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Номер 126, страница 52 - гдз по геометрии 11 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2020 - 2026
ISBN: 978-601-317-528-7
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 11 классе
I. Многогранники. 5. Многогранный угол и его свойства - номер 126, страница 52.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1144326 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "52" "field_page_end" => "53" "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-126" "field_display_title" => "126" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #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" => 1144165 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Многогранники" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "15" "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 {#1078 #items: array:1 [ 0 => App\Models\Book {#1048 #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" => 4313 "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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1049 #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 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "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 [ "field_accusative_case" => "геометрию" "field_creative_case" => "геометрией" "field_dative_case" => "геометрии" "field_genitive_case" => "геометрии" "field_nominative_case" => "геометрия" "field_prepositional_case" => "геометрии" ] "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 {#1051 #items: array:1 [ 0 => App\Models\Term {#1052 #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" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ "field_accusative_case" => "одиннадцатый" "field_creative_case" => "одиннадцатым" "field_dative_case" => "одиннадцатому" "field_genitive_case" => "одинадцатого" "field_nominative_case" => "одинадцатый" "field_prepositional_case" => "одиннадцатом" ] "field_translit" => "odinadcatyj" ] #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 {#1053 #items: array:1 [ 0 => App\Models\Term {#1054 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "Kokshetau" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1055 #items: array:3 [ 0 => App\Models\Term {#1056 #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" => 5836 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Г" "field_patronymic" => "Н" "field_surname_rp" => null ] #original: array:12 [ "id" => 5836 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Г" "field_patronymic" => "Н" "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 {#1057 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1058 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060 #items: array:1 [ 0 => App\Models\Term {#1061 #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 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "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 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "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 {#1062 #items: array:1 [ 0 => App\Models\Term {#1063 #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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "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 {#1066 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2020" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "для учащихся 10 (ч. 1), 11 (ч. 2) классов естественно-математического направления общеобразовательной школы" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "679" "field_priority" => "8" "field_default_folder" => "/geometrija_11/Kokshetau-u20/" "field_isbn" => "978-601-317-528-7" "field_cover" => array:1 [ 0 => "/media/geometrija_11/Kokshetau-u20/covers/cover1.webp?ts=1753338288" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_11/Kokshetau-u20/covers/cover1.webp?ts=1753338288" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075 #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" => 382 "subject" => 543 "class_subject" => 393 ] ] #original: array:50 [ "id" => 4313 "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 {#1050} "field_class" => Illuminate\Database\Eloquent\Collection {#1051} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1053} "field_author" => Illuminate\Database\Eloquent\Collection {#1055} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1059} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060} "field_country" => Illuminate\Database\Eloquent\Collection {#1062} "field_city" => Illuminate\Database\Eloquent\Collection {#1064} "field_series" => Illuminate\Database\Eloquent\Collection {#1066} "field_umk" => Illuminate\Database\Eloquent\Collection {#1067} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2020" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "для учащихся 10 (ч. 1), 11 (ч. 2) классов естественно-математического направления общеобразовательной школы" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "679" "field_priority" => "8" "field_default_folder" => "/geometrija_11/Kokshetau-u20/" "field_isbn" => "978-601-317-528-7" "field_cover" => array:1 [ 0 => "/media/geometrija_11/Kokshetau-u20/covers/cover1.webp?ts=1753338288" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/geometrija_11/Kokshetau-u20/covers/cover1.webp?ts=1753338288" "alt" => "" "width" => "1632" "height" => "2472" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074} "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 382 "subject" => 543 "class_subject" => 393 ] ] #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 {#1076 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1144165 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Многогранники" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_page_start" => "15" "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 {#1078} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #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 {#1105 #items: array:2 [ 0 => App\Models\Branch {#1113 #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" => 1144165 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Многогранники" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1114 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "15" "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 {#1115 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1144165 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "I. Многогранники" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1114} "field_page_start" => "15" "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 {#1115} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1117 #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" => 1144170 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "5. Многогранный угол и его свойства" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1118 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "49" "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 {#1119 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1144170 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "5. Многогранный угол и его свойства" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1118} "field_page_start" => "49" "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 {#1119} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1120} ] #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 {#1102 #items: array:3 [ 0 => App\Models\Element {#1095 #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" => 1379546 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ 0 => App\Models\Edition {#1087 #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" => 5039 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ …1] #escapeWhenCastingToString: false } "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 {#1041 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1081 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5039 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1043} "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 {#1041} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1081} "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" => "1144326" "type" => "task" ] "text" => "<p>126.</p><p><strong>a)</strong> В трехгранном угле $PABC$ все плоские углы равны по $60^\circ$. Найдите двугранные углы при его ребрах $PB$ и $PC$.</p><p><strong>б)</strong> В трехгранном угле $PABC$ двугранные углы при его ребрах $PB$ и $PC$ равны по $60^\circ$, а плоский угол $CPB$ равен $120^\circ$. Найдите два других его плоских угла.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "126-1.jpg" "alt" => null "width" => "1595" "height" => 216 "path" => "/media/geometrija_11/Kokshetau-u20/0-upr/126-1.webp?ts=1754600302" ] 1 => array:5 [ "name" => "126-2.jpg" "alt" => null "width" => "1531" "height" => 260 "path" => "/media/geometrija_11/Kokshetau-u20/0-upr/126-2.webp?ts=1754600302" ] ] ] #original: array:7 [ "id" => 1379546 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1085} "task" => array:2 [ "refs" => "1144326" "type" => "task" ] "text" => "<p>126.</p><p><strong>a)</strong> В трехгранном угле $PABC$ все плоские углы равны по $60^\circ$. Найдите двугранные углы при его ребрах $PB$ и $PC$.</p><p><strong>б)</strong> В трехгранном угле $PABC$ двугранные углы при его ребрах $PB$ и $PC$ равны по $60^\circ$, а плоский угол $CPB$ равен $120^\circ$. Найдите два других его плоских угла.</p>" "img" => array:2 [ 0 => array:5 [ "name" => "126-1.jpg" "alt" => null "width" => "1595" "height" => 216 "path" => "/media/geometrija_11/Kokshetau-u20/0-upr/126-1.webp?ts=1754600302" ] 1 => array:5 [ "name" => "126-2.jpg" "alt" => null "width" => "1531" "height" => 260 "path" => "/media/geometrija_11/Kokshetau-u20/0-upr/126-2.webp?ts=1754600302" ] ] ] #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 {#1093 #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" => 1379942 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Edition {#1092 #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" => 5040 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088 #items: array:1 [ …1] #escapeWhenCastingToString: false } "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 {#1090 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1091 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5040 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1088} "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 {#1090} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1089} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1091} "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" => "1144326" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "126-1.jpg" "alt" => null "width" => "1574" "height" => 541 "path" => "/media/geometrija_11/Kokshetau-u20/1-upr/126-1.webp?ts=1754600356" ] ] ] #original: array:6 [ "id" => 1379942 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079} "task" => array:2 [ "refs" => "1144326" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "126-1.jpg" "alt" => null "width" => "1574" "height" => 541 "path" => "/media/geometrija_11/Kokshetau-u20/1-upr/126-1.webp?ts=1754600356" ] ] ] #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 {#1084 #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" => 1562431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Edition {#1080 #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" => 5685 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1034 #items: array:1 [ …1] #escapeWhenCastingToString: false } "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 {#1104 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1112 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1108 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5685 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1034} "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 {#1104} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1112} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1108} "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" => "1144326" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p><strong>Дано:</strong></p><p>Трехгранный угол $PABC$.