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Номер 634, страница 185 - 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. 24. Основные тригонометрические тождества - номер 634, страница 185.
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" => 1108370 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-634" "field_display_title" => "634" "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 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1035 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1050 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037} "field_author" => Illuminate\Database\Eloquent\Collection {#1039} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044} "field_country" => Illuminate\Database\Eloquent\Collection {#1046} "field_city" => Illuminate\Database\Eloquent\Collection {#1048} "field_series" => Illuminate\Database\Eloquent\Collection {#1050} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107666 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "24. Основные тригонометрические тождества" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "183" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Book {#1065 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1066} "field_class" => Illuminate\Database\Eloquent\Collection {#1068} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070} "field_author" => Illuminate\Database\Eloquent\Collection {#1072} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077} "field_country" => Illuminate\Database\Eloquent\Collection {#1079} "field_city" => Illuminate\Database\Eloquent\Collection {#1081} "field_series" => Illuminate\Database\Eloquent\Collection {#1083} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Branch {#1094 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 …2} ] #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 {#1097 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107666 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "24. Основные тригонометрические тождества" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "183" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1064} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:3 [ 0 => App\Models\Element {#1099 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1276989 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1101 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108370" "type" => "task" ] "text" => "<p><strong>634.</strong> Найдите значения остальных тригонометрических функций, если известно, что:</p><p><strong>a)</strong> $ \sin \alpha = 0,6 $, $ 0^\circ < \alpha < 90^\circ $;</p><p><strong>б)</strong> $ \cos \alpha = -\frac{1}{3} $, $ 90^\circ < \alpha < 180^\circ $;</p><p><strong>в)</strong> $ \text{tg} \alpha = 2 $, $ 180^\circ < \alpha < 270^\circ $;</p><p><strong>г)</strong> $ \text{ctg} \alpha = -3 $, $ 270^\circ < \alpha < 360^\circ $.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "634-1.jpg" "alt" => null "width" => "1549" "height" => 375 "path" => "/media/algebra_09/soltan-u19/0-1/634-1.webp?ts=1752834644" ] ] ] #original: array:7 [ "id" => 1276989 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1108370" "type" => "task" ] "text" => "<p><strong>634.</strong> Найдите значения остальных тригонометрических функций, если известно, что:</p><p><strong>a)</strong> $ \sin \alpha = 0,6 $, $ 0^\circ < \alpha < 90^\circ $;</p><p><strong>б)</strong> $ \cos \alpha = -\frac{1}{3} $, $ 90^\circ < \alpha < 180^\circ $;</p><p><strong>в)</strong> $ \text{tg} \alpha = 2 $, $ 180^\circ < \alpha < 270^\circ $;</p><p><strong>г)</strong> $ \text{ctg} \alpha = -3 $, $ 270^\circ < \alpha < 360^\circ $.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "634-1.jpg" "alt" => null "width" => "1549" "height" => 375 "path" => "/media/algebra_09/soltan-u19/0-1/634-1.webp?ts=1752834644" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1107 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1278280 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1109 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108370" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "634-1.jpg" "alt" => null "width" => "1427" "height" => 2369 "path" => "/media/algebra_09/soltan-u19/1-1/634-1.webp?ts=1752831420" ] ] ] #original: array:6 [ "id" => 1278280 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1108370" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "634-1.jpg" "alt" => null "width" => "1427" "height" => 2369 "path" => "/media/algebra_09/soltan-u19/1-1/634-1.webp?ts=1752831420" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1115 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1554235 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1117 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108370" "type" => "task" ] "text" => "<p><strong>а)</strong> Дано: $\sin \alpha = 0.6$ и $0^\circ < \alpha < 90^\circ$. Угол $\alpha$ находится в I четверти, где все тригонометрические функции ($\sin \alpha, \cos \alpha, \tan \alpha, \cot \alpha$) положительны.