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Номер 2.45, страница 65 - гдз по геометрии 8 класс учебник Шыныбеков, Шыныбеков
Авторы: Шыныбеков А. Н., Шыныбеков Д. А., Жумабаев Р. Н.
Тип: Учебник
Издательство: Атамұра
Год издания: 2018 - 2026
ISBN: 978-601-331-161-6
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 8 классе
Раздел 2. Соотношения между сторонами и углами прямоугольного треугольника. 2.2. Синус, тангенс и котангенс острого угла - номер 2.45, страница 65.
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"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" => 1122381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Соотношения между сторонами и углами прямоугольного треугольника" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: array:1 [ 0 => App\Models\Term {#1033 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 30 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Раздел" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 30 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Раздел" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "52" "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 {#1030 #items: array:1 [ 0 => App\Models\Book {#1032 #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" => 2592 "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 {#1034 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1036 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1038 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1040 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_under_the_edition" => 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…2} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1060 …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" => 2592 "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 {#1034 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1036 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1038 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1040 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1049 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "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 {#1057 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1060 …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 {#1061 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1122381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Соотношения между сторонами и углами прямоугольного треугольника" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "52" "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 {#1030} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1061} ] #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 {#1062 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1063 #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" => 1122383 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.2. Синус, тангенс и котангенс острого угла" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1064 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "60" "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 {#1065 #items: array:1 [ 0 => App\Models\Book {#1066 #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" => 2592 "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 {#1067 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1069 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "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 {#1090 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1093 …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" => 2592 "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 {#1067 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1069 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1071 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1073 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1088 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "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 {#1090 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1093 …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 {#1094 #items: array:1 [ 0 => App\Models\Branch {#1095 #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" => 1122381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Соотношения между сторонами и углами прямоугольного треугольника" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_page_start" => "52" "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 {#1098 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1099 …2} ] #original: array:24 [ "id" => 1122381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Соотношения между сторонами и углами прямоугольного треугольника" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1096 …2} "field_page_start" => "52" "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 {#1098 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1099 …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" => 1122383 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "2.2. Синус, тангенс и котангенс острого угла" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1064} "field_page_start" => "60" "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 {#1065} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1094} ] #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 {#1100 #items: array:4 [ 0 => App\Models\Element {#1101 #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" => 1305115 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ 0 => App\Models\Edition {#1103 #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" => 5006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Учебник rus" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1104 …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" => "0rus-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Учебник rus" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1104 …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" => "0rus-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1108 …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" => "1122714" "type" => "task" ] "text" => "<p><strong>2.45.</strong> Через гипотенузу $c$ и острый угол $\alpha$ прямоугольного треугольника выразите катеты и второй острый угол.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "45-1.jpg" "alt" => null "width" => "1373" "height" => 200 "path" => "/media/geometrija_08/Shynybekov-u17/0rus-2/45-1.webp?ts=1753776392" ] ] ] #original: array:7 [ "id" => 1305115 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1102} "task" => array:2 [ "refs" => "1122714" "type" => "task" ] "text" => "<p><strong>2.45.</strong> Через гипотенузу $c$ и острый угол $\alpha$ прямоугольного треугольника выразите катеты и второй острый угол.