gdz.top
Номер 823, страница 226 - 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 823, страница 226.
App\Models\Task {#1024 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108569 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-823" "field_display_title" => "823" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107672 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "30. Упражнения на повторение раздела «Тригонометрия»" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "221" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Book {#1065 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Branch {#1094 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125 …2} ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107672 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "30. Упражнения на повторение раздела «Тригонометрия»" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "221" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1064} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:3 [ 0 => App\Models\Element {#1127 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1277178 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Edition {#1129 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1134 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108569" "type" => "task" ] "text" => "<p><strong>823.</strong> Докажите, что для треугольника ABC со сторонами a, b, c верно равенство:</p><p><strong>a)</strong> $ \frac{a+b}{c} = \frac{\cos \frac{A-B}{2}}{\sin \frac{C}{2}} $;</p><p><strong>б)</strong> $ \frac{a-b}{c} = \frac{\sin \frac{A-B}{2}}{\cos \frac{C}{2}} $ (формулы немецкого математика К. Моллвейде (1774–1825)).</p>" "img" => array:1 [ 0 => array:5 [ "name" => "823-1.jpg" "alt" => null "width" => "1550" "height" => 489 "path" => "/media/algebra_09/soltan-u19/0-1/823-1.webp?ts=1752834807" ] ] ] #original: array:7 [ "id" => 1277178 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128} "task" => array:2 [ "refs" => "1108569" "type" => "task" ] "text" => "<p><strong>823.</strong> Докажите, что для треугольника ABC со сторонами a, b, c верно равенство:</p><p><strong>a)</strong> $ \frac{a+b}{c} = \frac{\cos \frac{A-B}{2}}{\sin \frac{C}{2}} $;</p><p><strong>б)</strong> $ \frac{a-b}{c} = \frac{\sin \frac{A-B}{2}}{\cos \frac{C}{2}} $ (формулы немецкого математика К. Моллвейде (1774–1825)).</p>" "img" => array:1 [ 0 => array:5 [ "name" => "823-1.jpg" "alt" => null "width" => "1550" "height" => 489 "path" => "/media/algebra_09/soltan-u19/0-1/823-1.webp?ts=1752834807" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1135 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1278469 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136 #items: array:1 [ 0 => App\Models\Edition {#1137 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1138 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108569" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "823-1.jpg" "alt" => null "width" => "1088" "height" => 1442 "path" => "/media/algebra_09/soltan-u19/1-1/823-1.webp?ts=1752831631" ] 1 => array:5 [ "name" => "823-2.jpg" "alt" => null "width" => "1439" "height" => 2018 "path" => "/media/algebra_09/soltan-u19/1-1/823-2.webp?ts=1752831631" ] ] ] #original: array:6 [ "id" => 1278469 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136} "task" => array:2 [ "refs" => "1108569" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "823-1.jpg" "alt" => null "width" => "1088" "height" => 1442 "path" => "/media/algebra_09/soltan-u19/1-1/823-1.webp?ts=1752831631" ] 1 => array:5 [ "name" => "823-2.jpg" "alt" => null "width" => "1439" "height" => 2018 "path" => "/media/algebra_09/soltan-u19/1-1/823-2.webp?ts=1752831631" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1143 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1554424 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:1 [ 0 => App\Models\Edition {#1145 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1146 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1148 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1150 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1146 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1148 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1150 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108569" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Для доказательства равенства $ \frac{a+b}{c} = \frac{\cos\frac{A-B}{2}}{\sin\frac{C}{2}} $ воспользуемся теоремой синусов.</p><p>Согласно теореме синусов, для любого треугольника со сторонами $a, b, c$ и противолежащими им углами $A, B, C$ верно соотношение $ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k $, где $k$ – постоянная величина (равная диаметру описанной окружности). Отсюда выразим стороны через синусы углов: $ a = k \sin A, b = k \sin B, c = k \sin C $.</p><p>Преобразуем левую часть доказываемого равенства, подставив в нее полученные выражения:<br>$ \frac{a+b}{c} = \frac{k \sin A + k \sin B}{k \sin C} = \frac{\sin A + \sin B}{\sin C} $.</p><p>Далее применим тригонометрические формулы преобразования:<br>1. Для числителя используем формулу суммы синусов: $ \sin A + \sin B = 2 \sin\frac{A+B}{2} \cos\frac{A-B}{2} $.<br>2. Для знаменателя используем формулу синуса двойного угла: $ \sin C = 2 \sin\frac{C}{2} \cos\frac{C}{2} $.