac{\alpha}{4} \cdot \cos\frac{\alpha}{4} \cdot \cos\frac{\alpha}{2}$</p><p>Используем формулу синуса двойного угла $2\sin x \cos x = \sin(2x)$, из которой следует, что $\sin x \cos x = \frac{1}{2}\sin(2x)$.</p><p>Сначала преобразуем произведение $\sin\frac{\alpha}{4} \cdot \cos\frac{\alpha}{4}$:</p><p>$\sin\frac{\alpha}{4} \cdot \cos\frac{\alpha}{4} = \frac{1}{2}\sin(2 \cdot \frac{\alpha}{4}) = \frac{1}{2}\sin\frac{\alpha}{2}$.</p><p>Теперь подставим это в исходное выражение:</p><p>$\frac{1}{2}\sin\frac{\alpha}{2} \cdot \cos\frac{\alpha}{2}$.</p><p>Снова применяем ту же формулу для угла $\frac{\alpha}{2}$:</p><p>$\frac{1}{2}(\frac{1}{2}\sin(2 \cdot \frac{\alpha}{2})) = \frac{1}{4}\sin\alpha$.</p><p><strong>Ответ:</strong> $\frac{1}{4}\sin\alpha$.</p><p><strong>б)</strong> $\cos^4\frac{\alpha}{2} - \sin^4\frac{\alpha}{2}$</p><p>Используем формулу разности квадратов $a^2 - b^2 = (a-b)(a+b)$:</p><p>$\cos^4\frac{\alpha}{2} - \sin^4\frac{\alpha}{2} = (\cos^2\frac{\alpha}{2} - \sin^2\frac{\alpha}{2})(\cos^2\frac{\alpha}{2} + \sin^2\frac{\alpha}{2})$.</p><p>Применяем основное тригонометрическое тождество $\sin^2 x + \cos^2 x = 1$ и формулу косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$.</p><p>Первая скобка: $\cos^2\frac{\alpha}{2} - \sin^2\frac{\alpha}{2} = \cos(2 \cdot \frac{\alpha}{2}) = \cos\alpha$.</p><p>Вторая скобка: $\cos^2\frac{\alpha}{2} + \sin^2\frac{\alpha}{2} = 1$.</p><p>В результате получаем:</p><p>$\cos\alpha \cdot 1 = \cos\alpha$.</p><p><strong>Ответ:</strong> $\cos\alpha$.</p><p><strong>в)</strong> $\cos^2\alpha - 4\sin^2\frac{\alpha}{2} \cdot \cos^2\frac{\alpha}{2}$</p><p>Преобразуем вторую часть выражения:</p><p>$4\sin^2\frac{\alpha}{2} \cdot \cos^2\frac{\alpha}{2} = (2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2})^2$.</p><p>Используем формулу синуса двойного угла $2\sin x \cos x = \sin(2x)$:</p><p>$(2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2})^2 = (\sin(2 \cdot \frac{\alpha}{2}))^2 = (\sin\alpha)^2 = \sin^2\alpha$.</p><p>Подставляем обратно в исходное выражение:</p><p>$\cos^2\alpha - \sin^2\alpha$.</p><p>Это формула косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$.</p><p>$\cos^2\alpha - \sin^2\alpha = \cos(2\alpha)$.</p><p><strong>Ответ:</strong> $\cos(2\alpha)$.</p><p><strong>г)</strong> $1 - 8\sin^2\alpha \cdot \cos^2\alpha$</p><p>Преобразуем вторую часть выражения:</p><p>$8\sin^2\alpha \cdot \cos^2\alpha = 2 \cdot (4\sin^2\alpha \cos^2\alpha) = 2 \cdot (2\sin\alpha\cos\alpha)^2$.</p><p>Используем формулу синуса двойного угла $2\sin x \cos x = \sin(2x)$:</p><p>$2 \cdot (\sin(2\alpha))^2 = 2\sin^2(2\alpha)$.</p><p>Подставляем обратно в исходное выражение:</p><p>$1 - 2\sin^2(2\alpha)$.</p><p>Используем формулу косинуса двойного угла $\cos(2x) = 1 - 2\sin^2 x$. В нашем случае $x = 2\alpha$.</p><p>$1 - 2\sin^2(2\alpha) = \cos(2 \cdot 2\alpha) = \cos(4\alpha)$.</p><p><strong>Ответ:</strong> $\cos(4\alpha)$.</p><p><strong>д)</strong> $(\text{ctg}\frac{\alpha}{2} - \text{tg}\frac{\alpha}{2}) \cdot \sin\alpha$</p><p>Сначала упростим выражение в скобках. Представим котангенс и тангенс через синус и косинус:</p><p>$\text{ctg}\frac{\alpha}{2} - \text{tg}\frac{\alpha}{2} = \frac{\cos\frac{\alpha}{2}}{\sin\frac{\alpha}{2}} - \frac{\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}}$.</p><p>Приводим к общему знаменателю:</p><p>$\frac{\cos^2\frac{\alpha}{2} - \sin^2\frac{\alpha}{2}}{\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}}$.</p><p>Числитель - это формула косинуса двойного угла: $\cos^2\frac{\alpha}{2} - \sin^2\frac{\alpha}{2} = \cos\alpha$.</p><p>Знаменатель - это половина синуса двойного угла: $\sin\frac{\alpha}{2}\cos\frac{\alpha}{2} = \frac{1}{2}\sin\alpha$.</p><p>Выражение в скобках равно: $\frac{\cos\alpha}{\frac{1}{2}\sin\alpha} = \frac{2\cos\alpha}{\sin\alpha}$.</p><p>Теперь умножим это на $\sin\alpha$:</p><p>$\frac{2\cos\alpha}{\sin\alpha} \cdot \sin\alpha = 2\cos\alpha$.</p><p><strong>Ответ:</strong> $2\cos\alpha$.</p><p><strong>е)</strong> $(2\text{ctg}\frac{\alpha}{2} \cdot \sin^2\frac{\alpha}{2}) : (\sin^2\frac{\alpha}{2} - \cos^2\frac{\alpha}{2})$</p><p>Упростим делимое: $2\text{ctg}\frac{\alpha}{2} \cdot \sin^2\frac{\alpha}{2}$.