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "45-1.jpg" "alt" => null "width" => "1373" "height" => 200 "path" => "/media/geometrija_08/Shynybekov-u17/0rus-2/45-1.webp?ts=1753776392" ] ] ] #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 {#1109 #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" => 1305462 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Edition {#1111 #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" => 5007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Учебник kz" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "image" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0kz-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Учебник kz" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "image" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0kz-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1114 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1116 …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" => "1122714" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "45-1.jpg" "alt" => null "width" => "1746" "height" => 185 "path" => "/media/geometrija_08/Shynybekov-u17/0kz-2/45-1.webp?ts=1753776440" ] ] ] #original: array:6 [ "id" => 1305462 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1110} "task" => array:2 [ "refs" => "1122714" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "45-1.jpg" "alt" => null "width" => "1746" "height" => 185 "path" => "/media/geometrija_08/Shynybekov-u17/0kz-2/45-1.webp?ts=1753776440" ] ] ] #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 {#1117 #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" => 1306115 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1118 #items: array:1 [ 0 => App\Models\Edition {#1119 #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" => 5008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120 …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 {#1122 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120 …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 {#1122 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1124 …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" => "1122714" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "45-1.jpg" "alt" => null "width" => "1574" "height" => 224 "path" => "/media/geometrija_08/Shynybekov-u17/1-2/45-1.webp?ts=1753776500" ] ] ] #original: array:6 [ "id" => 1306115 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1118} "task" => array:2 [ "refs" => "1122714" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "45-1.jpg" "alt" => null "width" => "1574" "height" => 224 "path" => "/media/geometrija_08/Shynybekov-u17/1-2/45-1.webp?ts=1753776500" ] ] ] #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\Element {#1125 #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" => 1480948 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\Edition {#1127 #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" => 5435 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 rus" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1128 …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 {#1130 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5435 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 rus" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1128 …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 {#1130 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1132 …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" => "1122714" "type" => "task" ] "text" => "<p>Пусть дан прямоугольный треугольник, в котором $c$ — длина гипотенузы, а $\alpha$ — один из острых углов. Обозначим второй острый угол как $\beta$, катет, противолежащий углу $\alpha$, как $a$, а катет, прилежащий к углу $\alpha$, как $b$.</p><p><strong>Второй острый угол</strong></p><p>Сумма углов в любом треугольнике равна $180^\circ$. Так как в прямоугольном треугольнике один угол равен $90^\circ$, то сумма двух острых углов составляет $180^\circ - 90^\circ = 90^\circ$.<br>Отсюда, зная один острый угол $\alpha$, можно найти второй острый угол $\beta$:<br>$\beta = 90^\circ - \alpha$</p><p>Ответ: $90^\circ - \alpha$.</p><p><strong>Катеты</strong></p><p>Для нахождения длин катетов используются определения основных тригонометрических функций — синуса и косинуса.<br><br><em>Катет, противолежащий углу $\alpha$ (катет $a$):</em><br>Синус острого угла в прямоугольном треугольнике — это отношение противолежащего катета к гипотенузе.<br>$\sin(\alpha) = \frac{a}{c}$<br>Из этой формулы выражаем катет $a$:<br>$a = c \cdot \sin(\alpha)$<br><br><em>Катет, прилежащий к углу $\alpha$ (катет $b$):</em><br>Косинус острого угла в прямоугольном треугольнике — это отношение прилежащего катета к гипотенузе.<br>$\cos(\alpha) = \frac{b}{c}$<br>Из этой формулы выражаем катет $b$:<br>$b = c \cdot \cos(\alpha)$</p><p>Ответ: Катет, противолежащий углу $\alpha$, равен $c \cdot \sin(\alpha)$; катет, прилежащий к углу $\alpha$, равен $c \cdot \cos(\alpha)$.</p>" ] #original: array:6 [ "id" => 1480948 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1126} "task" => array:2 [ "refs" => "1122714" "type" => "task" ] "text" => "<p>Пусть дан прямоугольный треугольник, в котором $c$ — длина гипотенузы, а $\alpha$ — один из острых углов. Обозначим второй острый угол как $\beta$, катет, противолежащий углу $\alpha$, как $a$, а катет, прилежащий к углу $\alpha$, как $b$.</p><p><strong>Второй острый угол</strong></p><p>Сумма углов в любом треугольнике равна $180^\circ$. Так как в прямоугольном треугольнике один угол равен $90^\circ$, то сумма двух острых углов составляет $180^\circ - 90^\circ = 90^\circ$.<br>Отсюда, зная один острый угол $\alpha$, можно найти второй острый угол $\beta$:<br>$\beta = 90^\circ - \alpha$</p><p>Ответ: $90^\circ - \alpha$.</p><p><strong>Катеты</strong></p><p>Для нахождения длин катетов используются определения основных тригонометрических функций — синуса и косинуса.<br><br><em>Катет, противолежащий углу $\alpha$ (катет $a$):</em><br>Синус острого угла в прямоугольном треугольнике — это отношение противолежащего катета к гипотенузе.<br>$\sin(\alpha) = \frac{a}{c}$<br>Из этой формулы выражаем катет $a$:<br>$a = c \cdot \sin(\alpha)$<br><br><em>Катет, прилежащий к углу $\alpha$ (катет $b$):</em><br>Косинус острого угла в прямоугольном треугольнике — это отношение прилежащего катета к гипотенузе.