<br>1. Найдем $\cos \alpha$ из основного тригонометрического тождества $\sin^2 \alpha + \cos^2 \alpha = 1$.<br>$\cos^2 \alpha = 1 - \sin^2 \alpha = 1 - (0.6)^2 = 1 - 0.36 = 0.64$.<br>Так как угол $\alpha$ находится в I четверти, $\cos \alpha$ положителен: $\cos \alpha = \sqrt{0.64} = 0.8$.<br>2. Найдем $\tan \alpha$ по определению: $\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$.<br>$\tan \alpha = \frac{0.6}{0.8} = \frac{6}{8} = \frac{3}{4} = 0.75$.<br>3. Найдем $\cot \alpha$ по определению: $\cot \alpha = \frac{1}{\tan \alpha}$.<br>$\cot \alpha = \frac{1}{3/4} = \frac{4}{3}$.<br>Ответ: $\cos \alpha = 0.8$, $\tan \alpha = 0.75$, $\cot \alpha = \frac{4}{3}$.</p><p><strong>б)</strong> Дано: $\cos \alpha = -\frac{1}{3}$ и $90^\circ < \alpha < 180^\circ$. Угол $\alpha$ находится во II четверти, где синус положителен ($\sin \alpha > 0$), а тангенс и котангенс отрицательны ($\tan \alpha < 0$, $\cot \alpha < 0$).<br>1. Найдем $\sin \alpha$ из основного тригонометрического тождества $\sin^2 \alpha + \cos^2 \alpha = 1$.<br>$\sin^2 \alpha = 1 - \cos^2 \alpha = 1 - \left(-\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9}$.<br>Так как угол $\alpha$ находится во II четверти, $\sin \alpha$ положителен: $\sin \alpha = \sqrt{\frac{8}{9}} = \frac{\sqrt{8}}{3} = \frac{2\sqrt{2}}{3}$.<br>2. Найдем $\tan \alpha$ по определению: $\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$.<br>$\tan \alpha = \frac{2\sqrt{2}/3}{-1/3} = -2\sqrt{2}$.<br>3. Найдем $\cot \alpha$ по определению: $\cot \alpha = \frac{1}{\tan \alpha}$.<br>$\cot \alpha = \frac{1}{-2\sqrt{2}} = -\frac{1}{2\sqrt{2}} = -\frac{\sqrt{2}}{2\sqrt{2} \cdot \sqrt{2}} = -\frac{\sqrt{2}}{4}$.<br>Ответ: $\sin \alpha = \frac{2\sqrt{2}}{3}$, $\tan \alpha = -2\sqrt{2}$, $\cot \alpha = -\frac{\sqrt{2}}{4}$.</p><p><strong>в)</strong> Дано: $\tan \alpha = 2$ и $180^\circ < \alpha < 270^\circ$. Угол $\alpha$ находится в III четверти, где синус и косинус отрицательны ($\sin \alpha < 0, \cos \alpha < 0$), а котангенс положителен ($\cot \alpha > 0$).<br>1. Найдем $\cot \alpha$: $\cot \alpha = \frac{1}{\tan \alpha} = \frac{1}{2}$.<br>2. Найдем $\cos \alpha$ из тождества $1 + \tan^2 \alpha = \frac{1}{\cos^2 \alpha}$.<br>$\cos^2 \alpha = \frac{1}{1 + \tan^2 \alpha} = \frac{1}{1 + 2^2} = \frac{1}{5}$.<br>Так как угол $\alpha$ находится в III четверти, $\cos \alpha$ отрицателен: $\cos \alpha = -\sqrt{\frac{1}{5}} = -\frac{1}{\sqrt{5}} = -\frac{\sqrt{5}}{5}$.<br>3. Найдем $\sin \alpha$ из соотношения $\sin \alpha = \tan \alpha \cdot \cos \alpha$.<br>$\sin \alpha = 2 \cdot \left(-\frac{\sqrt{5}}{5}\right) = -\frac{2\sqrt{5}}{5}$.<br>Ответ: $\sin \alpha = -\frac{2\sqrt{5}}{5}$, $\cos \alpha = -\frac{\sqrt{5}}{5}$, $\cot \alpha = \frac{1}{2}$.</p><p><strong>г)</strong> Дано: $\cot \alpha = -3$ и $270^\circ < \alpha < 360^\circ$. Угол $\alpha$ находится в IV четверти, где косинус положителен ($\cos \alpha > 0$), а синус и тангенс отрицательны ($\sin \alpha < 0$, $\tan \alpha < 0$).<br>1. Найдем $\tan \alpha$: $\tan \alpha = \frac{1}{\cot \alpha} = -\frac{1}{3}$.<br>2. Найдем $\sin \alpha$ из тождества $1 + \cot^2 \alpha = \frac{1}{\sin^2 \alpha}$.<br>$\sin^2 \alpha = \frac{1}{1 + \cot^2 \alpha} = \frac{1}{1 + (-3)^2} = \frac{1}{10}$.<br>Так как угол $\alpha$ находится в IV четверти, $\sin \alpha$ отрицателен: $\sin \alpha = -\sqrt{\frac{1}{10}} = -\frac{1}{\sqrt{10}} = -\frac{\sqrt{10}}{10}$.<br>3. Найдем $\cos \alpha$ из соотношения $\cos \alpha = \cot \alpha \cdot \sin \alpha$.<br>$\cos \alpha = (-3) \cdot \left(-\frac{\sqrt{10}}{10}\right) = \frac{3\sqrt{10}}{10}$.<br>Ответ: $\sin \alpha = -\frac{\sqrt{10}}{10}$, $\cos \alpha = \frac{3\sqrt{10}}{10}$, $\tan \alpha = -\frac{1}{3}$.</p>" ] #original: array:6 [ "id" => 1554235 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1108370" "type" => "task" ] "text" => "<p><strong>а)</strong> Дано: $\sin \alpha = 0.6$ и $0^\circ < \alpha < 90^\circ$. Угол $\alpha$ находится в I четверти, где все тригонометрические функции ($\sin \alpha, \cos \alpha, \tan \alpha, \cot \alpha$) положительны.