</p><p>$2 \cdot \frac{\cos\frac{\alpha}{2}}{\sin\frac{\alpha}{2}} \cdot \sin^2\frac{\alpha}{2} = 2\cos\frac{\alpha}{2}\sin\frac{\alpha}{2}$.</p><p>Это формула синуса двойного угла: $2\cos\frac{\alpha}{2}\sin\frac{\alpha}{2} = \sin(2 \cdot \frac{\alpha}{2}) = \sin\alpha$.</p><p>Теперь упростим делитель: $\sin^2\frac{\alpha}{2} - \cos^2\frac{\alpha}{2}$.</p><p>Вынесем минус за скобки: $-(\cos^2\frac{\alpha}{2} - \sin^2\frac{\alpha}{2})$.</p><p>В скобках формула косинуса двойного угла: $\cos(2 \cdot \frac{\alpha}{2}) = \cos\alpha$.</p><p>Значит, делитель равен $-\cos\alpha$.</p><p>Теперь выполним деление:</p><p>$\sin\alpha : (-\cos\alpha) = \frac{\sin\alpha}{-\cos\alpha} = -\text{tg}\alpha$.</p><p><strong>Ответ:</strong> $-\text{tg}\alpha$.</p>" ] #original: array:6 [ "id" => 1554404 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203} "task" => array:2 [ "refs" => "1108549" "type" => "task" ] "text" => "<p><strong>а)</strong> $\sin\frac{\alpha}{4} \cdot \cos\frac{\alpha}{4} \cdot \cos\frac{\alpha}{2}$</p><p>Используем формулу синуса двойного угла $2\sin x \cos x = \sin(2x)$, из которой следует, что $\sin x \cos x = \frac{1}{2}\sin(2x)$.</p><p>Сначала преобразуем произведение $\sin\frac{\alpha}{4} \cdot \cos\frac{\alpha}{4}$:</p><p>$\sin\frac{\alpha}{4} \cdot \cos\frac{\alpha}{4} = \frac{1}{2}\sin(2 \cdot \frac{\alpha}{4}) = \frac{1}{2}\sin\frac{\alpha}{2}$.</p><p>Теперь подставим это в исходное выражение:</p><p>$\frac{1}{2}\sin\frac{\alpha}{2} \cdot \cos\frac{\alpha}{2}$.</p><p>Снова применяем ту же формулу для угла $\frac{\alpha}{2}$:</p><p>$\frac{1}{2}(\frac{1}{2}\sin(2 \cdot \frac{\alpha}{2})) = \frac{1}{4}\sin\alpha$.</p><p><strong>Ответ:</strong> $\frac{1}{4}\sin\alpha$.</p><p><strong>б)</strong> $\cos^4\frac{\alpha}{2} - \sin^4\frac{\alpha}{2}$</p><p>Используем формулу разности квадратов $a^2 - b^2 = (a-b)(a+b)$:</p><p>$\cos^4\frac{\alpha}{2} - \sin^4\frac{\alpha}{2} = (\cos^2\frac{\alpha}{2} - \sin^2\frac{\alpha}{2})(\cos^2\frac{\alpha}{2} + \sin^2\frac{\alpha}{2})$.</p><p>Применяем основное тригонометрическое тождество $\sin^2 x + \cos^2 x = 1$ и формулу косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$.</p><p>Первая скобка: $\cos^2\frac{\alpha}{2} - \sin^2\frac{\alpha}{2} = \cos(2 \cdot \frac{\alpha}{2}) = \cos\alpha$.</p><p>Вторая скобка: $\cos^2\frac{\alpha}{2} + \sin^2\frac{\alpha}{2} = 1$.</p><p>В результате получаем:</p><p>$\cos\alpha \cdot 1 = \cos\alpha$.</p><p><strong>Ответ:</strong> $\cos\alpha$.</p><p><strong>в)</strong> $\cos^2\alpha - 4\sin^2\frac{\alpha}{2} \cdot \cos^2\frac{\alpha}{2}$</p><p>Преобразуем вторую часть выражения:</p><p>$4\sin^2\frac{\alpha}{2} \cdot \cos^2\frac{\alpha}{2} = (2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2})^2$.</p><p>Используем формулу синуса двойного угла $2\sin x \cos x = \sin(2x)$:</p><p>$(2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2})^2 = (\sin(2 \cdot \frac{\alpha}{2}))^2 = (\sin\alpha)^2 = \sin^2\alpha$.</p><p>Подставляем обратно в исходное выражение:</p><p>$\cos^2\alpha - \sin^2\alpha$.</p><p>Это формула косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$.</p><p>$\cos^2\alpha - \sin^2\alpha = \cos(2\alpha)$.</p><p><strong>Ответ:</strong> $\cos(2\alpha)$.</p><p><strong>г)</strong> $1 - 8\sin^2\alpha \cdot \cos^2\alpha$</p><p>Преобразуем вторую часть выражения:</p><p>$8\sin^2\alpha \cdot \cos^2\alpha = 2 \cdot (4\sin^2\alpha \cos^2\alpha) = 2 \cdot (2\sin\alpha\cos\alpha)^2$.</p><p>Используем формулу синуса двойного угла $2\sin x \cos x = \sin(2x)$:</p><p>$2 \cdot (\sin(2\alpha))^2 = 2\sin^2(2\alpha)$.</p><p>Подставляем обратно в исходное выражение:</p><p>$1 - 2\sin^2(2\alpha)$.</p><p>Используем формулу косинуса двойного угла $\cos(2x) = 1 - 2\sin^2 x$. В нашем случае $x = 2\alpha$.</p><p>$1 - 2\sin^2(2\alpha) = \cos(2 \cdot 2\alpha) = \cos(4\alpha)$.</p><p><strong>Ответ:</strong> $\cos(4\alpha)$.</p><p><strong>д)</strong> $(\text{ctg}\frac{\alpha}{2} - \text{tg}\frac{\alpha}{2}) \cdot \sin\alpha$</p><p>Сначала упростим выражение в скобках. Представим котангенс и тангенс через синус и косинус:</p><p>$\text{ctg}\frac{\alpha}{2} - \text{tg}\frac{\alpha}{2} = \frac{\cos\frac{\alpha}{2}}{\sin\frac{\alpha}{2}} - \frac{\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}}$.