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Номер 689, страница 199 - 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. 26. Формулы косинуса, синуса, тангенса и котангенса суммы и разности двух углов - номер 689, страница 199.
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" => 1108427 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-689" "field_display_title" => "689" "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" => 1107668 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "26. Формулы косинуса, синуса, тангенса и котангенса суммы и разности двух углов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "196" "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" => 1107668 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "26. Формулы косинуса, синуса, тангенса и котангенса суммы и разности двух углов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "196" "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" => 1277044 "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" => "1108427" "type" => "task" ] "text" => "<p><strong>689.</strong> Дано: $cos \alpha = \frac{1}{\sqrt{2}}$, $cos \beta = \frac{1}{3}$, $\alpha$ и $\beta$ – углы IV четверти. Вычислите:</p><p><strong>а)</strong> $sin(\alpha - \beta);$</p><p><strong>б)</strong> $cos(\alpha + \beta);$</p><p><strong>в)</strong> $tg(\alpha + \beta).$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "689-1.jpg" "alt" => null "width" => "1549" "height" => 269 "path" => "/media/algebra_09/soltan-u19/0-1/689-1.webp?ts=1752834689" ] ] ] #original: array:7 [ "id" => 1277044 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128} "task" => array:2 [ "refs" => "1108427" "type" => "task" ] "text" => "<p><strong>689.</strong> Дано: $cos \alpha = \frac{1}{\sqrt{2}}$, $cos \beta = \frac{1}{3}$, $\alpha$ и $\beta$ – углы IV четверти. Вычислите:</p><p><strong>а)</strong> $sin(\alpha - \beta);$</p><p><strong>б)</strong> $cos(\alpha + \beta);$</p><p><strong>в)</strong> $tg(\alpha + \beta).$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "689-1.jpg" "alt" => null "width" => "1549" "height" => 269 "path" => "/media/algebra_09/soltan-u19/0-1/689-1.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 {#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" => 1278335 "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" => "1108427" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "689-1.jpg" "alt" => null "width" => "1252" "height" => 1094 "path" => "/media/algebra_09/soltan-u19/1-1/689-1.webp?ts=1752831477" ] 1 => array:5 [ "name" => "689-2.jpg" "alt" => null "width" => "1270" "height" => 1560 "path" => "/media/algebra_09/soltan-u19/1-1/689-2.webp?ts=1752831477" ] ] ] #original: array:6 [ "id" => 1278335 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136} "task" => array:2 [ "refs" => "1108427" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "689-1.jpg" "alt" => null "width" => "1252" "height" => 1094 "path" => "/media/algebra_09/soltan-u19/1-1/689-1.webp?ts=1752831477" ] 1 => array:5 [ "name" => "689-2.jpg" "alt" => null "width" => "1270" "height" => 1560 "path" => "/media/algebra_09/soltan-u19/1-1/689-2.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 {#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" => 1554290 "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" => "1108427" "type" => "task" ] "text" => "<p>Для решения задачи нам понадобятся значения синусов углов $\alpha$ и $\beta$.</p><p>Поскольку углы $\alpha$ и $\beta$ принадлежат IV четверти, их косинусы положительны (что соответствует условию), а синусы отрицательны.</p><p>Используем основное тригонометрическое тождество $sin^2(x) + cos^2(x) = 1$, откуда $sin(x) = \pm\sqrt{1 - cos^2(x)}$.</p><p><strong>1.</strong> Найдем $sin(\alpha)$:</p><p>$sin(\alpha) = -\sqrt{1 - cos^2(\alpha)} = -\sqrt{1 - (\frac{1}{\sqrt{2}})^2} = -\sqrt{1 - \frac{1}{2}} = -\sqrt{\frac{1}{2}} = -\frac{1}{\sqrt{2}}$</p><p><strong>2.</strong> Найдем $sin(\beta)$:</p><p>$sin(\beta) = -\sqrt{1 - cos^2(\beta)} = -\sqrt{1 - (\frac{1}{3})^2} = -\sqrt{1 - \frac{1}{9}} = -\sqrt{\frac{8}{9}} = -\frac{\sqrt{8}}{3} = -\frac{2\sqrt{2}}{3}$</p><p>Теперь можем вычислить требуемые значения.