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Номер 680, страница 195 - 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. 25. Формулы приведения - номер 680, страница 195.
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" => 1108417 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-680" "field_display_title" => "680" "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" => 1107667 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "25. Формулы приведения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "189" "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" => 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"field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#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. 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Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107667 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "25. Формулы приведения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103} "field_page_start" => "189" "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 {#1164 #items: array:3 [ 0 => App\Models\Element {#1140 #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" => 1277035 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1141 #items: array:1 [ 0 => App\Models\Edition {#1139 #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 {#1142 …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 {#1144 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1145 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1146 …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 {#1142 …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 {#1144 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1145 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1146 …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" => "1108417" "type" => "task" ] "text" => "<p><strong>680.</strong> Дана задача «Боковые стороны AB и DC трапеции ABCD перпендикулярны. Найти площадь этой трапеции, если AB = 1 дм, $\angle BAC = \angle CDA = \alpha$». Ее решение Даша начала так: площадь данной трапеции равна сумме площадей треугольников ABC и ACD (рисунок 71):</p><p>$S_{ABCD} = \frac{1}{2}AB \cdot AC \cdot \sin \alpha + \frac{1}{2}AC \cdot AD \cdot \sin \angle CAD.$</p><p>Найдем неизвестные величины AC, AD, используя теорему синусов для треугольников ABC и ADC. Продолжите ее решение.</p><svg viewBox="0 0 320 120" xmlns="http://www.w3.org/2000/svg" font-family="serif"><!-- Quadrilateral ABCD --><path d="M0 100 L80 20 L220 40 L300 100 Z" stroke="black" fill="none"/><!-- Diagonal AC --><line x1="0" y1="100" x2="220" y2="40" stroke="black"/><!-- Labels for points --><text x="-5" y="105" font-size="12">A</text><text x="75" y="15" font-size="12">B</text><text x="225" y="45" font-size="12">C</text><text x="305" y="105" font-size="12">D</text><!-- Angle BAC = alpha (double arc) --><path d="M20 95 A20 20 0 0 1 25 80" stroke="black" fill="none"/><path d="M17 92 A17 17 0 0 1 22 83" stroke="black" fill="none"/><text x="20" y="85" font-size="10">$\alpha$</text><!-- Angle CAD = 90 - 2alpha (triple arc) --><path d="M40 98 A70 70 0 0 0 190 45" stroke="black" fill="none"/><path d="M45 96 A65 65 0 0 0 185 47" stroke="black" fill="none"/><path d="M50 94 A60 60 0 0 0 180 49" stroke="black" fill="none"/><text x="90" y="70" font-size="10">$90^\circ-2\alpha$</text><!-- Angle ABC = 90 + alpha (single arc) --><path d="M85 25 A15 15 0 0 1 95 35" stroke="black" fill="none"/><text x="90" y="30" font-size="10">$90^\circ+\alpha$</text><!-- Angle CDA = alpha (double arc) --><path d="M270 70 A20 20 0 0 1 285 95" stroke="black" fill="none"/><path d="M267 73 A17 17 0 0 1 282 92" stroke="black" fill="none"/><text x="270" y="80" font-size="10">$\alpha$</text><!-- Right angle symbol at C for angle BCA --><path d="M215 40 L215 45 L220 45" stroke="black" fill="none"/><rect x="215" y="40" width="5" height="5" stroke="black" fill="none"/><!-- "Рисунок 71" text --><text x="120" y="115" font-size="14">Рисунок 71</text></svg>" "img" => array:1 [ 0 => array:5 [ "name" => "680-1.jpg" "alt" => null "width" => "1610" "height" => 982 "path" => "/media/algebra_09/soltan-u19/0-1/680-1.webp?ts=1752834689" ] ] ] #original: array:7 [ "id" => 1277035 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1141} "task" => array:2 [ "refs" => "1108417" "type" => "task" ] "text" => "<p><strong>680.