<br>1. Найдем $\cos \alpha$ из основного тригонометрического тождества $\sin^2 \alpha + \cos^2 \alpha = 1$.<br>$\cos^2 \alpha = 1 - \sin^2 \alpha = 1 - (0.6)^2 = 1 - 0.36 = 0.64$.<br>Так как угол $\alpha$ находится в I четверти, $\cos \alpha$ положителен: $\cos \alpha = \sqrt{0.64} = 0.8$.<br>2. Найдем $\tan \alpha$ по определению: $\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$.<br>$\tan \alpha = \frac{0.6}{0.8} = \frac{6}{8} = \frac{3}{4} = 0.75$.<br>3. Найдем $\cot \alpha$ по определению: $\cot \alpha = \frac{1}{\tan \alpha}$.<br>$\cot \alpha = \frac{1}{3/4} = \frac{4}{3}$.<br>Ответ: $\cos \alpha = 0.8$, $\tan \alpha = 0.75$, $\cot \alpha = \frac{4}{3}$.</p><p><strong>б)</strong> Дано: $\cos \alpha = -\frac{1}{3}$ и $90^\circ < \alpha < 180^\circ$. Угол $\alpha$ находится во II четверти, где синус положителен ($\sin \alpha > 0$), а тангенс и котангенс отрицательны ($\tan \alpha < 0$, $\cot \alpha < 0$).<br>1. Найдем $\sin \alpha$ из основного тригонометрического тождества $\sin^2 \alpha + \cos^2 \alpha = 1$.<br>$\sin^2 \alpha = 1 - \cos^2 \alpha = 1 - \left(-\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9}$.<br>Так как угол $\alpha$ находится во II четверти, $\sin \alpha$ положителен: $\sin \alpha = \sqrt{\frac{8}{9}} = \frac{\sqrt{8}}{3} = \frac{2\sqrt{2}}{3}$.<br>2. Найдем $\tan \alpha$ по определению: $\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$.<br>$\tan \alpha = \frac{2\sqrt{2}/3}{-1/3} = -2\sqrt{2}$.<br>3. Найдем $\cot \alpha$ по определению: $\cot \alpha = \frac{1}{\tan \alpha}$.<br>$\cot \alpha = \frac{1}{-2\sqrt{2}} = -\frac{1}{2\sqrt{2}} = -\frac{\sqrt{2}}{2\sqrt{2} \cdot \sqrt{2}} = -\frac{\sqrt{2}}{4}$.<br>Ответ: $\sin \alpha = \frac{2\sqrt{2}}{3}$, $\tan \alpha = -2\sqrt{2}$, $\cot \alpha = -\frac{\sqrt{2}}{4}$.</p><p><strong>в)</strong> Дано: $\tan \alpha = 2$ и $180^\circ < \alpha < 270^\circ$. Угол $\alpha$ находится в III четверти, где синус и косинус отрицательны ($\sin \alpha < 0, \cos \alpha < 0$), а котангенс положителен ($\cot \alpha > 0$).<br>1. Найдем $\cot \alpha$: $\cot \alpha = \frac{1}{\tan \alpha} = \frac{1}{2}$.<br>2. Найдем $\cos \alpha$ из тождества $1 + \tan^2 \alpha = \frac{1}{\cos^2 \alpha}$.<br>$\cos^2 \alpha = \frac{1}{1 + \tan^2 \alpha} = \frac{1}{1 + 2^2} = \frac{1}{5}$.<br>Так как угол $\alpha$ находится в III четверти, $\cos \alpha$ отрицателен: $\cos \alpha = -\sqrt{\frac{1}{5}} = -\frac{1}{\sqrt{5}} = -\frac{\sqrt{5}}{5}$.<br>3. Найдем $\sin \alpha$ из соотношения $\sin \alpha = \tan \alpha \cdot \cos \alpha$.<br>$\sin \alpha = 2 \cdot \left(-\frac{\sqrt{5}}{5}\right) = -\frac{2\sqrt{5}}{5}$.<br>Ответ: $\sin \alpha = -\frac{2\sqrt{5}}{5}$, $\cos \alpha = -\frac{\sqrt{5}}{5}$, $\cot \alpha = \frac{1}{2}$.</p><p><strong>г)</strong> Дано: $\cot \alpha = -3$ и $270^\circ < \alpha < 360^\circ$. Угол $\alpha$ находится в IV четверти, где косинус положителен ($\cos \alpha > 0$), а синус и тангенс отрицательны ($\sin \alpha < 0$, $\tan \alpha < 0$).<br>1. Найдем $\tan \alpha$: $\tan \alpha = \frac{1}{\cot \alpha} = -\frac{1}{3}$.<br>2. Найдем $\sin \alpha$ из тождества $1 + \cot^2 \alpha = \frac{1}{\sin^2 \alpha}$.<br>$\sin^2 \alpha = \frac{1}{1 + \cot^2 \alpha} = \frac{1}{1 + (-3)^2} = \frac{1}{10}$.<br>Так как угол $\alpha$ находится в IV четверти, $\sin \alpha$ отрицателен: $\sin \alpha = -\sqrt{\frac{1}{10}} = -\frac{1}{\sqrt{10}} = -\frac{\sqrt{10}}{10}$.<br>3. Найдем $\cos \alpha$ из соотношения $\cos \alpha = \cot \alpha \cdot \sin \alpha$.<br>$\cos \alpha = (-3) \cdot \left(-\frac{\sqrt{10}}{10}\right) = \frac{3\sqrt{10}}{10}$.<br>Ответ: $\sin \alpha = -\frac{\sqrt{10}}{10}$, $\cos \alpha = \frac{3\sqrt{10}}{10}$, $\tan \alpha = -\frac{1}{3}$.</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" => "1108371" "type" => "task" ] "previous" => array:2 [ "refs" => "1108369" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Book {#1124 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 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"field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-185" "field_display_title" => "185" "field_folder" => "folder1" "field_image_name" => "185" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1167 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Book {#1168 #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 {#1170 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1191 …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 {#1192 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1195 …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 {#1170 