</p><p>Приводим к общему знаменателю:</p><p>$\frac{\cos^2\frac{\alpha}{2} - \sin^2\frac{\alpha}{2}}{\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}}$.</p><p>Числитель - это формула косинуса двойного угла: $\cos^2\frac{\alpha}{2} - \sin^2\frac{\alpha}{2} = \cos\alpha$.</p><p>Знаменатель - это половина синуса двойного угла: $\sin\frac{\alpha}{2}\cos\frac{\alpha}{2} = \frac{1}{2}\sin\alpha$.</p><p>Выражение в скобках равно: $\frac{\cos\alpha}{\frac{1}{2}\sin\alpha} = \frac{2\cos\alpha}{\sin\alpha}$.</p><p>Теперь умножим это на $\sin\alpha$:</p><p>$\frac{2\cos\alpha}{\sin\alpha} \cdot \sin\alpha = 2\cos\alpha$.</p><p><strong>Ответ:</strong> $2\cos\alpha$.</p><p><strong>е)</strong> $(2\text{ctg}\frac{\alpha}{2} \cdot \sin^2\frac{\alpha}{2}) : (\sin^2\frac{\alpha}{2} - \cos^2\frac{\alpha}{2})$</p><p>Упростим делимое: $2\text{ctg}\frac{\alpha}{2} \cdot \sin^2\frac{\alpha}{2}$.</p><p>$2 \cdot \frac{\cos\frac{\alpha}{2}}{\sin\frac{\alpha}{2}} \cdot \sin^2\frac{\alpha}{2} = 2\cos\frac{\alpha}{2}\sin\frac{\alpha}{2}$.</p><p>Это формула синуса двойного угла: $2\cos\frac{\alpha}{2}\sin\frac{\alpha}{2} = \sin(2 \cdot \frac{\alpha}{2}) = \sin\alpha$.</p><p>Теперь упростим делитель: $\sin^2\frac{\alpha}{2} - \cos^2\frac{\alpha}{2}$.</p><p>Вынесем минус за скобки: $-(\cos^2\frac{\alpha}{2} - \sin^2\frac{\alpha}{2})$.</p><p>В скобках формула косинуса двойного угла: $\cos(2 \cdot \frac{\alpha}{2}) = \cos\alpha$.</p><p>Значит, делитель равен $-\cos\alpha$.</p><p>Теперь выполним деление:</p><p>$\sin\alpha : (-\cos\alpha) = \frac{\sin\alpha}{-\cos\alpha} = -\text{tg}\alpha$.</p><p><strong>Ответ:</strong> $-\text{tg}\alpha$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108550" "type" => "task" ] "previous" => array:2 [ "refs" => "1108548" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:1 [ 0 => App\Models\Book {#1175 #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 {#1168 #items: array:1 [ 0 => App\Models\Term {#1167 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1174 #items: array:1 [ 0 => App\Models\Term {#1176 #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 {#1172 #items: array:1 [ 0 => App\Models\Term {#1170 #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 {#1173 #items: array:3 [ 0 => App\Models\Term {#1171 #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 {#1201 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array: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 {#1202 #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 {#1204 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1205 #items: array:1 [ 0 => App\Models\Term {#1206 #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 {#1207 #items: array:1 [ 0 => App\Models\Term {#1208 #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 {#1209 #items: array:1 [ 0 => App\Models\Term {#1210 #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 {#1211 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1212 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1213 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1214 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1215 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1216 #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 {#1217 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1218 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1219 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1220 #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 {#1168} "field_class" => Illuminate\Database\Eloquent\Collection {#1174} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1172} "field_author" => Illuminate\Database\Eloquent\Collection {#1173} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1204} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1205} "field_country" => Illuminate\Database\Eloquent\Collection {#1207} "field_city" => Illuminate\Database\Eloquent\Collection {#1209} "field_series" => Illuminate\Database\Eloquent\Collection {#1211} "field_umk" => Illuminate\Database\Eloquent\Collection {#1212} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1213} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1214} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1215} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1216} "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 {#1217} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1218} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1219} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1220} "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 {#1228 #items: array:1 [ 0 => App\Models\BookPage {#1227 #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" => 1097965 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "223" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-223" "field_display_title" => "223" "field_folder" => "folder1" "field_image_name" => "223" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1229 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1230 #items: array:1 [ 0 => App\Models\Book {#1175} ] #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 {#1231 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1241 #items: array:1 [ 0 => App\Models\Element {#1240 #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" => 1261116 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1242 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261116 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1242 …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" => "1097966" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097964" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1425 #items: array:7 [ 0 => App\Models\Task {#1439 #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" => 1108544 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "223" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-798" "field_display_title" => "798" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1440 …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 {#1441 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1442 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1443 …2} "content" => Illuminate\Database\Eloquent\Collection {#1446 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1452 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108544 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "223" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-798" "field_display_title" => "798" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1440 …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 {#1441 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1442 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1443 …2} "content" => Illuminate\Database\Eloquent\Collection {#1446 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1452 …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 {#1445 #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" => 1108545 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "223" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-799" "field_display_title" => "799" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1444 …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 {#1451 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1453 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1449 …2} "content" => Illuminate\Database\Eloquent\Collection {#1448 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1466 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108545 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "223" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-799" "field_display_title" => "799" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1444 …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 {#1451 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1453 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1449 …2} "content" => Illuminate\Database\Eloquent\Collection {#1448 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1466 …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 {#1450 #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" => 1108546 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "223" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-800" "field_display_title" => "800" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1447 …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 {#1465 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1467 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1463 …2} "content" => Illuminate\Database\Eloquent\Collection {#1462 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1480 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108546 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "223" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-800" "field_display_title" => "800" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1447 …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 {#1465 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1467 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1463 …2} "content" => Illuminate\Database\Eloquent\Collection {#1462 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1480 …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 {#1464 #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" => 1108547 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "223" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-801" "field_display_title" => "801" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1461 …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 {#1479 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1481 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1477 …2} "content" => Illuminate\Database\Eloquent\Collection {#1476 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1494 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108547 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "223" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-801" "field_display_title" => "801" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1461 …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 {#1479 …2} "top_parent_branch" => 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[] #guarded: array:1 [ 0 => "*" ] }
№803 (с. 223)
Условие. №803 (с. 223)
скриншот условия
803. Упростите выражение:
а) $\sin\frac{\alpha}{4} \cdot \cos\frac{\alpha}{4} \cdot \cos\frac{\alpha}{2}$;
б) $\cos^4\frac{\alpha}{2} - \sin^4\frac{\alpha}{2}$;
в) $\cos^2\alpha - 4\sin^2\frac{\alpha}{2} \cdot \cos^2\frac{\alpha}{2}$;
г) $1 - 8\sin^2\alpha \cdot \cos^2\alpha$;
д) $(\cot\frac{\alpha}{2} - \tan\frac{\alpha}{2}) \cdot \sin\alpha$;
е) $(2\cot\frac{\alpha}{2} \cdot \sin^2\frac{\alpha}{2}) : (\sin^2\frac{\alpha}{2} - \cos^2\frac{\alpha}{2})$.
Решение. №803 (с. 223)
Решение 2 (rus). №803 (с. 223)
а) $\sin\frac{\alpha}{4} \cdot \cos\frac{\alpha}{4} \cdot \cos\frac{\alpha}{2}$
Используем формулу синуса двойного угла $2\sin x \cos x = \sin(2x)$, из которой следует, что $\sin x \cos x = \frac{1}{2}\sin(2x)$.
Сначала преобразуем произведение $\sin\frac{\alpha}{4} \cdot \cos\frac{\alpha}{4}$:
$\sin\frac{\alpha}{4} \cdot \cos\frac{\alpha}{4} = \frac{1}{2}\sin(2 \cdot \frac{\alpha}{4}) = \frac{1}{2}\sin\frac{\alpha}{2}$.
Теперь подставим это в исходное выражение:
$\frac{1}{2}\sin\frac{\alpha}{2} \cdot \cos\frac{\alpha}{2}$.
Снова применяем ту же формулу для угла $\frac{\alpha}{2}$:
$\frac{1}{2}(\frac{1}{2}\sin(2 \cdot \frac{\alpha}{2})) = \frac{1}{4}\sin\alpha$.
Ответ: $\frac{1}{4}\sin\alpha$.
б) $\cos^4\frac{\alpha}{2} - \sin^4\frac{\alpha}{2}$
Используем формулу разности квадратов $a^2 - b^2 = (a-b)(a+b)$:
$\cos^4\frac{\alpha}{2} - \sin^4\frac{\alpha}{2} = (\cos^2\frac{\alpha}{2} - \sin^2\frac{\alpha}{2})(\cos^2\frac{\alpha}{2} + \sin^2\frac{\alpha}{2})$.
Применяем основное тригонометрическое тождество $\sin^2 x + \cos^2 x = 1$ и формулу косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$.
Первая скобка: $\cos^2\frac{\alpha}{2} - \sin^2\frac{\alpha}{2} = \cos(2 \cdot \frac{\alpha}{2}) = \cos\alpha$.
Вторая скобка: $\cos^2\frac{\alpha}{2} + \sin^2\frac{\alpha}{2} = 1$.
В результате получаем:
$\cos\alpha \cdot 1 = \cos\alpha$.
Ответ: $\cos\alpha$.
в) $\cos^2\alpha - 4\sin^2\frac{\alpha}{2} \cdot \cos^2\frac{\alpha}{2}$
Преобразуем вторую часть выражения:
$4\sin^2\frac{\alpha}{2} \cdot \cos^2\frac{\alpha}{2} = (2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2})^2$.