</strong> Дана задача «Боковые стороны AB и DC трапеции ABCD перпендикулярны. Найти площадь этой трапеции, если AB = 1 дм, $\angle BAC = \angle CDA = \alpha$». Ее решение Даша начала так: площадь данной трапеции равна сумме площадей треугольников ABC и ACD (рисунок 71):</p><p>$S_{ABCD} = \frac{1}{2}AB \cdot AC \cdot \sin \alpha + \frac{1}{2}AC \cdot AD \cdot \sin \angle CAD.$</p><p>Найдем неизвестные величины AC, AD, используя теорему синусов для треугольников ABC и ADC. Продолжите ее решение.</p><svg viewBox="0 0 320 120" xmlns="http://www.w3.org/2000/svg" font-family="serif"><!-- Quadrilateral ABCD --><path d="M0 100 L80 20 L220 40 L300 100 Z" stroke="black" fill="none"/><!-- Diagonal AC --><line x1="0" y1="100" x2="220" y2="40" stroke="black"/><!-- Labels for points --><text x="-5" y="105" font-size="12">A</text><text x="75" y="15" font-size="12">B</text><text x="225" y="45" font-size="12">C</text><text x="305" y="105" font-size="12">D</text><!-- Angle BAC = alpha (double arc) --><path d="M20 95 A20 20 0 0 1 25 80" stroke="black" fill="none"/><path d="M17 92 A17 17 0 0 1 22 83" stroke="black" fill="none"/><text x="20" y="85" font-size="10">$\alpha$</text><!-- Angle CAD = 90 - 2alpha (triple arc) --><path d="M40 98 A70 70 0 0 0 190 45" stroke="black" fill="none"/><path d="M45 96 A65 65 0 0 0 185 47" stroke="black" fill="none"/><path d="M50 94 A60 60 0 0 0 180 49" stroke="black" fill="none"/><text x="90" y="70" font-size="10">$90^\circ-2\alpha$</text><!-- Angle ABC = 90 + alpha (single arc) --><path d="M85 25 A15 15 0 0 1 95 35" stroke="black" fill="none"/><text x="90" y="30" font-size="10">$90^\circ+\alpha$</text><!-- Angle CDA = alpha (double arc) --><path d="M270 70 A20 20 0 0 1 285 95" stroke="black" fill="none"/><path d="M267 73 A17 17 0 0 1 282 92" stroke="black" fill="none"/><text x="270" y="80" font-size="10">$\alpha$</text><!-- Right angle symbol at C for angle BCA --><path d="M215 40 L215 45 L220 45" stroke="black" fill="none"/><rect x="215" y="40" width="5" height="5" stroke="black" fill="none"/><!-- "Рисунок 71" text --><text x="120" y="115" font-size="14">Рисунок 71</text></svg>" "img" => array:1 [ 0 => array:5 [ "name" => "680-1.jpg" "alt" => null "width" => "1610" "height" => 982 "path" => "/media/algebra_09/soltan-u19/0-1/680-1.webp?ts=1752834689" ] ] ] #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 {#1147 #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" => 1278326 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1148 #items: array:1 [ 0 => App\Models\Edition {#1149 #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 {#1150 …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 {#1152 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1154 …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 {#1150 …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 {#1152 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1154 …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" => "1108417" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "680-1.jpg" "alt" => null "width" => "1424" "height" => 1840 "path" => "/media/algebra_09/soltan-u19/1-1/680-1.webp?ts=1752831477" ] ] ] #original: array:6 [ "id" => 1278326 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1148} "task" => array:2 [ "refs" => "1108417" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "680-1.jpg" "alt" => null "width" => "1424" "height" => 1840 "path" => "/media/algebra_09/soltan-u19/1-1/680-1.webp?ts=1752831477" ] ] ] #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 {#1155 #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" => 1554281 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1156 #items: array:1 [ 0 => App\Models\Edition {#1157 #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 {#1158 …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 {#1160 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1161 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1162 …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 {#1158 …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 {#1160 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1161 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1162 …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" => "1108417" "type" => "task" ] "text" => "<p>Продолжим решение задачи, начатое Дашей. Для этого нам необходимо найти неизвестные величины: длину стороны $AC$, длину стороны $AD$ и угол $\angle CAD$.