</p><p>Плоские углы: $\angle APB = 60^\circ$, $\angle BPC = 60^\circ$, $\angle CPA = 60^\circ$.</p><p><strong>Перевод в СИ:</strong></p><p>Углы заданы в градусах, что является стандартной единицей измерения углов в геометрии. Перевод в радианы не требуется для решения данной задачи.</p><p><strong>Найти:</strong></p><p>Двугранные углы при ребрах $PB$ и $PC$.</p><p><strong>Решение:</strong></p><p>Для трехгранного угла с плоскими углами $\alpha = \angle APB$, $\beta = \angle BPC$, $\gamma = \angle CPA$ и соответствующими им противоположными двугранными углами $C_1$ (при ребре $PC$), $A_1$ (при ребре $PA$), $B_1$ (при ребре $PB$), справедливы следующие соотношения (теорема косинусов для трехгранного угла):</p><p>$\cos \gamma = \cos \alpha \cos \beta + \sin \alpha \sin \beta \cos C_1$</p><p>$\cos \alpha = \cos \beta \cos \gamma + \sin \beta \sin \gamma \cos A_1$</p><p>$\cos \beta = \cos \alpha \cos \gamma + \sin \alpha \sin \gamma \cos B_1$</p><p>В данном случае все плоские углы равны $60^\circ$, т.е. $\alpha = \beta = \gamma = 60^\circ$. В силу симметрии, все двугранные углы также будут равны.</p><p>Найдем двугранный угол при ребре $PB$, который обозначается как $B_1$ (он лежит напротив плоского угла $\angle CPA = \gamma$).</p><p>Используем формулу $\cos \gamma = \cos \alpha \cos \beta + \sin \alpha \sin \beta \cos B_1$ (используем $\gamma$ как угол напротив $B_1$, что соответствует стандартным обозначениям для сферического треугольника, где $\gamma$ - это сторона, а $B_1$ - угол при вершине, обозначаемый как $B$). В данном случае, $\alpha$, $\beta$, $\gamma$ - это плоские углы, а $A_1$, $B_1$, $C_1$ - двугранные углы при соответствующих ребрах. Таким образом, $B_1$ находится напротив плоского угла $\gamma = \angle CPA$.</p><p>Подставим значения:</p><p>$\cos 60^\circ = \cos 60^\circ \cos 60^\circ + \sin 60^\circ \sin 60^\circ \cos B_1$</p><p>$\frac{1}{2} = \frac{1}{2} \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} \cos B_1$</p><p>$\frac{1}{2} = \frac{1}{4} + \frac{3}{4} \cos B_1$</p><p>$\frac{1}{2} - \frac{1}{4} = \frac{3}{4} \cos B_1$</p><p>$\frac{1}{4} = \frac{3}{4} \cos B_1$</p><p>$\cos B_1 = \frac{1}{3}$</p><p>$B_1 = \arccos\left(\frac{1}{3}\right)$</p><p>Аналогично, двугранный угол при ребре $PC$, обозначаемый как $C_1$ (он лежит напротив плоского угла $\angle APB = \alpha$), будет равен:</p><p>$\cos \alpha = \cos \beta \cos \gamma + \sin \beta \sin \gamma \cos C_1$</p><p>$\cos 60^\circ = \cos 60^\circ \cos 60^\circ + \sin 60^\circ \sin 60^\circ \cos C_1$</p><p>$\frac{1}{2} = \frac{1}{4} + \frac{3}{4} \cos C_1$</p><p>$\cos C_1 = \frac{1}{3}$</p><p>$C_1 = \arccos\left(\frac{1}{3}\right)$</p><p><strong>Ответ:</strong> Двугранные углы при ребрах $PB$ и $PC$ равны $\arccos\left(\frac{1}{3}\right)$.</p><p><strong>б)</strong></p><p><strong>Дано:</strong></p><p>Трехгранный угол $PABC$.</p><p>Двугранные углы при ребрах $PB$ и $PC$ равны $60^\circ$.</p><p>Плоский угол $\angle CPB = 120^\circ$.</p><p>Обозначим плоские углы: $\alpha = \angle APB$, $\beta = \angle BPC$, $\gamma = \angle CPA$.</p><p>Обозначим двугранные углы: $A_1$ (при ребре $PA$, напротив $\beta$), $B_1$ (при ребре $PB$, напротив $\gamma$), $C_1$ (при ребре $PC$, напротив $\alpha$).</p><p>Таким образом, дано: $B_1 = 60^\circ$, $C_1 = 60^\circ$, $\beta = 120^\circ$.</p><p><strong>Перевод в СИ:</strong></p><p>Углы заданы в градусах, что является стандартной единицей измерения углов в геометрии. Перевод в радианы не требуется для решения данной задачи.</p><p><strong>Найти:</strong></p><p>Плоские углы $\alpha = \angle APB$ и $\gamma = \angle CPA$.</p><p><strong>Решение:</strong></p><p>Для трехгранного угла справедлива вторая теорема косинусов для сферического треугольника (или дуальная теорема косинусов):</p><p>$\cos A_1 = -\cos B_1 \cos C_1 + \sin B_1 \sin C_1 \cos \alpha$</p><p>$\cos B_1 = -\cos A_1 \cos C_1 + \sin A_1 \sin C_1 \cos \beta$</p><p>$\cos C_1 = -\cos A_1 \cos B_1 + \sin A_1 \sin B_1 \cos \gamma$</p><p>Известны $B_1 = 60^\circ$, $C_1 = 60^\circ$ и $\beta = 120^\circ$.</p><p>Сначала найдем двугранный угол $A_1$ при ребре $PA$, используя формулу, содержащую известный плоский угол $\beta$:</p><p>$\cos A_1 = -\cos B_1 \cos C_1 + \sin B_1 \sin C_1 \cos \beta$ (Осторожно, в стандартных обозначениях сферического треугольника $A$ - это угол, а $a$ - противолежащая сторона. Здесь $A_1$ - двугранный угол при ребре $PA$, а $\beta = \angle BPC$ - противолежащий плоский угол).</p><p>$\cos A_1 = -\cos 60^\circ \cos 60^\circ + \sin 60^\circ \sin 60^\circ \cos 120^\circ$</p><p>$\cos A_1 = -\left(\frac{1}{2}\right)\left(\frac{1}{2}\right) + \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)\left(-\frac{1}{2}\right)$</p><p>$\cos A_1 = -\frac{1}{4} + \frac{3}{4} \cdot \left(-\frac{1}{2}\right)$</p><p>$\cos A_1 = -\frac{1}{4} - \frac{3}{8}$</p><p>$\cos A_1 = -\frac{2}{8} - \frac{3}{8}$</p><p>$\cos A_1 = -\frac{5}{8}$</p><p>Теперь найдем плоские углы $\alpha$ и $\gamma$. Поскольку двугранные углы $B_1$ и $C_1$ равны, то и противолежащие им плоские углы $\gamma$ и $\alpha$ будут равны, т.е. $\alpha = \gamma$.</p><p>Используем формулу для $\cos C_1$ для нахождения $\gamma$:</p><p>$\cos C_1 = -\cos A_1 \cos B_1 + \sin A_1 \sin B_1 \cos \gamma$</p><p>$\cos 60^\circ = -\left(-\frac{5}{8}\right) \cos 60^\circ + \sin A_1 \sin 60^\circ \cos \gamma$</p><p>$\frac{1}{2} = \frac{5}{8} \cdot \frac{1}{2} + \sin A_1 \cdot \frac{\sqrt{3}}{2} \cos \gamma$</p><p>$\frac{1}{2} = \frac{5}{16} + \frac{\sqrt{3}}{2} \sin A_1 \cos \gamma$</p><p>Найдем $\sin A_1$ из $\cos A_1 = -\frac{5}{8}$:</p><p>$\sin^2 A_1 = 1 - \cos^2 A_1 = 1 - \left(-\frac{5}{8}\right)^2 = 1 - \frac{25}{64} = \frac{64 - 25}{64} = \frac{39}{64}$</p><p>Так как $A_1$ является углом в трехгранном угле ($0 < A_1 < 180^\circ$), то $\sin A_1 > 0$.