</p><p>После подстановки выражение примет вид:<br>$ \frac{2 \sin\frac{A+B}{2} \cos\frac{A-B}{2}}{2 \sin\frac{C}{2} \cos\frac{C}{2}} = \frac{\sin\frac{A+B}{2} \cos\frac{A-B}{2}}{\sin\frac{C}{2} \cos\frac{C}{2}} $.</p><p>Сумма углов в треугольнике равна $ \pi $, то есть $ A+B+C = \pi $. Отсюда следует, что $ A+B = \pi - C $, и, разделив на 2, получим $ \frac{A+B}{2} = \frac{\pi}{2} - \frac{C}{2} $.<br>Применяя формулу приведения, находим: $ \sin\frac{A+B}{2} = \sin(\frac{\pi}{2} - \frac{C}{2}) = \cos\frac{C}{2} $.</p><p>Подставим это в наше выражение:<br>$ \frac{\cos\frac{C}{2} \cos\frac{A-B}{2}}{\sin\frac{C}{2} \cos\frac{C}{2}} $.</p><p>Сократив общий множитель $ \cos\frac{C}{2} $ (что допустимо, так как для любого угла треугольника $C$ выполняется $ 0 < C < \pi $, а значит $ 0 < \frac{C}{2} < \frac{\pi}{2} $ и $ \cos\frac{C}{2} \neq 0 $), мы получаем правую часть исходного равенства:<br>$ \frac{\cos\frac{A-B}{2}}{\sin\frac{C}{2}} $.<br>Таким образом, равенство доказано.</p><p><strong>Ответ:</strong> Равенство $ \frac{a+b}{c} = \frac{\cos\frac{A-B}{2}}{\sin\frac{C}{2}} $ доказано.</p><p><strong>б)</strong></p><p>Для доказательства равенства $ \frac{a-b}{c} = \frac{\sin\frac{A-B}{2}}{\cos\frac{C}{2}} $ будем действовать аналогично предыдущему пункту.</p><p>Используя выражения для сторон из теоремы синусов ($ a = k \sin A, b = k \sin B, c = k \sin C $), преобразуем левую часть равенства:<br>$ \frac{a-b}{c} = \frac{k \sin A - k \sin B}{k \sin C} = \frac{\sin A - \sin B}{\sin C} $.</p><p>Применим тригонометрические формулы:<br>1. Для числителя используем формулу разности синусов: $ \sin A - \sin B = 2 \sin\frac{A-B}{2} \cos\frac{A+B}{2} $.<br>2. Для знаменателя используем формулу синуса двойного угла: $ \sin C = 2 \sin\frac{C}{2} \cos\frac{C}{2} $.</p><p>После подстановки получим:<br>$ \frac{2 \sin\frac{A-B}{2} \cos\frac{A+B}{2}}{2 \sin\frac{C}{2} \cos\frac{C}{2}} = \frac{\sin\frac{A-B}{2} \cos\frac{A+B}{2}}{\sin\frac{C}{2} \cos\frac{C}{2}} $.</p><p>Как и ранее, используем соотношение $ \frac{A+B}{2} = \frac{\pi}{2} - \frac{C}{2} $.<br>Применим формулу приведения: $ \cos\frac{A+B}{2} = \cos(\frac{\pi}{2} - \frac{C}{2}) = \sin\frac{C}{2} $.</p><p>Подставим полученный результат в наше выражение:<br>$ \frac{\sin\frac{A-B}{2} \sin\frac{C}{2}}{\sin\frac{C}{2} \cos\frac{C}{2}} $.</p><p>Сократив общий множитель $ \sin\frac{C}{2} $ (что допустимо, так как $ 0 < \frac{C}{2} < \frac{\pi}{2} $ и $ \sin\frac{C}{2} \neq 0 $), мы получаем правую часть исходного равенства:<br>$ \frac{\sin\frac{A-B}{2}}{\cos\frac{C}{2}} $.<br>Таким образом, равенство доказано.</p><p><strong>Ответ:</strong> Равенство $ \frac{a-b}{c} = \frac{\sin\frac{A-B}{2}}{\cos\frac{C}{2}} $ доказано.</p>" ] #original: array:6 [ "id" => 1554424 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144} "task" => array:2 [ "refs" => "1108569" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Для доказательства равенства $ \frac{a+b}{c} = \frac{\cos\frac{A-B}{2}}{\sin\frac{C}{2}} $ воспользуемся теоремой синусов.</p><p>Согласно теореме синусов, для любого треугольника со сторонами $a, b, c$ и противолежащими им углами $A, B, C$ верно соотношение $ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k $, где $k$ – постоянная величина (равная диаметру описанной окружности). Отсюда выразим стороны через синусы углов: $ a = k \sin A, b = k \sin B, c = k \sin C $.</p><p>Преобразуем левую часть доказываемого равенства, подставив в нее полученные выражения:<br>$ \frac{a+b}{c} = \frac{k \sin A + k \sin B}{k \sin C} = \frac{\sin A + \sin B}{\sin C} $.</p><p>Далее применим тригонометрические формулы преобразования:<br>1. Для числителя используем формулу суммы синусов: $ \sin A + \sin B = 2 \sin\frac{A+B}{2} \cos\frac{A-B}{2} $.<br>2. Для знаменателя используем формулу синуса двойного угла: $ \sin C = 2 \sin\frac{C}{2} \cos\frac{C}{2} $.</p><p>После подстановки выражение примет вид:<br>$ \frac{2 \sin\frac{A+B}{2} \cos\frac{A-B}{2}}{2 \sin\frac{C}{2} \cos\frac{C}{2}} = \frac{\sin\frac{A+B}{2} \cos\frac{A-B}{2}}{\sin\frac{C}{2} \cos\frac{C}{2}} $.</p><p>Сумма углов в треугольнике равна $ \pi $, то есть $ A+B+C = \pi $. Отсюда следует, что $ A+B = \pi - C $, и, разделив на 2, получим $ \frac{A+B}{2} = \frac{\pi}{2} - \frac{C}{2} $.<br>Применяя формулу приведения, находим: $ \sin\frac{A+B}{2} = \sin(\frac{\pi}{2} - \frac{C}{2}) = \cos\frac{C}{2} $.</p><p>Подставим это в наше выражение:<br>$ \frac{\cos\frac{C}{2} \cos\frac{A-B}{2}}{\sin\frac{C}{2} \cos\frac{C}{2}} $.</p><p>Сократив общий множитель $ \cos\frac{C}{2} $ (что допустимо, так как для любого угла треугольника $C$ выполняется $ 0 < C < \pi $, а значит $ 0 < \frac{C}{2} < \frac{\pi}{2} $ и $ \cos\frac{C}{2} \neq 0 $), мы получаем правую часть исходного равенства:<br>$ \frac{\cos\frac{A-B}{2}}{\sin\frac{C}{2}} $.<br>Таким образом, равенство доказано.</p><p><strong>Ответ:</strong> Равенство $ \frac{a+b}{c} = \frac{\cos\frac{A-B}{2}}{\sin\frac{C}{2}} $ доказано.</p><p><strong>б)</strong></p><p>Для доказательства равенства $ \frac{a-b}{c} = \frac{\sin\frac{A-B}{2}}{\cos\frac{C}{2}} $ будем действовать аналогично предыдущему пункту.</p><p>Используя выражения для сторон из теоремы синусов ($ a = k \sin A, b = k \sin B, c = k \sin C $), преобразуем левую часть равенства:<br>$ \frac{a-b}{c} = \frac{k \sin A - k \sin B}{k \sin C} = \frac{\sin A - \sin B}{\sin C} $.</p><p>Применим тригонометрические формулы:<br>1. Для числителя используем формулу разности синусов: $ \sin A - \sin B = 2 \sin\frac{A-B}{2} \cos\frac{A+B}{2} $.