</p><p><strong>1.</strong> <strong>Найдем углы в трапеции.</strong></p><p>По условию, боковые стороны $AB$ и $DC$ трапеции перпендикулярны. Это означает, что прямые, содержащие эти стороны, пересекаются под прямым углом. Обозначим точку их пересечения как $K$. Тогда $\triangle AKD$ является прямоугольным с $\angle AKD = 90°$.</p><p>В прямоугольном $\triangle AKD$ один из острых углов равен $\angle KDA = \angle CDA = \alpha$. Следовательно, другой острый угол $\angle KAD = 90° - \alpha$.</p><p>Угол $\angle BAD$ трапеции равен углу $\angle KAD$, то есть $\angle BAD = 90° - \alpha$.Угол $\angle BAD$ также является суммой двух углов: $\angle BAD = \angle BAC + \angle CAD$.По условию $\angle BAC = \alpha$, значит, мы можем найти $\angle CAD$:$90° - \alpha = \alpha + \angle CAD$$\angle CAD = 90° - 2\alpha$.</p><p>Поскольку $ABCD$ — трапеция, ее основания $BC$ и $AD$ параллельны ($BC \parallel AD$). Прямая $AC$ является секущей. Поэтому накрест лежащие углы равны: $\angle BCA = \angle CAD = 90° - 2\alpha$.</p><p>Теперь рассмотрим $\triangle ABC$. Сумма его углов равна $180°$. Мы знаем два угла: $\angle BAC = \alpha$ и $\angle BCA = 90° - 2\alpha$. Найдем третий угол, $\angle ABC$:$\angle ABC = 180° - (\angle BAC + \angle BCA) = 180° - (\alpha + 90° - 2\alpha) = 180° - (90° - \alpha) = 90° + \alpha$.</p><p>Аналогично найдем угол $\angle ACD$ в $\triangle ADC$. Мы знаем $\angle CAD = 90° - 2\alpha$ и $\angle CDA = \alpha$.$\angle ACD = 180° - (\angle CAD + \angle CDA) = 180° - (90° - 2\alpha + \alpha) = 180° - (90° - \alpha) = 90° + \alpha$.</p><p><strong>2.</strong> <strong>Найдем длины сторон $AC$ и $AD$ по теореме синусов.</strong></p><p>Применим теорему синусов для $\triangle ABC$:$\frac{AB}{\sin(\angle BCA)} = \frac{AC}{\sin(\angle ABC)}$Подставим известные значения ($AB=1$ дм) и выражения для углов:$\frac{1}{\sin(90° - 2\alpha)} = \frac{AC}{\sin(90° + \alpha)}$Используя формулы приведения ($\sin(90° - x) = \cos(x)$ и $\sin(90° + x) = \cos(x)$), получаем:$\frac{1}{\cos(2\alpha)} = \frac{AC}{\cos(\alpha)}$Отсюда выражаем $AC$:$AC = \frac{\cos(\alpha)}{\cos(2\alpha)}$.</p><p>Теперь применим теорему синусов для $\triangle ADC$:$\frac{AC}{\sin(\angle CDA)} = \frac{AD}{\sin(\angle ACD)}$Подставим известные выражения для сторон и углов:$\frac{AC}{\sin(\alpha)} = \frac{AD}{\sin(90° + \alpha)}$$\frac{AC}{\sin(\alpha)} = \frac{AD}{\cos(\alpha)}$Выражаем $AD$:$AD = AC \cdot \frac{\cos(\alpha)}{\sin(\alpha)} = \frac{\cos(\alpha)}{\cos(2\alpha)} \cdot \frac{\cos(\alpha)}{\sin(\alpha)} = \frac{\cos^2(\alpha)}{\sin(\alpha)\cos(2\alpha)}$.</p><p><strong>3.</strong> <strong>Вычислим площадь трапеции.</strong></p><p>Подставим найденные величины в формулу Даши:$S_{ABCD} = S_{ABC} + S_{ACD} = \frac{1}{2}AB \cdot AC \cdot \sin(\angle BAC) + \frac{1}{2}AC \cdot AD \cdot \sin(\angle CAD)$$S_{ABCD} = \frac{1}{2} \cdot 1 \cdot \left(\frac{\cos(\alpha)}{\cos(2\alpha)}\right) \cdot \sin(\alpha) + \frac{1}{2}\left(\frac{\cos(\alpha)}{\cos(2\alpha)}\right) \cdot \left(\frac{\cos^2(\alpha)}{\sin(\alpha)\cos(2\alpha)}\right) \cdot \sin(90° - 2\alpha)$Упростим выражение. Заметим, что $\sin(90° - 2\alpha) = \cos(2\alpha)$.$S_{ABCD} = \frac{\sin(\alpha)\cos(\alpha)}{2\cos(2\alpha)} + \frac{1}{2} \frac{\cos^3(\alpha)}{\sin(\alpha)\cos^2(2\alpha)} \cdot \cos(2\alpha)$$S_{ABCD} = \frac{\sin(\alpha)\cos(\alpha)}{2\cos(2\alpha)} + \frac{\cos^3(\alpha)}{2\sin(\alpha)\cos(2\alpha)}$Приведем к общему знаменателю $2\sin(\alpha)\cos(2\alpha)$:$S_{ABCD} = \frac{\sin^2(\alpha)\cos(\alpha) + \cos^3(\alpha)}{2\sin(\alpha)\cos(2\alpha)}$В числителе вынесем $\cos(\alpha)$ за скобки:$S_{ABCD} = \frac{\cos(\alpha)(\sin^2(\alpha) + \cos^2(\alpha))}{2\sin(\alpha)\cos(2\alpha)}$Используя основное тригонометрическое тождество $\sin^2(\alpha) + \cos^2(\alpha) = 1$, получаем:$S_{ABCD} = \frac{\cos(\alpha)}{2\sin(\alpha)\cos(2\alpha)}$Так как $\frac{\cos(\alpha)}{\sin(\alpha)} = \cot(\alpha)$, окончательное выражение для площади:$S_{ABCD} = \frac{\cot(\alpha)}{2\cos(2\alpha)}$.</p><p><strong>Ответ:</strong> $S_{ABCD} = \frac{\cot(\alpha)}{2\cos(2\alpha)}$ дм$^2$.