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1191 …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 {#1192 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1195 …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 {#1196 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1207 #items: array:1 [ 0 => App\Models\Element {#1206 #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" => 1261078 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1216 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261078 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1216 …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" => "1097928" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097926" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#2008 #items: array:6 [ 0 => App\Models\Task {#1199 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108368 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-632" "field_display_title" => "632" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1197 …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 {#1203 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1201 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1245 …2} "content" => Illuminate\Database\Eloquent\Collection {#1282 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1307 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108368 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-632" "field_display_title" => "632" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1197 …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 {#1203 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1201 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1245 …2} "content" => Illuminate\Database\Eloquent\Collection {#1282 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1307 …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 {#1336 #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" => 1108369 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-633" "field_display_title" => "633" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1337 …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 {#1339 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1340 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1373 …2} "content" => Illuminate\Database\Eloquent\Collection {#1410 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1435 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108369 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-633" "field_display_title" => "633" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1337 …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 {#1339 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1340 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1373 …2} "content" => Illuminate\Database\Eloquent\Collection {#1410 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1435 …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 {#1464 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108370 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-634" "field_display_title" => "634" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1465 …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 {#1467 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1468 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1501 …2} "content" => Illuminate\Database\Eloquent\Collection {#1538 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1563 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108370 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-634" "field_display_title" => "634" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1465 …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 {#1467 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1468 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1501 …2} "content" => Illuminate\Database\Eloquent\Collection {#1538 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1563 …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 {#1592 #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" => 1108371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-635" "field_display_title" => "635" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1593 …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 {#1595 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1596 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1629 …2} "content" => Illuminate\Database\Eloquent\Collection {#1666 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1691 