<br>2. Для знаменателя используем формулу синуса двойного угла: $ \sin C = 2 \sin\frac{C}{2} \cos\frac{C}{2} $.</p><p>После подстановки получим:<br>$ \frac{2 \sin\frac{A-B}{2} \cos\frac{A+B}{2}}{2 \sin\frac{C}{2} \cos\frac{C}{2}} = \frac{\sin\frac{A-B}{2} \cos\frac{A+B}{2}}{\sin\frac{C}{2} \cos\frac{C}{2}} $.</p><p>Как и ранее, используем соотношение $ \frac{A+B}{2} = \frac{\pi}{2} - \frac{C}{2} $.<br>Применим формулу приведения: $ \cos\frac{A+B}{2} = \cos(\frac{\pi}{2} - \frac{C}{2}) = \sin\frac{C}{2} $.</p><p>Подставим полученный результат в наше выражение:<br>$ \frac{\sin\frac{A-B}{2} \sin\frac{C}{2}}{\sin\frac{C}{2} \cos\frac{C}{2}} $.</p><p>Сократив общий множитель $ \sin\frac{C}{2} $ (что допустимо, так как $ 0 < \frac{C}{2} < \frac{\pi}{2} $ и $ \sin\frac{C}{2} \neq 0 $), мы получаем правую часть исходного равенства:<br>$ \frac{\sin\frac{A-B}{2}}{\cos\frac{C}{2}} $.<br>Таким образом, равенство доказано.</p><p><strong>Ответ:</strong> Равенство $ \frac{a-b}{c} = \frac{\sin\frac{A-B}{2}}{\cos\frac{C}{2}} $ доказано.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108570" "type" => "task" ] "previous" => array:2 [ "refs" => "1108568" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1151 #items: array:1 [ 0 => App\Models\Book {#1152 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1153 #items: array:1 [ 0 => App\Models\Term {#1154 #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 {#1155 #items: array:1 [ 0 => App\Models\Term {#1156 #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 {#1157 #items: array:1 [ 0 => App\Models\Term {#1158 #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 {#1159 #items: array:3 [ 0 => App\Models\Term {#1160 #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 {#1161 #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 {#1162 #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 {#1163 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1164 #items: array:1 [ 0 => App\Models\Term {#1165 #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 {#1166 #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: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 {#1168 #items: array:1 [ 0 => App\Models\Term {#1169 #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 {#1170 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1171 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1172 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1173 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1174 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1175 #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 {#1176 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1177 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1178 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1179 #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 {#1153} "field_class" => Illuminate\Database\Eloquent\Collection {#1155} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1157} "field_author" => Illuminate\Database\Eloquent\Collection {#1159} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1163} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1164} "field_country" => Illuminate\Database\Eloquent\Collection {#1166} "field_city" => Illuminate\Database\Eloquent\Collection {#1168} "field_series" => Illuminate\Database\Eloquent\Collection {#1170} "field_umk" => Illuminate\Database\Eloquent\Collection {#1171} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1172} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1173} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1174} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1175} "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 {#1176} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1177} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1178} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1179} "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 {#1194 #items: array:1 [ 0 => App\Models\BookPage {#1193 #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" => 1097968 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-226" "field_display_title" => "226" "field_folder" => "folder1" "field_image_name" => "226" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1195 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:1 [ 0 => App\Models\Book {#1196 #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 {#1198 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1201 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1203 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1216 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1217 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1218 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1219 …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 {#1220 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1221 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1222 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1223 …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 {#1198 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1201 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1203 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1216 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1217 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1218 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1219 …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 {#1220 