</p>" ] #original: array:6 [ "id" => 1554281 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1156} "task" => array:2 [ "refs" => "1108417" "type" => "task" ] "text" => "<p>Продолжим решение задачи, начатое Дашей. Для этого нам необходимо найти неизвестные величины: длину стороны $AC$, длину стороны $AD$ и угол $\angle CAD$.</p><p><strong>1.</strong> <strong>Найдем углы в трапеции.</strong></p><p>По условию, боковые стороны $AB$ и $DC$ трапеции перпендикулярны. Это означает, что прямые, содержащие эти стороны, пересекаются под прямым углом. Обозначим точку их пересечения как $K$. Тогда $\triangle AKD$ является прямоугольным с $\angle AKD = 90°$.</p><p>В прямоугольном $\triangle AKD$ один из острых углов равен $\angle KDA = \angle CDA = \alpha$. Следовательно, другой острый угол $\angle KAD = 90° - \alpha$.</p><p>Угол $\angle BAD$ трапеции равен углу $\angle KAD$, то есть $\angle BAD = 90° - \alpha$.Угол $\angle BAD$ также является суммой двух углов: $\angle BAD = \angle BAC + \angle CAD$.По условию $\angle BAC = \alpha$, значит, мы можем найти $\angle CAD$:$90° - \alpha = \alpha + \angle CAD$$\angle CAD = 90° - 2\alpha$.</p><p>Поскольку $ABCD$ — трапеция, ее основания $BC$ и $AD$ параллельны ($BC \parallel AD$). Прямая $AC$ является секущей. Поэтому накрест лежащие углы равны: $\angle BCA = \angle CAD = 90° - 2\alpha$.</p><p>Теперь рассмотрим $\triangle ABC$. Сумма его углов равна $180°$. Мы знаем два угла: $\angle BAC = \alpha$ и $\angle BCA = 90° - 2\alpha$. Найдем третий угол, $\angle ABC$:$\angle ABC = 180° - (\angle BAC + \angle BCA) = 180° - (\alpha + 90° - 2\alpha) = 180° - (90° - \alpha) = 90° + \alpha$.</p><p>Аналогично найдем угол $\angle ACD$ в $\triangle ADC$. Мы знаем $\angle CAD = 90° - 2\alpha$ и $\angle CDA = \alpha$.$\angle ACD = 180° - (\angle CAD + \angle CDA) = 180° - (90° - 2\alpha + \alpha) = 180° - (90° - \alpha) = 90° + \alpha$.</p><p><strong>2.</strong> <strong>Найдем длины сторон $AC$ и $AD$ по теореме синусов.</strong></p><p>Применим теорему синусов для $\triangle ABC$:$\frac{AB}{\sin(\angle BCA)} = \frac{AC}{\sin(\angle ABC)}$Подставим известные значения ($AB=1$ дм) и выражения для углов:$\frac{1}{\sin(90° - 2\alpha)} = \frac{AC}{\sin(90° + \alpha)}$Используя формулы приведения ($\sin(90° - x) = \cos(x)$ и $\sin(90° + x) = \cos(x)$), получаем:$\frac{1}{\cos(2\alpha)} = \frac{AC}{\cos(\alpha)}$Отсюда выражаем $AC$:$AC = \frac{\cos(\alpha)}{\cos(2\alpha)}$.</p><p>Теперь применим теорему синусов для $\triangle ADC$:$\frac{AC}{\sin(\angle CDA)} = \frac{AD}{\sin(\angle ACD)}$Подставим известные выражения для сторон и углов:$\frac{AC}{\sin(\alpha)} = \frac{AD}{\sin(90° + \alpha)}$$\frac{AC}{\sin(\alpha)} = \frac{AD}{\cos(\alpha)}$Выражаем $AD$:$AD = AC \cdot \frac{\cos(\alpha)}{\sin(\alpha)} = \frac{\cos(\alpha)}{\cos(2\alpha)} \cdot \frac{\cos(\alpha)}{\sin(\alpha)} = \frac{\cos^2(\alpha)}{\sin(\alpha)\cos(2\alpha)}$.</p><p><strong>3.</strong> <strong>Вычислим площадь трапеции.</strong></p><p>Подставим найденные величины в формулу Даши:$S_{ABCD} = S_{ABC} + S_{ACD} = \frac{1}{2}AB \cdot AC \cdot \sin(\angle BAC) + \frac{1}{2}AC \cdot AD \cdot \sin(\angle CAD)$$S_{ABCD} = \frac{1}{2} \cdot 1 \cdot \left(\frac{\cos(\alpha)}{\cos(2\alpha)}\right) \cdot \sin(\alpha) + \frac{1}{2}\left(\frac{\cos(\alpha)}{\cos(2\alpha)}\right) \cdot \left(\frac{\cos^2(\alpha)}{\sin(\alpha)\cos(2\alpha)}\right) \cdot \sin(90° - 2\alpha)$Упростим выражение. Заметим, что $\sin(90° - 2\alpha) = \cos(2\alpha)$.$S_{ABCD} = \frac{\sin(\alpha)\cos(\alpha)}{2\cos(2\alpha)} + \frac{1}{2} \frac{\cos^3(\alpha)}{\sin(\alpha)\cos^2(2\alpha)} \cdot \cos(2\alpha)$$S_{ABCD} = \frac{\sin(\alpha)\cos(\alpha)}{2\cos(2\alpha)} + \frac{\cos^3(\alpha)}{2\sin(\alpha)\cos(2\alpha)}$Приведем к общему знаменателю $2\sin(\alpha)\cos(2\alpha)$:$S_{ABCD} = \frac{\sin^2(\alpha)\cos(\alpha) + \cos^3(\alpha)}{2\sin(\alpha)\cos(2\alpha)}$В числителе вынесем $\cos(\alpha)$ за скобки:$S_{ABCD} = \frac{\cos(\alpha)(\sin^2(\alpha) + \cos^2(\alpha))}{2\sin(\alpha)\cos(2\alpha)}$Используя основное тригонометрическое тождество $\sin^2(\alpha) + \cos^2(\alpha) = 1$, получаем:$S_{ABCD} = \frac{\cos(\alpha)}{2\sin(\alpha)\cos(2\alpha)}$Так как $\frac{\cos(\alpha)}{\sin(\alpha)} = \cot(\alpha)$, окончательное выражение для площади:$S_{ABCD} = \frac{\cot(\alpha)}{2\cos(2\alpha)}$.