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-635" "field_display_title" => "635" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1593 …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 {#1595 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1596 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1629 …2} "content" => Illuminate\Database\Eloquent\Collection {#1666 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1691 …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 {#1720 #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" => 1108372 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-636" "field_display_title" => "636" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1721 …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 {#1723 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1724 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1757 …2} "content" => Illuminate\Database\Eloquent\Collection {#1794 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1819 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108372 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-636" "field_display_title" => "636" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1721 …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 {#1723 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1724 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1757 …2} "content" => Illuminate\Database\Eloquent\Collection {#1794 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1819 …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 {#1848 #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" => 1108373 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-637" "field_display_title" => "637" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1849 …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 {#1851 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1852 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1885 …2} "content" => Illuminate\Database\Eloquent\Collection {#1954 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1979 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108373 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-637" "field_display_title" => "637" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1849 …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 {#1851 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1852 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1885 …2} "content" => Illuminate\Database\Eloquent\Collection {#1954 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1979 …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 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1097927 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "185" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-185" "field_display_title" => "185" "field_folder" => "folder1" "field_image_name" => "185" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1167} 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№634 (с. 185)
Условие. №634 (с. 185)
скриншот условия
634. Найдите значения остальных тригонометрических функций, если известно, что:
a) $ \sin \alpha = 0,6 $, $ 0^\circ < \alpha < 90^\circ $;
б) $ \cos \alpha = -\frac{1}{3} $, $ 90^\circ < \alpha < 180^\circ $;
в) $ \text{tg} \alpha = 2 $, $ 180^\circ < \alpha < 270^\circ $;
г) $ \text{ctg} \alpha = -3 $, $ 270^\circ < \alpha < 360^\circ $.
Решение. №634 (с. 185)
Решение 2 (rus). №634 (с. 185)
а) Дано: $\sin \alpha = 0.6$ и $0^\circ < \alpha < 90^\circ$. Угол $\alpha$ находится в I четверти, где все тригонометрические функции ($\sin \alpha, \cos \alpha, \tan \alpha, \cot \alpha$) положительны.
1. Найдем $\cos \alpha$ из основного тригонометрического тождества $\sin^2 \alpha + \cos^2 \alpha = 1$.
$\cos^2 \alpha = 1 - \sin^2 \alpha = 1 - (0.6)^2 = 1 - 0.36 = 0.64$.
Так как угол $\alpha$ находится в I четверти, $\cos \alpha$ положителен: $\cos \alpha = \sqrt{0.64} = 0.8$.
2. Найдем $\tan \alpha$ по определению: $\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$.
$\tan \alpha = \frac{0.6}{0.8} = \frac{6}{8} = \frac{3}{4} = 0.75$.
3. Найдем $\cot \alpha$ по определению: $\cot \alpha = \frac{1}{\tan \alpha}$.
$\cot \alpha = \frac{1}{3/4} = \frac{4}{3}$.
Ответ: $\cos \alpha = 0.8$, $\tan \alpha = 0.75$, $\cot \alpha = \frac{4}{3}$.
б) Дано: $\cos \alpha = -\frac{1}{3}$ и $90^\circ < \alpha < 180^\circ$. Угол $\alpha$ находится во II четверти, где синус положителен ($\sin \alpha > 0$), а тангенс и котангенс отрицательны ($\tan \alpha < 0$, $\cot \alpha < 0$).