</p><p>$\sin A_1 = \sqrt{\frac{39}{64}} = \frac{\sqrt{39}}{8}$</p><p>Подставим $\sin A_1$ в уравнение для $\cos \gamma$:</p><p>$\frac{1}{2} = \frac{5}{16} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{39}}{8} \cos \gamma$</p><p>$\frac{1}{2} - \frac{5}{16} = \frac{\sqrt{3 \cdot 39}}{16} \cos \gamma$</p><p>$\frac{8}{16} - \frac{5}{16} = \frac{\sqrt{117}}{16} \cos \gamma$</p><p>$\frac{3}{16} = \frac{\sqrt{9 \cdot 13}}{16} \cos \gamma$</p><p>$\frac{3}{16} = \frac{3\sqrt{13}}{16} \cos \gamma$</p><p>$\cos \gamma = \frac{3}{3\sqrt{13}} = \frac{1}{\sqrt{13}}$</p><p>$\cos \gamma = \frac{\sqrt{13}}{13}$</p><p>$\gamma = \arccos\left(\frac{\sqrt{13}}{13}\right)$</p><p>Поскольку $\alpha = \gamma$, то $\alpha = \arccos\left(\frac{\sqrt{13}}{13}\right)$.</p><p><strong>Ответ:</strong> Два других плоских угла равны $\arccos\left(\frac{\sqrt{13}}{13}\right)$.</p>" ] #original: array:6 [ "id" => 1562431 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "1144326" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p><strong>Дано:</strong></p><p>Трехгранный угол $PABC$.</p><p>Плоские углы: $\angle APB = 60^\circ$, $\angle BPC = 60^\circ$, $\angle CPA = 60^\circ$.</p><p><strong>Перевод в СИ:</strong></p><p>Углы заданы в градусах, что является стандартной единицей измерения углов в геометрии. Перевод в радианы не требуется для решения данной задачи.</p><p><strong>Найти:</strong></p><p>Двугранные углы при ребрах $PB$ и $PC$.</p><p><strong>Решение:</strong></p><p>Для трехгранного угла с плоскими углами $\alpha = \angle APB$, $\beta = \angle BPC$, $\gamma = \angle CPA$ и соответствующими им противоположными двугранными углами $C_1$ (при ребре $PC$), $A_1$ (при ребре $PA$), $B_1$ (при ребре $PB$), справедливы следующие соотношения (теорема косинусов для трехгранного угла):</p><p>$\cos \gamma = \cos \alpha \cos \beta + \sin \alpha \sin \beta \cos C_1$</p><p>$\cos \alpha = \cos \beta \cos \gamma + \sin \beta \sin \gamma \cos A_1$</p><p>$\cos \beta = \cos \alpha \cos \gamma + \sin \alpha \sin \gamma \cos B_1$</p><p>В данном случае все плоские углы равны $60^\circ$, т.е. $\alpha = \beta = \gamma = 60^\circ$. В силу симметрии, все двугранные углы также будут равны.</p><p>Найдем двугранный угол при ребре $PB$, который обозначается как $B_1$ (он лежит напротив плоского угла $\angle CPA = \gamma$).</p><p>Используем формулу $\cos \gamma = \cos \alpha \cos \beta + \sin \alpha \sin \beta \cos B_1$ (используем $\gamma$ как угол напротив $B_1$, что соответствует стандартным обозначениям для сферического треугольника, где $\gamma$ - это сторона, а $B_1$ - угол при вершине, обозначаемый как $B$). В данном случае, $\alpha$, $\beta$, $\gamma$ - это плоские углы, а $A_1$, $B_1$, $C_1$ - двугранные углы при соответствующих ребрах. Таким образом, $B_1$ находится напротив плоского угла $\gamma = \angle CPA$.</p><p>Подставим значения:</p><p>$\cos 60^\circ = \cos 60^\circ \cos 60^\circ + \sin 60^\circ \sin 60^\circ \cos B_1$</p><p>$\frac{1}{2} = \frac{1}{2} \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} \cos B_1$</p><p>$\frac{1}{2} = \frac{1}{4} + \frac{3}{4} \cos B_1$</p><p>$\frac{1}{2} - \frac{1}{4} = \frac{3}{4} \cos B_1$</p><p>$\frac{1}{4} = \frac{3}{4} \cos B_1$</p><p>$\cos B_1 = \frac{1}{3}$</p><p>$B_1 = \arccos\left(\frac{1}{3}\right)$</p><p>Аналогично, двугранный угол при ребре $PC$, обозначаемый как $C_1$ (он лежит напротив плоского угла $\angle APB = \alpha$), будет равен:</p><p>$\cos \alpha = \cos \beta \cos \gamma + \sin \beta \sin \gamma \cos C_1$</p><p>$\cos 60^\circ = \cos 60^\circ \cos 60^\circ + \sin 60^\circ \sin 60^\circ \cos C_1$</p><p>$\frac{1}{2} = \frac{1}{4} + \frac{3}{4} \cos C_1$</p><p>$\cos C_1 = \frac{1}{3}$</p><p>$C_1 = \arccos\left(\frac{1}{3}\right)$</p><p><strong>Ответ:</strong> Двугранные углы при ребрах $PB$ и $PC$ равны $\arccos\left(\frac{1}{3}\right)$.</p><p><strong>б)</strong></p><p><strong>Дано:</strong></p><p>Трехгранный угол $PABC$.</p><p>Двугранные углы при ребрах $PB$ и $PC$ равны $60^\circ$.</p><p>Плоский угол $\angle CPB = 120^\circ$.</p><p>Обозначим плоские углы: $\alpha = \angle APB$, $\beta = \angle BPC$, $\gamma = \angle CPA$.</p><p>Обозначим двугранные углы: $A_1$ (при ребре $PA$, напротив $\beta$), $B_1$ (при ребре $PB$, напротив $\gamma$), $C_1$ (при ребре $PC$, напротив $\alpha$).</p><p>Таким образом, дано: $B_1 = 60^\circ$, $C_1 = 60^\circ$, $\beta = 120^\circ$.</p><p><strong>Перевод в СИ:</strong></p><p>Углы заданы в градусах, что является стандартной единицей измерения углов в геометрии. Перевод в радианы не требуется для решения данной задачи.</p><p><strong>Найти:</strong></p><p>Плоские углы $\alpha = \angle APB$ и $\gamma = \angle CPA$.</p><p><strong>Решение:</strong></p><p>Для трехгранного угла справедлива вторая теорема косинусов для сферического треугольника (или дуальная теорема косинусов):</p><p>$\cos A_1 = -\cos B_1 \cos C_1 + \sin B_1 \sin C_1 \cos \alpha$</p><p>$\cos B_1 = -\cos A_1 \cos C_1 + \sin A_1 \sin C_1 \cos \beta$</p><p>$\cos C_1 = -\cos A_1 \cos B_1 + \sin A_1 \sin B_1 \cos \gamma$</p><p>Известны $B_1 = 60^\circ$, $C_1 = 60^\circ$ и $\beta = 120^\circ$.</p><p>Сначала найдем двугранный угол $A_1$ при ребре $PA$, используя формулу, содержащую известный плоский угол $\beta$:</p><p>$\cos A_1 = -\cos B_1 \cos C_1 + \sin B_1 \sin C_1 \cos \beta$ (Осторожно, в стандартных обозначениях сферического треугольника $A$ - это угол, а $a$ - противолежащая сторона. Здесь $A_1$ - двугранный угол при ребре $PA$, а $\beta = \angle BPC$ - противолежащий плоский угол).