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1221 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1222 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1223 …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 {#1224 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:1 [ 0 => App\Models\Element {#1234 #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" => 1261119 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1244 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261119 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1244 …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" => "1097969" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097967" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1705 #items: array:7 [ 0 => App\Models\Task {#1719 #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" => 1108565 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "225" "field_page_end" => "226" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-819" "field_display_title" => "819" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1723 …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 {#1722 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1757 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1852 …2} "content" => Illuminate\Database\Eloquent\Collection {#1870 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1877 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108565 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "225" "field_page_end" => "226" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-819" "field_display_title" => "819" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1723 …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 {#1722 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1757 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1852 …2} "content" => Illuminate\Database\Eloquent\Collection {#1870 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1877 …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 {#1861 #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" => 1108566 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-820" "field_display_title" => "820" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1862 …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 {#1863 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1868 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1867 …2} "content" => Illuminate\Database\Eloquent\Collection {#1875 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1903 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108566 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-820" "field_display_title" => "820" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1862 …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 {#1863 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1868 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1867 …2} "content" => Illuminate\Database\Eloquent\Collection {#1875 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1903 …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 {#1858 #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" => 1108567 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-821" "field_display_title" => "821" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1856 …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 {#1857 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1874 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1878 …2} "content" => Illuminate\Database\Eloquent\Collection {#1901 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1917 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108567 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-821" "field_display_title" => "821" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1856 …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 {#1857 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1874 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1878 …2} "content" => Illuminate\Database\Eloquent\Collection {#1901 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1917 …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 {#1853 #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" => 1108568 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-822" "field_display_title" => "822" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1854 …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 {#1855 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1900 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1904 …2} "content" => Illuminate\Database\Eloquent\Collection {#1915 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1931 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108568 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-822" "field_display_title" => "822" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1854 …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 {#1855 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1900 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1904 …2} "content" => Illuminate\Database\Eloquent\Collection {#1915 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1931 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1864 #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" => 1108569 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-823" "field_display_title" => "823" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1866 …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 {#1865 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1914 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1918 …2} "content" => Illuminate\Database\Eloquent\Collection {#1929 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1945 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108569 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-823" "field_display_title" => "823" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1866 …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 {#1865 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1914 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1918 …2} "content" => Illuminate\Database\Eloquent\Collection {#1929 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1945 …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 => "*" ] } 5 => App\Models\Task {#1871 #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" => 1108570 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-824" "field_display_title" => "824" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1876 …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 {#1869 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1928 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1932 …2} "content" => Illuminate\Database\Eloquent\Collection {#1943 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1959 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108570 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-824" "field_display_title" => "824" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1876 …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 {#1869 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1928 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1932 …2} "content" => Illuminate\Database\Eloquent\Collection {#1943 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1959 …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 => "*" ] } 6 => App\Models\Task {#1873 #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" => 1108571 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-825" "field_display_title" => "825" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1902 …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 {#1872 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1942 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1946 …2} "content" => Illuminate\Database\Eloquent\Collection {#1957 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1973 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108571 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_page_end" => null …19 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1097968 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-226" "field_display_title" => "226" "field_folder" => "folder1" "field_image_name" => "226" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1195} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1226} "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 {#1224} "content" => Illuminate\Database\Eloquent\Collection {#1235} "next" => array:2 [ "refs" => "1097969" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097967" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1705} ] #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" => 1108569 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "226" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-823" "field_display_title" => "823" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1061} "content" => Illuminate\Database\Eloquent\Collection {#1126} "next" => array:2 [ "refs" => "1108570" "type" => "task" ] "previous" => array:2 [ "refs" => "1108568" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1151} "page" => array:2 [ "refs" => "1097968" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№823 (с. 226)
Условие. №823 (с. 226)
скриншот условия
823. Докажите, что для треугольника ABC со сторонами a, b, c верно равенство:
a) $ \frac{a+b}{c} = \frac{\cos \frac{A-B}{2}}{\sin \frac{C}{2}} $;
б) $ \frac{a-b}{c} = \frac{\sin \frac{A-B}{2}}{\cos \frac{C}{2}} $ (формулы немецкого математика К. Моллвейде (1774–1825)).
Решение. №823 (с. 226)
Решение 2 (rus). №823 (с. 226)
а)
Для доказательства равенства $ \frac{a+b}{c} = \frac{\cos\frac{A-B}{2}}{\sin\frac{C}{2}} $ воспользуемся теоремой синусов.
Согласно теореме синусов, для любого треугольника со сторонами $a, b, c$ и противолежащими им углами $A, B, C$ верно соотношение $ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k $, где $k$ – постоянная величина (равная диаметру описанной окружности). Отсюда выразим стороны через синусы углов: $ a = k \sin A, b = k \sin B, c = k \sin C $.
Преобразуем левую часть доказываемого равенства, подставив в нее полученные выражения:
$ \frac{a+b}{c} = \frac{k \sin A + k \sin B}{k \sin C} = \frac{\sin A + \sin B}{\sin C} $.
Далее применим тригонометрические формулы преобразования:
1. Для числителя используем формулу суммы синусов: $ \sin A + \sin B = 2 \sin\frac{A+B}{2} \cos\frac{A-B}{2} $.
2. Для знаменателя используем формулу синуса двойного угла: $ \sin C = 2 \sin\frac{C}{2} \cos\frac{C}{2} $.
После подстановки выражение примет вид:
$ \frac{2 \sin\frac{A+B}{2} \cos\frac{A-B}{2}}{2 \sin\frac{C}{2} \cos\frac{C}{2}} = \frac{\sin\frac{A+B}{2} \cos\frac{A-B}{2}}{\sin\frac{C}{2} \cos\frac{C}{2}} $.