</p><p><strong>а) $sin(\alpha - \beta)$</strong></p><p>Используем формулу синуса разности: $sin(\alpha - \beta) = sin(\alpha)cos(\beta) - cos(\alpha)sin(\beta)$.</p><p>Подставляем известные значения:</p><p>$sin(\alpha - \beta) = (-\frac{1}{\sqrt{2}}) \cdot \frac{1}{3} - \frac{1}{\sqrt{2}} \cdot (-\frac{2\sqrt{2}}{3}) = -\frac{1}{3\sqrt{2}} + \frac{2\sqrt{2}}{3\sqrt{2}} = \frac{2\sqrt{2} - 1}{3\sqrt{2}}$</p><p>Избавимся от иррациональности в знаменателе:</p><p>$\frac{2\sqrt{2} - 1}{3\sqrt{2}} = \frac{(2\sqrt{2} - 1) \cdot \sqrt{2}}{3\sqrt{2} \cdot \sqrt{2}} = \frac{2\sqrt{2} \cdot \sqrt{2} - 1 \cdot \sqrt{2}}{3 \cdot 2} = \frac{4 - \sqrt{2}}{6}$</p><p><strong>Ответ:</strong> $\frac{4 - \sqrt{2}}{6}$</p><p><strong>б) $cos(\alpha + \beta)$</strong></p><p>Используем формулу косинуса суммы: $cos(\alpha + \beta) = cos(\alpha)cos(\beta) - sin(\alpha)sin(\beta)$.</p><p>Подставляем известные значения:</p><p>$cos(\alpha + \beta) = \frac{1}{\sqrt{2}} \cdot \frac{1}{3} - (-\frac{1}{\sqrt{2}}) \cdot (-\frac{2\sqrt{2}}{3}) = \frac{1}{3\sqrt{2}} - \frac{2\sqrt{2}}{3\sqrt{2}} = \frac{1 - 2\sqrt{2}}{3\sqrt{2}}$</p><p>Избавимся от иррациональности в знаменателе:</p><p>$\frac{1 - 2\sqrt{2}}{3\sqrt{2}} = \frac{(1 - 2\sqrt{2}) \cdot \sqrt{2}}{3\sqrt{2} \cdot \sqrt{2}} = \frac{1 \cdot \sqrt{2} - 2\sqrt{2} \cdot \sqrt{2}}{3 \cdot 2} = \frac{\sqrt{2} - 4}{6}$</p><p><strong>Ответ:</strong> $\frac{\sqrt{2} - 4}{6}$</p><p><strong>в) $tg(\alpha + \beta)$</strong></p><p>Для нахождения тангенса суммы воспользуемся формулой $tg(\alpha + \beta) = \frac{tg(\alpha) + tg(\beta)}{1 - tg(\alpha)tg(\beta)}$. Сначала найдем $tg(\alpha)$ и $tg(\beta)$.</p><p>$tg(\alpha) = \frac{sin(\alpha)}{cos(\alpha)} = \frac{-1/\sqrt{2}}{1/\sqrt{2}} = -1$</p><p>$tg(\beta) = \frac{sin(\beta)}{cos(\beta)} = \frac{-2\sqrt{2}/3}{1/3} = -2\sqrt{2}$</p><p>Теперь подставляем в формулу тангенса суммы:</p><p>$tg(\alpha + \beta) = \frac{-1 + (-2\sqrt{2})}{1 - (-1)(-2\sqrt{2})} = \frac{-1 - 2\sqrt{2}}{1 - 2\sqrt{2}}$</p><p>Умножим числитель и знаменатель на сопряженное знаменателю выражение $(1 + 2\sqrt{2})$:</p><p>$\frac{(-1 - 2\sqrt{2})(1 + 2\sqrt{2})}{(1 - 2\sqrt{2})(1 + 2\sqrt{2})} = \frac{-(1 + 2\sqrt{2})^2}{1^2 - (2\sqrt{2})^2} = \frac{-(1 + 4\sqrt{2} + 8)}{1 - 8} = \frac{-(9 + 4\sqrt{2})}{-7} = \frac{9 + 4\sqrt{2}}{7}$</p><p><strong>Ответ:</strong> $\frac{9 + 4\sqrt{2}}{7}$</p>" ] #original: array:6 [ "id" => 1554290 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144} "task" => array:2 [ "refs" => "1108427" "type" => "task" ] "text" => "<p>Для решения задачи нам понадобятся значения синусов углов $\alpha$ и $\beta$.</p><p>Поскольку углы $\alpha$ и $\beta$ принадлежат IV четверти, их косинусы положительны (что соответствует условию), а синусы отрицательны.</p><p>Используем основное тригонометрическое тождество $sin^2(x) + cos^2(x) = 1$, откуда $sin(x) = \pm\sqrt{1 - cos^2(x)}$.</p><p><strong>1.</strong> Найдем $sin(\alpha)$:</p><p>$sin(\alpha) = -\sqrt{1 - cos^2(\alpha)} = -\sqrt{1 - (\frac{1}{\sqrt{2}})^2} = -\sqrt{1 - \frac{1}{2}} = -\sqrt{\frac{1}{2}} = -\frac{1}{\sqrt{2}}$</p><p><strong>2.</strong> Найдем $sin(\beta)$:</p><p>$sin(\beta) = -\sqrt{1 - cos^2(\beta)} = -\sqrt{1 - (\frac{1}{3})^2} = -\sqrt{1 - \frac{1}{9}} = -\sqrt{\frac{8}{9}} = -\frac{\sqrt{8}}{3} = -\frac{2\sqrt{2}}{3}$</p><p>Теперь можем вычислить требуемые значения.</p><p><strong>а) $sin(\alpha - \beta)$</strong></p><p>Используем формулу синуса разности: $sin(\alpha - \beta) = sin(\alpha)cos(\beta) - cos(\alpha)sin(\beta)$.</p><p>Подставляем известные значения:</p><p>$sin(\alpha - \beta) = (-\frac{1}{\sqrt{2}}) \cdot \frac{1}{3} - \frac{1}{\sqrt{2}} \cdot (-\frac{2\sqrt{2}}{3}) = -\frac{1}{3\sqrt{2}} + \frac{2\sqrt{2}}{3\sqrt{2}} = \frac{2\sqrt{2} - 1}{3\sqrt{2}}$</p><p>Избавимся от иррациональности в знаменателе:</p><p>$\frac{2\sqrt{2} - 1}{3\sqrt{2}} = \frac{(2\sqrt{2} - 1) \cdot \sqrt{2}}{3\sqrt{2} \cdot \sqrt{2}} = \frac{2\sqrt{2} \cdot \sqrt{2} - 1 \cdot \sqrt{2}}{3 \cdot 2} = \frac{4 - \sqrt{2}}{6}$</p><p><strong>Ответ:</strong> $\frac{4 - \sqrt{2}}{6}$</p><p><strong>б) $cos(\alpha + \beta)$</strong></p><p>Используем формулу косинуса суммы: $cos(\alpha + \beta) = cos(\alpha)cos(\beta) - sin(\alpha)sin(\beta)$.