</p><p>$\cos A_1 = -\cos 60^\circ \cos 60^\circ + \sin 60^\circ \sin 60^\circ \cos 120^\circ$</p><p>$\cos A_1 = -\left(\frac{1}{2}\right)\left(\frac{1}{2}\right) + \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)\left(-\frac{1}{2}\right)$</p><p>$\cos A_1 = -\frac{1}{4} + \frac{3}{4} \cdot \left(-\frac{1}{2}\right)$</p><p>$\cos A_1 = -\frac{1}{4} - \frac{3}{8}$</p><p>$\cos A_1 = -\frac{2}{8} - \frac{3}{8}$</p><p>$\cos A_1 = -\frac{5}{8}$</p><p>Теперь найдем плоские углы $\alpha$ и $\gamma$. Поскольку двугранные углы $B_1$ и $C_1$ равны, то и противолежащие им плоские углы $\gamma$ и $\alpha$ будут равны, т.е. $\alpha = \gamma$.</p><p>Используем формулу для $\cos C_1$ для нахождения $\gamma$:</p><p>$\cos C_1 = -\cos A_1 \cos B_1 + \sin A_1 \sin B_1 \cos \gamma$</p><p>$\cos 60^\circ = -\left(-\frac{5}{8}\right) \cos 60^\circ + \sin A_1 \sin 60^\circ \cos \gamma$</p><p>$\frac{1}{2} = \frac{5}{8} \cdot \frac{1}{2} + \sin A_1 \cdot \frac{\sqrt{3}}{2} \cos \gamma$</p><p>$\frac{1}{2} = \frac{5}{16} + \frac{\sqrt{3}}{2} \sin A_1 \cos \gamma$</p><p>Найдем $\sin A_1$ из $\cos A_1 = -\frac{5}{8}$:</p><p>$\sin^2 A_1 = 1 - \cos^2 A_1 = 1 - \left(-\frac{5}{8}\right)^2 = 1 - \frac{25}{64} = \frac{64 - 25}{64} = \frac{39}{64}$</p><p>Так как $A_1$ является углом в трехгранном угле ($0 < A_1 < 180^\circ$), то $\sin A_1 > 0$.</p><p>$\sin A_1 = \sqrt{\frac{39}{64}} = \frac{\sqrt{39}}{8}$</p><p>Подставим $\sin A_1$ в уравнение для $\cos \gamma$:</p><p>$\frac{1}{2} = \frac{5}{16} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{39}}{8} \cos \gamma$</p><p>$\frac{1}{2} - \frac{5}{16} = \frac{\sqrt{3 \cdot 39}}{16} \cos \gamma$</p><p>$\frac{8}{16} - \frac{5}{16} = \frac{\sqrt{117}}{16} \cos \gamma$</p><p>$\frac{3}{16} = \frac{\sqrt{9 \cdot 13}}{16} \cos \gamma$</p><p>$\frac{3}{16} = \frac{3\sqrt{13}}{16} \cos \gamma$</p><p>$\cos \gamma = \frac{3}{3\sqrt{13}} = \frac{1}{\sqrt{13}}$</p><p>$\cos \gamma = \frac{\sqrt{13}}{13}$</p><p>$\gamma = \arccos\left(\frac{\sqrt{13}}{13}\right)$</p><p>Поскольку $\alpha = \gamma$, то $\alpha = \arccos\left(\frac{\sqrt{13}}{13}\right)$.</p><p><strong>Ответ:</strong> Два других плоских угла равны $\arccos\left(\frac{\sqrt{13}}{13}\right)$.</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" => "1144327" "type" => "task" ] "previous" => array:2 [ "refs" => "1144325" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:1 [ 0 => App\Models\Book {#1048} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\BookPage {#1039 #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" => 1116682 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "52" "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/page-52" "field_display_title" => "52" "field_folder" => "1" "field_image_name" => "52" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1042 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1106 #items: array:1 [ 0 => App\Models\Book {#1048} ] #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 {#1109 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Element {#1129 #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" => 1294947 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1116682" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1294947 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1131} "book_page" => array:2 [ "refs" => "1116682" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #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" => "1116683" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1116681" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:9 [ 0 => App\Models\Task {#1214 #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" => 1144318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "51" "field_page_end" => "52" "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/vopr-5" "field_display_title" => "Вопросы?" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1215 #items: [] #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 {#1216 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1144319" "type" => "task" ] "previous" => array:2 [ "refs" => "1144317" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1227 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1116681" "type" => "book_page" ] ] #original: array:24 [ "id" => 1144318 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "51" "field_page_end" => "52" "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/vopr-5" "field_display_title" => "Вопросы?" "field_outside_task" => "0" "field_task_type" => Illuminate\Database\Eloquent\Collection {#1215} "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 {#1216} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1217} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1218} "content" => Illuminate\Database\Eloquent\Collection {#1221} "next" => array:2 [ "refs" => "1144319" "type" => "task" ] "previous" => array:2 [ "refs" => "1144317" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1227} "page" => array:2 [ "refs" => "1116681" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1220 #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" => 1144319 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "52" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-119" "field_display_title" => "119" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ …1] #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 {#1226 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1228 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1223 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1144320" "type" => "task" ] "previous" => array:2 [ "refs" => "1144318" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1241 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1116682" "type" => "book_page" ] ] #original: array:24 [ "id" => 1144319 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "52" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-119" "field_display_title" => "119" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219} "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 {#1226} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1228} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1224} "content" => Illuminate\Database\Eloquent\Collection {#1223} "next" => array:2 [ "refs" => "1144320" "type" => "task" ] "previous" => array:2 [ "refs" => "1144318" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1241} "page" => array:2 [ "refs" => "1116682" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1225 #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" => 1144320 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "52" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-120" "field_display_title" => "120" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1222 #items: array:1 [ …1] #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 {#1240 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1242 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1238 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1237 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1144321" "type" => "task" ] "previous" => array:2 [ "refs" => "1144319" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1255 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1116682" "type" => "book_page" ] ] #original: array:24 [ "id" => 1144320 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "52" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-120" "field_display_title" => "120" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1222} "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 {#1240} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1242} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1238} "content" => Illuminate\Database\Eloquent\Collection {#1237} "next" => array:2 [ "refs" => "1144321" "type" => "task" ] "previous" => array:2 [ "refs" => "1144319" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1255} "page" => array:2 [ "refs" => "1116682" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1239 #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" => 1144321 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "52" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-121" "field_display_title" => "121" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1236 #items: array:1 [ …1] #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 {#1254 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1256 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1252 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1251 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1144322" "type" => "task" ] "previous" => array:2 [ "refs" => "1144320" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1269 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1116682" "type" => "book_page" ] ] #original: array:24 [ "id" => 1144321 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "52" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-121" "field_display_title" => "121" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1236} "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 {#1254} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1256} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1252} "content" => Illuminate\Database\Eloquent\Collection {#1251} "next" => array:2 [ "refs" => "1144322" "type" => "task" ] "previous" => array:2 [ "refs" => "1144320" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1269} "page" => array:2 [ "refs" => "1116682" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1253 #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" => 1144322 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "52" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-122" "field_display_title" => "122" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1250 #items: array:1 [ …1] #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 {#1268 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1270 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1266 …2} "content" => Illuminate\Database\Eloquent\Collection {#1265 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1283 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1144322 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "52" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-122" "field_display_title" => "122" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1250} "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 {#1268} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1270} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1266 …2} "content" => Illuminate\Database\Eloquent\Collection {#1265 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1283 …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 {#1267 #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" => 1144323 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "52" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-123" "field_display_title" => "123" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1264 …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 {#1282 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1284 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1280 …2} "content" => Illuminate\Database\Eloquent\Collection {#1279 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1297 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1144323 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "52" "field_page_end" => null "field_url" => "/11-klass/geometrija/kokshetau-soltan-uchebnik/upr-123" "field_display_title" => "123" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1264 …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 {#1282 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1284 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1280 …2} "content" => Illuminate\Database\Eloquent\Collection {#1279 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1297 …2} "page" => array:2 [ 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№126 (с. 52)
Условие. №126 (с. 52)
скриншот условия
126.