</p><p><strong>Ответ:</strong> $S_{ABCD} = \frac{\cot(\alpha)}{2\cos(2\alpha)}$ дм$^2$.</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" => "1108418" "type" => "task" ] "previous" => array:2 [ "refs" => "1108416" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1194 #items: array:1 [ 0 => App\Models\Book {#1165 #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 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"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 {#1174 #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 {#1175 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1176 #items: array:1 [ 0 => App\Models\Term {#1177 #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 {#1178 #items: array:1 [ 0 => App\Models\Term {#1179 #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 {#1180 #items: array:1 [ 0 => App\Models\Term {#1181 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1182 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1183 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1184 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1185 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1186 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1187 #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 {#1188 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1189 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1190 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1191 #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 {#1166} "field_class" => Illuminate\Database\Eloquent\Collection {#1167} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1169} "field_author" => Illuminate\Database\Eloquent\Collection {#1171} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1175} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1176} "field_country" => Illuminate\Database\Eloquent\Collection {#1178} "field_city" => Illuminate\Database\Eloquent\Collection {#1180} "field_series" => Illuminate\Database\Eloquent\Collection {#1182} "field_umk" => Illuminate\Database\Eloquent\Collection {#1183} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1184} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1185} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1186} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1187} "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 {#1188} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1189} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1190} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1191} "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 {#1425 #items: array:1 [ 0 => App\Models\BookPage {#1032 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1097937 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-195" "field_display_title" => "195" "field_folder" => "folder1" "field_image_name" => "195" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1023 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1026 #items: array:1 [ 0 => App\Models\Book {#1192 #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 {#1193 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1196 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1198 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1204 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1205 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1209 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1211 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1216 …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 {#1217 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1218 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1220 …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 {#1193 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1196 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1198 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1204 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1205 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1209 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1211 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1216 …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 {#1217 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1218 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1220 …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 {#1221 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1222 #items: array:1 [ 0 => App\Models\Element {#1223 #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" => 1261088 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1224 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261088 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1224 …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" => "1097938" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097936" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1231 #items: array:6 [ 0 => App\Models\Task {#1232 #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" => 1108413 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-676" "field_display_title" => "676" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1233 …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 {#1235 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1236 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1269 …2} "content" => Illuminate\Database\Eloquent\Collection {#1338 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1357 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108413 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-676" "field_display_title" => "676" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1233 …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 {#1235 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1236 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1269 …2} "content" => Illuminate\Database\Eloquent\Collection {#1338 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1357 …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 {#1358 #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" => 1108414 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "195" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-677" "field_display_title" => "677" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1359 …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 {#1360 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1361 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1362 …2} "content" => Illuminate\Database\Eloquent\Collection {#1363 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1370 …2} "page" => array:2 [ …2] ] 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№680 (с. 195)
Условие. №680 (с. 195)
скриншот условия
680. Дана задача «Боковые стороны AB и DC трапеции ABCD перпендикулярны. Найти площадь этой трапеции, если AB = 1 дм, $\angle BAC = \angle CDA = \alpha$». Ее решение Даша начала так: площадь данной трапеции равна сумме площадей треугольников ABC и ACD (рисунок 71):
$S_{ABCD} = \frac{1}{2}AB \cdot AC \cdot \sin \alpha + \frac{1}{2}AC \cdot AD \cdot \sin \angle CAD.$
Найдем неизвестные величины AC, AD, используя теорему синусов для треугольников ABC и ADC. Продолжите ее решение.
Решение. №680 (с. 195)
Решение 2 (rus). №680 (с. 195)
Продолжим решение задачи, начатое Дашей. Для этого нам необходимо найти неизвестные величины: длину стороны $AC$, длину стороны $AD$ и угол $\angle CAD$.
1. Найдем углы в трапеции.
По условию, боковые стороны $AB$ и $DC$ трапеции перпендикулярны. Это означает, что прямые, содержащие эти стороны, пересекаются под прямым углом. Обозначим точку их пересечения как $K$. Тогда $\triangle AKD$ является прямоугольным с $\angle AKD = 90°$.
В прямоугольном $\triangle AKD$ один из острых углов равен $\angle KDA = \angle CDA = \alpha$. Следовательно, другой острый угол $\angle KAD = 90° - \alpha$.
Угол $\angle BAD$ трапеции равен углу $\angle KAD$, то есть $\angle BAD = 90° - \alpha$.Угол $\angle BAD$ также является суммой двух углов: $\angle BAD = \angle BAC + \angle CAD$.По условию $\angle BAC = \alpha$, значит, мы можем найти $\angle CAD$:$90° - \alpha = \alpha + \angle CAD$$\angle CAD = 90° - 2\alpha$.
Поскольку $ABCD$ — трапеция, ее основания $BC$ и $AD$ параллельны ($BC \parallel AD$). Прямая $AC$ является секущей. Поэтому накрест лежащие углы равны: $\angle BCA = \angle CAD = 90° - 2\alpha$.
Теперь рассмотрим $\triangle ABC$. Сумма его углов равна $180°$. Мы знаем два угла: $\angle BAC = \alpha$ и $\angle BCA = 90° - 2\alpha$. Найдем третий угол, $\angle ABC$:$\angle ABC = 180° - (\angle BAC + \angle BCA) = 180° - (\alpha + 90° - 2\alpha) = 180° - (90° - \alpha) = 90° + \alpha$.