</p><p>Подставляем известные значения:</p><p>$cos(\alpha + \beta) = \frac{1}{\sqrt{2}} \cdot \frac{1}{3} - (-\frac{1}{\sqrt{2}}) \cdot (-\frac{2\sqrt{2}}{3}) = \frac{1}{3\sqrt{2}} - \frac{2\sqrt{2}}{3\sqrt{2}} = \frac{1 - 2\sqrt{2}}{3\sqrt{2}}$</p><p>Избавимся от иррациональности в знаменателе:</p><p>$\frac{1 - 2\sqrt{2}}{3\sqrt{2}} = \frac{(1 - 2\sqrt{2}) \cdot \sqrt{2}}{3\sqrt{2} \cdot \sqrt{2}} = \frac{1 \cdot \sqrt{2} - 2\sqrt{2} \cdot \sqrt{2}}{3 \cdot 2} = \frac{\sqrt{2} - 4}{6}$</p><p><strong>Ответ:</strong> $\frac{\sqrt{2} - 4}{6}$</p><p><strong>в) $tg(\alpha + \beta)$</strong></p><p>Для нахождения тангенса суммы воспользуемся формулой $tg(\alpha + \beta) = \frac{tg(\alpha) + tg(\beta)}{1 - tg(\alpha)tg(\beta)}$. Сначала найдем $tg(\alpha)$ и $tg(\beta)$.</p><p>$tg(\alpha) = \frac{sin(\alpha)}{cos(\alpha)} = \frac{-1/\sqrt{2}}{1/\sqrt{2}} = -1$</p><p>$tg(\beta) = \frac{sin(\beta)}{cos(\beta)} = \frac{-2\sqrt{2}/3}{1/3} = -2\sqrt{2}$</p><p>Теперь подставляем в формулу тангенса суммы:</p><p>$tg(\alpha + \beta) = \frac{-1 + (-2\sqrt{2})}{1 - (-1)(-2\sqrt{2})} = \frac{-1 - 2\sqrt{2}}{1 - 2\sqrt{2}}$</p><p>Умножим числитель и знаменатель на сопряженное знаменателю выражение $(1 + 2\sqrt{2})$:</p><p>$\frac{(-1 - 2\sqrt{2})(1 + 2\sqrt{2})}{(1 - 2\sqrt{2})(1 + 2\sqrt{2})} = \frac{-(1 + 2\sqrt{2})^2}{1^2 - (2\sqrt{2})^2} = \frac{-(1 + 4\sqrt{2} + 8)}{1 - 8} = \frac{-(9 + 4\sqrt{2})}{-7} = \frac{9 + 4\sqrt{2}}{7}$</p><p><strong>Ответ:</strong> $\frac{9 + 4\sqrt{2}}{7}$</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] 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"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" => 1097941 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-199" "field_display_title" => "199" "field_folder" => "folder1" "field_image_name" => "199" "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" => 1261092 "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" => 1261092 "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" => "1097942" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097940" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#2516 #items: array:8 [ 0 => App\Models\Task {#1227 #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" => 1108421 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-683" "field_display_title" => "683" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1225 …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 {#1231 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1229 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1273 …2} "content" => Illuminate\Database\Eloquent\Collection {#1370 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1395 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108421 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-683" "field_display_title" => "683" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1225 …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 {#1231 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1229 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1273 …2} "content" => Illuminate\Database\Eloquent\Collection {#1370 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1395 …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 {#1424 #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" => 1108422 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-684" "field_display_title" => "684" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1425 …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 {#1427 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1428 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1461 …2} "content" => Illuminate\Database\Eloquent\Collection {#1526 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1551 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108422 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-684" "field_display_title" => "684" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1425 …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 {#1427 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1428 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1461 …2} "content" => Illuminate\Database\Eloquent\Collection {#1526 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1551 …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 {#1580 #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" => 1108423 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-685" "field_display_title" => "685" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1581 …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 {#1583 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1584 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1617 …2} "content" => Illuminate\Database\Eloquent\Collection {#1682 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1707 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108423 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-685" "field_display_title" => "685" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1581 …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 {#1583 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1584 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1617 …2} "content" => Illuminate\Database\Eloquent\Collection {#1682 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1707 …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 {#1736 #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" => 1108424 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-686" "field_display_title" => "686" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1737 …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 {#1739 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1740 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1773 …2} "content" => Illuminate\Database\Eloquent\Collection {#1838 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1863 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108424 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-686" "field_display_title" => "686" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1737 …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 {#1739 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1740 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1773 …2} "content" => Illuminate\Database\Eloquent\Collection {#1838 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1863 …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 {#1892 #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" => 1108425 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-687" "field_display_title" => "687" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1893 …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 {#1895 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1896 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1929 …2} "content" => Illuminate\Database\Eloquent\Collection {#1994 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2019 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108425 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "199" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-687" "field_display_title" => "687" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1893 …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 {#1895 …2} "top_parent_branch" => 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№689 (с. 199)
Условие. №689 (с. 199)
скриншот условия
689. Дано: $cos \alpha = \frac{1}{\sqrt{2}}$, $cos \beta = \frac{1}{3}$, $\alpha$ и $\beta$ – углы IV четверти. Вычислите:
а) $sin(\alpha - \beta);$
б) $cos(\alpha + \beta);$
в) $tg(\alpha + \beta).$
Решение. №689 (с. 199)
Решение 2 (rus). №689 (с. 199)
Для решения задачи нам понадобятся значения синусов углов $\alpha$ и $\beta$.
Поскольку углы $\alpha$ и $\beta$ принадлежат IV четверти, их косинусы положительны (что соответствует условию), а синусы отрицательны.
Используем основное тригонометрическое тождество $sin^2(x) + cos^2(x) = 1$, откуда $sin(x) = \pm\sqrt{1 - cos^2(x)}$.