a) В трехгранном угле $PABC$ все плоские углы равны по $60^\circ$. Найдите двугранные углы при его ребрах $PB$ и $PC$.
б) В трехгранном угле $PABC$ двугранные углы при его ребрах $PB$ и $PC$ равны по $60^\circ$, а плоский угол $CPB$ равен $120^\circ$. Найдите два других его плоских угла.
Решение. №126 (с. 52)
Решение 2 (rus). №126 (с. 52)
а)
Дано:
Трехгранный угол $PABC$.
Плоские углы: $\angle APB = 60^\circ$, $\angle BPC = 60^\circ$, $\angle CPA = 60^\circ$.
Перевод в СИ:
Углы заданы в градусах, что является стандартной единицей измерения углов в геометрии. Перевод в радианы не требуется для решения данной задачи.
Найти:
Двугранные углы при ребрах $PB$ и $PC$.
Решение:
Для трехгранного угла с плоскими углами $\alpha = \angle APB$, $\beta = \angle BPC$, $\gamma = \angle CPA$ и соответствующими им противоположными двугранными углами $C_1$ (при ребре $PC$), $A_1$ (при ребре $PA$), $B_1$ (при ребре $PB$), справедливы следующие соотношения (теорема косинусов для трехгранного угла):
$\cos \gamma = \cos \alpha \cos \beta + \sin \alpha \sin \beta \cos C_1$
$\cos \alpha = \cos \beta \cos \gamma + \sin \beta \sin \gamma \cos A_1$
$\cos \beta = \cos \alpha \cos \gamma + \sin \alpha \sin \gamma \cos B_1$
В данном случае все плоские углы равны $60^\circ$, т.е. $\alpha = \beta = \gamma = 60^\circ$. В силу симметрии, все двугранные углы также будут равны.
Найдем двугранный угол при ребре $PB$, который обозначается как $B_1$ (он лежит напротив плоского угла $\angle CPA = \gamma$).
Используем формулу $\cos \gamma = \cos \alpha \cos \beta + \sin \alpha \sin \beta \cos B_1$ (используем $\gamma$ как угол напротив $B_1$, что соответствует стандартным обозначениям для сферического треугольника, где $\gamma$ - это сторона, а $B_1$ - угол при вершине, обозначаемый как $B$). В данном случае, $\alpha$, $\beta$, $\gamma$ - это плоские углы, а $A_1$, $B_1$, $C_1$ - двугранные углы при соответствующих ребрах. Таким образом, $B_1$ находится напротив плоского угла $\gamma = \angle CPA$.
Подставим значения:
$\cos 60^\circ = \cos 60^\circ \cos 60^\circ + \sin 60^\circ \sin 60^\circ \cos B_1$
$\frac{1}{2} = \frac{1}{2} \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} \cos B_1$
$\frac{1}{2} = \frac{1}{4} + \frac{3}{4} \cos B_1$
$\frac{1}{2} - \frac{1}{4} = \frac{3}{4} \cos B_1$
$\frac{1}{4} = \frac{3}{4} \cos B_1$
$\cos B_1 = \frac{1}{3}$
$B_1 = \arccos\left(\frac{1}{3}\right)$
Аналогично, двугранный угол при ребре $PC$, обозначаемый как $C_1$ (он лежит напротив плоского угла $\angle APB = \alpha$), будет равен:
$\cos \alpha = \cos \beta \cos \gamma + \sin \beta \sin \gamma \cos C_1$
$\cos 60^\circ = \cos 60^\circ \cos 60^\circ + \sin 60^\circ \sin 60^\circ \cos C_1$
$\frac{1}{2} = \frac{1}{4} + \frac{3}{4} \cos C_1$
$\cos C_1 = \frac{1}{3}$
$C_1 = \arccos\left(\frac{1}{3}\right)$
Ответ: Двугранные углы при ребрах $PB$ и $PC$ равны $\arccos\left(\frac{1}{3}\right)$.
б)
Дано:
Трехгранный угол $PABC$.
Двугранные углы при ребрах $PB$ и $PC$ равны $60^\circ$.
Плоский угол $\angle CPB = 120^\circ$.