Сумма углов в треугольнике равна $ \pi $, то есть $ A+B+C = \pi $. Отсюда следует, что $ A+B = \pi - C $, и, разделив на 2, получим $ \frac{A+B}{2} = \frac{\pi}{2} - \frac{C}{2} $.
Применяя формулу приведения, находим: $ \sin\frac{A+B}{2} = \sin(\frac{\pi}{2} - \frac{C}{2}) = \cos\frac{C}{2} $.
Подставим это в наше выражение:
$ \frac{\cos\frac{C}{2} \cos\frac{A-B}{2}}{\sin\frac{C}{2} \cos\frac{C}{2}} $.
Сократив общий множитель $ \cos\frac{C}{2} $ (что допустимо, так как для любого угла треугольника $C$ выполняется $ 0 < C < \pi $, а значит $ 0 < \frac{C}{2} < \frac{\pi}{2} $ и $ \cos\frac{C}{2} \neq 0 $), мы получаем правую часть исходного равенства:
$ \frac{\cos\frac{A-B}{2}}{\sin\frac{C}{2}} $.
Таким образом, равенство доказано.
Ответ: Равенство $ \frac{a+b}{c} = \frac{\cos\frac{A-B}{2}}{\sin\frac{C}{2}} $ доказано.
б)
Для доказательства равенства $ \frac{a-b}{c} = \frac{\sin\frac{A-B}{2}}{\cos\frac{C}{2}} $ будем действовать аналогично предыдущему пункту.
Используя выражения для сторон из теоремы синусов ($ a = k \sin A, b = k \sin B, c = k \sin C $), преобразуем левую часть равенства:
$ \frac{a-b}{c} = \frac{k \sin A - k \sin B}{k \sin C} = \frac{\sin A - \sin B}{\sin C} $.
Применим тригонометрические формулы:
1. Для числителя используем формулу разности синусов: $ \sin A - \sin B = 2 \sin\frac{A-B}{2} \cos\frac{A+B}{2} $.
2. Для знаменателя используем формулу синуса двойного угла: $ \sin C = 2 \sin\frac{C}{2} \cos\frac{C}{2} $.
После подстановки получим:
$ \frac{2 \sin\frac{A-B}{2} \cos\frac{A+B}{2}}{2 \sin\frac{C}{2} \cos\frac{C}{2}} = \frac{\sin\frac{A-B}{2} \cos\frac{A+B}{2}}{\sin\frac{C}{2} \cos\frac{C}{2}} $.
Как и ранее, используем соотношение $ \frac{A+B}{2} = \frac{\pi}{2} - \frac{C}{2} $.
Применим формулу приведения: $ \cos\frac{A+B}{2} = \cos(\frac{\pi}{2} - \frac{C}{2}) = \sin\frac{C}{2} $.
Подставим полученный результат в наше выражение:
$ \frac{\sin\frac{A-B}{2} \sin\frac{C}{2}}{\sin\frac{C}{2} \cos\frac{C}{2}} $.
Сократив общий множитель $ \sin\frac{C}{2} $ (что допустимо, так как $ 0 < \frac{C}{2} < \frac{\pi}{2} $ и $ \sin\frac{C}{2} \neq 0 $), мы получаем правую часть исходного равенства:
$ \frac{\sin\frac{A-B}{2}}{\cos\frac{C}{2}} $.
Таким образом, равенство доказано.
Ответ: Равенство $ \frac{a-b}{c} = \frac{\sin\frac{A-B}{2}}{\cos\frac{C}{2}} $ доказано.
Другие задания:
Помогло решение? Оставьте отзыв в комментариях ниже.
Присоединяйтесь к Телеграм-группе @top_gdz
ПрисоединитьсяМы подготовили для вас ответ c подробным объяснением домашего задания по алгебре за 9 класс, для упражнения номер 823 расположенного на странице 226 к учебнику 2019 года издания для учащихся школ и гимназий.
Теперь на нашем сайте ГДЗ.ТОП вы всегда легко и бесплатно найдёте условие с правильным ответом на вопрос «Как решить ДЗ» и «Как сделать» задание по алгебре к упражнению №823 (с. 226), авторов: Солтан (Генадий Николаевич), Солтан (Алла Евгеньевна), Жумадилова (Аманбала Жумадиловна), учебного пособия издательства Кокшетау.