<br>$\cos(\alpha) = \frac{b}{c}$<br>Из этой формулы выражаем катет $b$:<br>$b = c \cdot \cos(\alpha)$</p><p>Ответ: Катет, противолежащий углу $\alpha$, равен $c \cdot \sin(\alpha)$; катет, прилежащий к углу $\alpha$, равен $c \cdot \cos(\alpha)$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1122715" "type" => "task" ] "previous" => array:2 [ "refs" => "1122713" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1133 #items: array:1 [ 0 => App\Models\Book {#1134 #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" => 2592 "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 {#1135 #items: array:1 [ 0 => App\Models\Term {#1136 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #original: array:10 [ "id" => 6477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Геометрия" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "geometrija" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1137 #items: array:1 [ 0 => App\Models\Term {#1138 #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" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ …6] "field_translit" => "vosmoj" ] #original: array:6 [ "id" => 5457 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "8" "field_cases" => array:6 [ …6] "field_translit" => "vosmoj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1139 #items: array:1 [ 0 => App\Models\Term {#1140 #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" => 6991 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атамұра" "field_cases" => array:6 [ …6] "field_translit" => "atamu'ra" ] #original: array:6 [ "id" => 6991 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Атамұра" "field_cases" => array:6 [ …6] "field_translit" => "atamu'ra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1141 #items: array:3 [ 0 => App\Models\Term {#1142 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6992 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Шыныбеков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Абдухали" "field_patronymic" => "Насырович" "field_surname_rp" => null ] #original: array:12 [ "id" => 6992 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Шыныбеков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Абдухали" "field_patronymic" => "Насырович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1143 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6993 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Шыныбеков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Данияр" "field_patronymic" => "Абдухалиевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 6993 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Шыныбеков" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Данияр" "field_patronymic" => "Абдухалиевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1144 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7001 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумабаев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Ринат" "field_patronymic" => "Нурланович" "field_surname_rp" => null ] #original: array:12 [ "id" => 7001 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумабаев" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Ринат" "field_patronymic" => "Нурланович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Term {#1147 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1148 #items: array:1 [ 0 => App\Models\Term {#1149 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1150 #items: array:1 [ 0 => App\Models\Term {#1151 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1152 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1153 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1154 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1155 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1156 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1157 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "field_cover" => array:1 [ 0 => "/media/geometrija_08/Shynybekov-u17/covers/cover1.webp?ts=1753258124" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_08/Shynybekov-u17/covers/cover1.webp?ts=1753258124" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1158 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1159 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1160 #items: [] #escapeWhenCastingToString: false } "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1161 #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" => 379 "subject" => 543 "class_subject" => 390 ] ] #original: array:50 [ "id" => 2592 "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 {#1135} "field_class" => Illuminate\Database\Eloquent\Collection {#1137} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1139} "field_author" => Illuminate\Database\Eloquent\Collection {#1141} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1145} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_country" => Illuminate\Database\Eloquent\Collection {#1148} "field_city" => Illuminate\Database\Eloquent\Collection {#1150} "field_series" => Illuminate\Database\Eloquent\Collection {#1152} "field_umk" => Illuminate\Database\Eloquent\Collection {#1153} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1154} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1155} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1156} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1157} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "field_cover" => array:1 [ 0 => "/media/geometrija_08/Shynybekov-u17/covers/cover1.webp?ts=1753258124" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_08/Shynybekov-u17/covers/cover1.webp?ts=1753258124" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1158} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1159} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1160} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1161} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 379 "subject" => 543 "class_subject" => 390 ] ] #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 {#1446 #items: array:1 [ 0 => App\Models\BookPage {#1165 #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" => 1114267 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "65" "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/page-65" "field_display_title" => "65" "field_folder" => "1" "field_image_name" => "65" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1169 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1168 #items: array:1 [ 0 => App\Models\Book {#1170 #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" => 2592 "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 {#1171 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1192 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "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 {#1194 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1196 …2} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => 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 "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 2592 "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 {#1171 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1192 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2018" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 8 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "538" "field_priority" => "6" "field_default_folder" => "/geometrija_08/Shynybekov-u17/" "field_isbn" => "978-601-331-161-6" "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 {#1194 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1196 …2} "field_url" => "/8-klass/geometrija/shynybekov-uchebnik" "field_cover_color" => 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 "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 {#1198 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1199 #items: array:2 [ 0 => App\Models\Element {#1200 #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" => 1290392 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1201 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1290392 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1201 …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 => "*" ] } 1 => App\Models\Element {#1208 #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" => 1290419 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1209 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1290419 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1209 …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" => "1114268" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114266" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1216 #items: array:8 [ 0 => App\Models\Task {#1217 #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" => 1122714 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "65" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-45" "field_display_title" => "2.45" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1218 …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 {#1220 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1221 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1255 …2} "content" => Illuminate\Database\Eloquent\Collection {#1293 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1326 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1122714 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "65" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-45" "field_display_title" => "2.45" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1218 …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 {#1220 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1221 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1255 …2} "content" => Illuminate\Database\Eloquent\Collection {#1293 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1326 …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 {#1327 #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" => 1122715 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "65" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-46" "field_display_title" => "2.46" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1328 …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 {#1329 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1330 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1331 …2} "content" => Illuminate\Database\Eloquent\Collection {#1332 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1353 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1122715 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "65" "field_page_end" => null …19 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 2 => App\Models\Task {#1354 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Task {#1369 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Task {#1384 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 5 => App\Models\Task {#1399 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1414 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 7 => App\Models\Task {#1429 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1114267 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "65" "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/page-65" "field_display_title" => "65" "field_folder" => "1" "field_image_name" => "65" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1169} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1168} "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 {#1198} "content" => Illuminate\Database\Eloquent\Collection {#1199} "next" => array:2 [ "refs" => "1114268" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1114266" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1216} ] #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" => 1122714 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "65" "field_page_end" => null "field_url" => "/8-klass/geometrija/shynybekov-uchebnik/2-45" "field_display_title" => "2.45" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342} "parent_branches" => 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№2.45 (с. 65)
Учебник rus. №2.45 (с. 65)
скриншот условия
2.45. Через гипотенузу $c$ и острый угол $\alpha$ прямоугольного треугольника выразите катеты и второй острый угол.
Учебник kz. №2.45 (с. 65)
Решение. №2.45 (с. 65)
Решение 2 rus. №2.45 (с. 65)
Пусть дан прямоугольный треугольник, в котором $c$ — длина гипотенузы, а $\alpha$ — один из острых углов. Обозначим второй острый угол как $\beta$, катет, противолежащий углу $\alpha$, как $a$, а катет, прилежащий к углу $\alpha$, как $b$.
Второй острый угол
Сумма углов в любом треугольнике равна $180^\circ$. Так как в прямоугольном треугольнике один угол равен $90^\circ$, то сумма двух острых углов составляет $180^\circ - 90^\circ = 90^\circ$.
Отсюда, зная один острый угол $\alpha$, можно найти второй острый угол $\beta$:
$\beta = 90^\circ - \alpha$
Ответ: $90^\circ - \alpha$.
Катеты
Для нахождения длин катетов используются определения основных тригонометрических функций — синуса и косинуса.
Катет, противолежащий углу $\alpha$ (катет $a$):
Синус острого угла в прямоугольном треугольнике — это отношение противолежащего катета к гипотенузе.
$\sin(\alpha) = \frac{a}{c}$
Из этой формулы выражаем катет $a$:
$a = c \cdot \sin(\alpha)$
Катет, прилежащий к углу $\alpha$ (катет $b$):
Косинус острого угла в прямоугольном треугольнике — это отношение прилежащего катета к гипотенузе.
$\cos(\alpha) = \frac{b}{c}$
Из этой формулы выражаем катет $b$:
$b = c \cdot \cos(\alpha)$
Ответ: Катет, противолежащий углу $\alpha$, равен $c \cdot \sin(\alpha)$; катет, прилежащий к углу $\alpha$, равен $c \cdot \cos(\alpha)$.
Другие задания:
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