Обозначим плоские углы: $\alpha = \angle APB$, $\beta = \angle BPC$, $\gamma = \angle CPA$.
Обозначим двугранные углы: $A_1$ (при ребре $PA$, напротив $\beta$), $B_1$ (при ребре $PB$, напротив $\gamma$), $C_1$ (при ребре $PC$, напротив $\alpha$).
Таким образом, дано: $B_1 = 60^\circ$, $C_1 = 60^\circ$, $\beta = 120^\circ$.
Перевод в СИ:
Углы заданы в градусах, что является стандартной единицей измерения углов в геометрии. Перевод в радианы не требуется для решения данной задачи.
Найти:
Плоские углы $\alpha = \angle APB$ и $\gamma = \angle CPA$.
Решение:
Для трехгранного угла справедлива вторая теорема косинусов для сферического треугольника (или дуальная теорема косинусов):
$\cos A_1 = -\cos B_1 \cos C_1 + \sin B_1 \sin C_1 \cos \alpha$
$\cos B_1 = -\cos A_1 \cos C_1 + \sin A_1 \sin C_1 \cos \beta$
$\cos C_1 = -\cos A_1 \cos B_1 + \sin A_1 \sin B_1 \cos \gamma$
Известны $B_1 = 60^\circ$, $C_1 = 60^\circ$ и $\beta = 120^\circ$.
Сначала найдем двугранный угол $A_1$ при ребре $PA$, используя формулу, содержащую известный плоский угол $\beta$:
$\cos A_1 = -\cos B_1 \cos C_1 + \sin B_1 \sin C_1 \cos \beta$ (Осторожно, в стандартных обозначениях сферического треугольника $A$ - это угол, а $a$ - противолежащая сторона. Здесь $A_1$ - двугранный угол при ребре $PA$, а $\beta = \angle BPC$ - противолежащий плоский угол).
$\cos A_1 = -\cos 60^\circ \cos 60^\circ + \sin 60^\circ \sin 60^\circ \cos 120^\circ$
$\cos A_1 = -\left(\frac{1}{2}\right)\left(\frac{1}{2}\right) + \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)\left(-\frac{1}{2}\right)$
$\cos A_1 = -\frac{1}{4} + \frac{3}{4} \cdot \left(-\frac{1}{2}\right)$
$\cos A_1 = -\frac{1}{4} - \frac{3}{8}$
$\cos A_1 = -\frac{2}{8} - \frac{3}{8}$
$\cos A_1 = -\frac{5}{8}$
Теперь найдем плоские углы $\alpha$ и $\gamma$. Поскольку двугранные углы $B_1$ и $C_1$ равны, то и противолежащие им плоские углы $\gamma$ и $\alpha$ будут равны, т.е. $\alpha = \gamma$.
Используем формулу для $\cos C_1$ для нахождения $\gamma$:
$\cos C_1 = -\cos A_1 \cos B_1 + \sin A_1 \sin B_1 \cos \gamma$
$\cos 60^\circ = -\left(-\frac{5}{8}\right) \cos 60^\circ + \sin A_1 \sin 60^\circ \cos \gamma$
$\frac{1}{2} = \frac{5}{8} \cdot \frac{1}{2} + \sin A_1 \cdot \frac{\sqrt{3}}{2} \cos \gamma$
$\frac{1}{2} = \frac{5}{16} + \frac{\sqrt{3}}{2} \sin A_1 \cos \gamma$
Найдем $\sin A_1$ из $\cos A_1 = -\frac{5}{8}$:
$\sin^2 A_1 = 1 - \cos^2 A_1 = 1 - \left(-\frac{5}{8}\right)^2 = 1 - \frac{25}{64} = \frac{64 - 25}{64} = \frac{39}{64}$
Так как $A_1$ является углом в трехгранном угле ($0 < A_1 < 180^\circ$), то $\sin A_1 > 0$.
$\sin A_1 = \sqrt{\frac{39}{64}} = \frac{\sqrt{39}}{8}$
Подставим $\sin A_1$ в уравнение для $\cos \gamma$:
$\frac{1}{2} = \frac{5}{16} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{39}}{8} \cos \gamma$
$\frac{1}{2} - \frac{5}{16} = \frac{\sqrt{3 \cdot 39}}{16} \cos \gamma$
$\frac{8}{16} - \frac{5}{16} = \frac{\sqrt{117}}{16} \cos \gamma$
$\frac{3}{16} = \frac{\sqrt{9 \cdot 13}}{16} \cos \gamma$
$\frac{3}{16} = \frac{3\sqrt{13}}{16} \cos \gamma$
$\cos \gamma = \frac{3}{3\sqrt{13}} = \frac{1}{\sqrt{13}}$
$\cos \gamma = \frac{\sqrt{13}}{13}$
$\gamma = \arccos\left(\frac{\sqrt{13}}{13}\right)$
Поскольку $\alpha = \gamma$, то $\alpha = \arccos\left(\frac{\sqrt{13}}{13}\right)$.
Ответ: Два других плоских угла равны $\arccos\left(\frac{\sqrt{13}}{13}\right)$.
Другие задания:
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