Используем формулу синуса двойного угла $2\sin x \cos x = \sin(2x)$:
$(2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2})^2 = (\sin(2 \cdot \frac{\alpha}{2}))^2 = (\sin\alpha)^2 = \sin^2\alpha$.
Подставляем обратно в исходное выражение:
$\cos^2\alpha - \sin^2\alpha$.
Это формула косинуса двойного угла $\cos(2x) = \cos^2 x - \sin^2 x$.
$\cos^2\alpha - \sin^2\alpha = \cos(2\alpha)$.
Ответ: $\cos(2\alpha)$.
г) $1 - 8\sin^2\alpha \cdot \cos^2\alpha$
Преобразуем вторую часть выражения:
$8\sin^2\alpha \cdot \cos^2\alpha = 2 \cdot (4\sin^2\alpha \cos^2\alpha) = 2 \cdot (2\sin\alpha\cos\alpha)^2$.
Используем формулу синуса двойного угла $2\sin x \cos x = \sin(2x)$:
$2 \cdot (\sin(2\alpha))^2 = 2\sin^2(2\alpha)$.
Подставляем обратно в исходное выражение:
$1 - 2\sin^2(2\alpha)$.
Используем формулу косинуса двойного угла $\cos(2x) = 1 - 2\sin^2 x$. В нашем случае $x = 2\alpha$.
$1 - 2\sin^2(2\alpha) = \cos(2 \cdot 2\alpha) = \cos(4\alpha)$.
Ответ: $\cos(4\alpha)$.
д) $(\text{ctg}\frac{\alpha}{2} - \text{tg}\frac{\alpha}{2}) \cdot \sin\alpha$
Сначала упростим выражение в скобках. Представим котангенс и тангенс через синус и косинус:
$\text{ctg}\frac{\alpha}{2} - \text{tg}\frac{\alpha}{2} = \frac{\cos\frac{\alpha}{2}}{\sin\frac{\alpha}{2}} - \frac{\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}}$.
Приводим к общему знаменателю:
$\frac{\cos^2\frac{\alpha}{2} - \sin^2\frac{\alpha}{2}}{\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}}$.
Числитель - это формула косинуса двойного угла: $\cos^2\frac{\alpha}{2} - \sin^2\frac{\alpha}{2} = \cos\alpha$.
Знаменатель - это половина синуса двойного угла: $\sin\frac{\alpha}{2}\cos\frac{\alpha}{2} = \frac{1}{2}\sin\alpha$.
Выражение в скобках равно: $\frac{\cos\alpha}{\frac{1}{2}\sin\alpha} = \frac{2\cos\alpha}{\sin\alpha}$.
Теперь умножим это на $\sin\alpha$:
$\frac{2\cos\alpha}{\sin\alpha} \cdot \sin\alpha = 2\cos\alpha$.
Ответ: $2\cos\alpha$.
е) $(2\text{ctg}\frac{\alpha}{2} \cdot \sin^2\frac{\alpha}{2}) : (\sin^2\frac{\alpha}{2} - \cos^2\frac{\alpha}{2})$
Упростим делимое: $2\text{ctg}\frac{\alpha}{2} \cdot \sin^2\frac{\alpha}{2}$.
$2 \cdot \frac{\cos\frac{\alpha}{2}}{\sin\frac{\alpha}{2}} \cdot \sin^2\frac{\alpha}{2} = 2\cos\frac{\alpha}{2}\sin\frac{\alpha}{2}$.
Это формула синуса двойного угла: $2\cos\frac{\alpha}{2}\sin\frac{\alpha}{2} = \sin(2 \cdot \frac{\alpha}{2}) = \sin\alpha$.
Теперь упростим делитель: $\sin^2\frac{\alpha}{2} - \cos^2\frac{\alpha}{2}$.
Вынесем минус за скобки: $-(\cos^2\frac{\alpha}{2} - \sin^2\frac{\alpha}{2})$.
В скобках формула косинуса двойного угла: $\cos(2 \cdot \frac{\alpha}{2}) = \cos\alpha$.
Значит, делитель равен $-\cos\alpha$.
Теперь выполним деление:
$\sin\alpha : (-\cos\alpha) = \frac{\sin\alpha}{-\cos\alpha} = -\text{tg}\alpha$.
Ответ: $-\text{tg}\alpha$.
Другие задания:
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