1. Найдем $sin(\alpha)$:
$sin(\alpha) = -\sqrt{1 - cos^2(\alpha)} = -\sqrt{1 - (\frac{1}{\sqrt{2}})^2} = -\sqrt{1 - \frac{1}{2}} = -\sqrt{\frac{1}{2}} = -\frac{1}{\sqrt{2}}$
2. Найдем $sin(\beta)$:
$sin(\beta) = -\sqrt{1 - cos^2(\beta)} = -\sqrt{1 - (\frac{1}{3})^2} = -\sqrt{1 - \frac{1}{9}} = -\sqrt{\frac{8}{9}} = -\frac{\sqrt{8}}{3} = -\frac{2\sqrt{2}}{3}$
Теперь можем вычислить требуемые значения.
а) $sin(\alpha - \beta)$
Используем формулу синуса разности: $sin(\alpha - \beta) = sin(\alpha)cos(\beta) - cos(\alpha)sin(\beta)$.
Подставляем известные значения:
$sin(\alpha - \beta) = (-\frac{1}{\sqrt{2}}) \cdot \frac{1}{3} - \frac{1}{\sqrt{2}} \cdot (-\frac{2\sqrt{2}}{3}) = -\frac{1}{3\sqrt{2}} + \frac{2\sqrt{2}}{3\sqrt{2}} = \frac{2\sqrt{2} - 1}{3\sqrt{2}}$
Избавимся от иррациональности в знаменателе:
$\frac{2\sqrt{2} - 1}{3\sqrt{2}} = \frac{(2\sqrt{2} - 1) \cdot \sqrt{2}}{3\sqrt{2} \cdot \sqrt{2}} = \frac{2\sqrt{2} \cdot \sqrt{2} - 1 \cdot \sqrt{2}}{3 \cdot 2} = \frac{4 - \sqrt{2}}{6}$
Ответ: $\frac{4 - \sqrt{2}}{6}$
б) $cos(\alpha + \beta)$
Используем формулу косинуса суммы: $cos(\alpha + \beta) = cos(\alpha)cos(\beta) - sin(\alpha)sin(\beta)$.
Подставляем известные значения:
$cos(\alpha + \beta) = \frac{1}{\sqrt{2}} \cdot \frac{1}{3} - (-\frac{1}{\sqrt{2}}) \cdot (-\frac{2\sqrt{2}}{3}) = \frac{1}{3\sqrt{2}} - \frac{2\sqrt{2}}{3\sqrt{2}} = \frac{1 - 2\sqrt{2}}{3\sqrt{2}}$
Избавимся от иррациональности в знаменателе:
$\frac{1 - 2\sqrt{2}}{3\sqrt{2}} = \frac{(1 - 2\sqrt{2}) \cdot \sqrt{2}}{3\sqrt{2} \cdot \sqrt{2}} = \frac{1 \cdot \sqrt{2} - 2\sqrt{2} \cdot \sqrt{2}}{3 \cdot 2} = \frac{\sqrt{2} - 4}{6}$
Ответ: $\frac{\sqrt{2} - 4}{6}$
в) $tg(\alpha + \beta)$
Для нахождения тангенса суммы воспользуемся формулой $tg(\alpha + \beta) = \frac{tg(\alpha) + tg(\beta)}{1 - tg(\alpha)tg(\beta)}$. Сначала найдем $tg(\alpha)$ и $tg(\beta)$.
$tg(\alpha) = \frac{sin(\alpha)}{cos(\alpha)} = \frac{-1/\sqrt{2}}{1/\sqrt{2}} = -1$
$tg(\beta) = \frac{sin(\beta)}{cos(\beta)} = \frac{-2\sqrt{2}/3}{1/3} = -2\sqrt{2}$
Теперь подставляем в формулу тангенса суммы:
$tg(\alpha + \beta) = \frac{-1 + (-2\sqrt{2})}{1 - (-1)(-2\sqrt{2})} = \frac{-1 - 2\sqrt{2}}{1 - 2\sqrt{2}}$
Умножим числитель и знаменатель на сопряженное знаменателю выражение $(1 + 2\sqrt{2})$:
$\frac{(-1 - 2\sqrt{2})(1 + 2\sqrt{2})}{(1 - 2\sqrt{2})(1 + 2\sqrt{2})} = \frac{-(1 + 2\sqrt{2})^2}{1^2 - (2\sqrt{2})^2} = \frac{-(1 + 4\sqrt{2} + 8)}{1 - 8} = \frac{-(9 + 4\sqrt{2})}{-7} = \frac{9 + 4\sqrt{2}}{7}$
Ответ: $\frac{9 + 4\sqrt{2}}{7}$
Другие задания:
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