Аналогично найдем угол $\angle ACD$ в $\triangle ADC$. Мы знаем $\angle CAD = 90° - 2\alpha$ и $\angle CDA = \alpha$.$\angle ACD = 180° - (\angle CAD + \angle CDA) = 180° - (90° - 2\alpha + \alpha) = 180° - (90° - \alpha) = 90° + \alpha$.
2. Найдем длины сторон $AC$ и $AD$ по теореме синусов.
Применим теорему синусов для $\triangle ABC$:$\frac{AB}{\sin(\angle BCA)} = \frac{AC}{\sin(\angle ABC)}$Подставим известные значения ($AB=1$ дм) и выражения для углов:$\frac{1}{\sin(90° - 2\alpha)} = \frac{AC}{\sin(90° + \alpha)}$Используя формулы приведения ($\sin(90° - x) = \cos(x)$ и $\sin(90° + x) = \cos(x)$), получаем:$\frac{1}{\cos(2\alpha)} = \frac{AC}{\cos(\alpha)}$Отсюда выражаем $AC$:$AC = \frac{\cos(\alpha)}{\cos(2\alpha)}$.
Теперь применим теорему синусов для $\triangle ADC$:$\frac{AC}{\sin(\angle CDA)} = \frac{AD}{\sin(\angle ACD)}$Подставим известные выражения для сторон и углов:$\frac{AC}{\sin(\alpha)} = \frac{AD}{\sin(90° + \alpha)}$$\frac{AC}{\sin(\alpha)} = \frac{AD}{\cos(\alpha)}$Выражаем $AD$:$AD = AC \cdot \frac{\cos(\alpha)}{\sin(\alpha)} = \frac{\cos(\alpha)}{\cos(2\alpha)} \cdot \frac{\cos(\alpha)}{\sin(\alpha)} = \frac{\cos^2(\alpha)}{\sin(\alpha)\cos(2\alpha)}$.
3. Вычислим площадь трапеции.
Подставим найденные величины в формулу Даши:$S_{ABCD} = S_{ABC} + S_{ACD} = \frac{1}{2}AB \cdot AC \cdot \sin(\angle BAC) + \frac{1}{2}AC \cdot AD \cdot \sin(\angle CAD)$$S_{ABCD} = \frac{1}{2} \cdot 1 \cdot \left(\frac{\cos(\alpha)}{\cos(2\alpha)}\right) \cdot \sin(\alpha) + \frac{1}{2}\left(\frac{\cos(\alpha)}{\cos(2\alpha)}\right) \cdot \left(\frac{\cos^2(\alpha)}{\sin(\alpha)\cos(2\alpha)}\right) \cdot \sin(90° - 2\alpha)$Упростим выражение. Заметим, что $\sin(90° - 2\alpha) = \cos(2\alpha)$.$S_{ABCD} = \frac{\sin(\alpha)\cos(\alpha)}{2\cos(2\alpha)} + \frac{1}{2} \frac{\cos^3(\alpha)}{\sin(\alpha)\cos^2(2\alpha)} \cdot \cos(2\alpha)$$S_{ABCD} = \frac{\sin(\alpha)\cos(\alpha)}{2\cos(2\alpha)} + \frac{\cos^3(\alpha)}{2\sin(\alpha)\cos(2\alpha)}$Приведем к общему знаменателю $2\sin(\alpha)\cos(2\alpha)$:$S_{ABCD} = \frac{\sin^2(\alpha)\cos(\alpha) + \cos^3(\alpha)}{2\sin(\alpha)\cos(2\alpha)}$В числителе вынесем $\cos(\alpha)$ за скобки:$S_{ABCD} = \frac{\cos(\alpha)(\sin^2(\alpha) + \cos^2(\alpha))}{2\sin(\alpha)\cos(2\alpha)}$Используя основное тригонометрическое тождество $\sin^2(\alpha) + \cos^2(\alpha) = 1$, получаем:$S_{ABCD} = \frac{\cos(\alpha)}{2\sin(\alpha)\cos(2\alpha)}$Так как $\frac{\cos(\alpha)}{\sin(\alpha)} = \cot(\alpha)$, окончательное выражение для площади:$S_{ABCD} = \frac{\cot(\alpha)}{2\cos(2\alpha)}$.
Ответ: $S_{ABCD} = \frac{\cot(\alpha)}{2\cos(2\alpha)}$ дм$^2$.
Другие задания:
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