1. Найдем $\sin \alpha$ из основного тригонометрического тождества $\sin^2 \alpha + \cos^2 \alpha = 1$.
$\sin^2 \alpha = 1 - \cos^2 \alpha = 1 - \left(-\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9}$.
Так как угол $\alpha$ находится во II четверти, $\sin \alpha$ положителен: $\sin \alpha = \sqrt{\frac{8}{9}} = \frac{\sqrt{8}}{3} = \frac{2\sqrt{2}}{3}$.
2. Найдем $\tan \alpha$ по определению: $\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$.
$\tan \alpha = \frac{2\sqrt{2}/3}{-1/3} = -2\sqrt{2}$.
3. Найдем $\cot \alpha$ по определению: $\cot \alpha = \frac{1}{\tan \alpha}$.
$\cot \alpha = \frac{1}{-2\sqrt{2}} = -\frac{1}{2\sqrt{2}} = -\frac{\sqrt{2}}{2\sqrt{2} \cdot \sqrt{2}} = -\frac{\sqrt{2}}{4}$.
Ответ: $\sin \alpha = \frac{2\sqrt{2}}{3}$, $\tan \alpha = -2\sqrt{2}$, $\cot \alpha = -\frac{\sqrt{2}}{4}$.
в) Дано: $\tan \alpha = 2$ и $180^\circ < \alpha < 270^\circ$. Угол $\alpha$ находится в III четверти, где синус и косинус отрицательны ($\sin \alpha < 0, \cos \alpha < 0$), а котангенс положителен ($\cot \alpha > 0$).
1. Найдем $\cot \alpha$: $\cot \alpha = \frac{1}{\tan \alpha} = \frac{1}{2}$.
2. Найдем $\cos \alpha$ из тождества $1 + \tan^2 \alpha = \frac{1}{\cos^2 \alpha}$.
$\cos^2 \alpha = \frac{1}{1 + \tan^2 \alpha} = \frac{1}{1 + 2^2} = \frac{1}{5}$.
Так как угол $\alpha$ находится в III четверти, $\cos \alpha$ отрицателен: $\cos \alpha = -\sqrt{\frac{1}{5}} = -\frac{1}{\sqrt{5}} = -\frac{\sqrt{5}}{5}$.
3. Найдем $\sin \alpha$ из соотношения $\sin \alpha = \tan \alpha \cdot \cos \alpha$.
$\sin \alpha = 2 \cdot \left(-\frac{\sqrt{5}}{5}\right) = -\frac{2\sqrt{5}}{5}$.
Ответ: $\sin \alpha = -\frac{2\sqrt{5}}{5}$, $\cos \alpha = -\frac{\sqrt{5}}{5}$, $\cot \alpha = \frac{1}{2}$.
г) Дано: $\cot \alpha = -3$ и $270^\circ < \alpha < 360^\circ$. Угол $\alpha$ находится в IV четверти, где косинус положителен ($\cos \alpha > 0$), а синус и тангенс отрицательны ($\sin \alpha < 0$, $\tan \alpha < 0$).
1. Найдем $\tan \alpha$: $\tan \alpha = \frac{1}{\cot \alpha} = -\frac{1}{3}$.
2. Найдем $\sin \alpha$ из тождества $1 + \cot^2 \alpha = \frac{1}{\sin^2 \alpha}$.
$\sin^2 \alpha = \frac{1}{1 + \cot^2 \alpha} = \frac{1}{1 + (-3)^2} = \frac{1}{10}$.
Так как угол $\alpha$ находится в IV четверти, $\sin \alpha$ отрицателен: $\sin \alpha = -\sqrt{\frac{1}{10}} = -\frac{1}{\sqrt{10}} = -\frac{\sqrt{10}}{10}$.
3. Найдем $\cos \alpha$ из соотношения $\cos \alpha = \cot \alpha \cdot \sin \alpha$.
$\cos \alpha = (-3) \cdot \left(-\frac{\sqrt{10}}{10}\right) = \frac{3\sqrt{10}}{10}$.
Ответ: $\sin \alpha = -\frac{\sqrt{10}}{10}$, $\cos \alpha = \frac{3\sqrt{10}}{10}$, $\tan \alpha = -\frac{1}{3}$.
Другие задания:
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