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Страница 92 - гдз по алгебре 10 класс учебник Шыныбеков, Шыныбеков
Авторы: Шыныбеков А. Н., Шыныбеков Д. А., Жумабаев Р. Н.
Тип: Учебник
Издательство: Атамұра
Год издания: 2019 - 2026
ISBN: 978-601-331-522-5
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Cтраница 92
App\Models\BookPage {#1030 // resources/views/models/book_page/default.blade.php #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:20 [ "id" => 1098154 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "92" "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/page-92" "field_display_title" => "92" "field_folder" => "1" "field_image_name" => "92" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1036 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1065 #items: array:1 [ 0 => App\Models\Book {#1034 #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" => 3423 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1035 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1038 #items: array:1 [ 0 => App\Models\Term {#1039 #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" => 5459 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #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 {#1040 #items: array:1 [ 0 => App\Models\Term {#1041 #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" => "2026-02-28 22:16:36" "updated_at" => null "name" => "Атамұра" "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" => "atamu'ra" ] #original: array:6 [ "id" => 6991 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "name" => "Атамұра" "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" => "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 {#1042 #items: array:3 [ 0 => App\Models\Term {#1043 #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" => "2026-02-28 22:16:36" "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" => "2026-02-28 22:16:36" "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 {#1044 #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" => "2026-02-28 22:16:36" "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" => "2026-02-28 22:16:36" "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 {#1045 #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" => "2026-02-28 22:16:36" "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" => "2026-02-28 22:16:36" "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 {#1046 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1047 #items: array:1 [ 0 => App\Models\Term {#1048 #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" => "2026-02-28 22:16:36" "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" => "2026-02-28 22:16:36" "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 {#1049 #items: array:1 [ 0 => App\Models\Term {#1050 #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" => "2026-02-28 22:16:36" "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" => "2026-02-28 22:16:36" "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 {#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" => 6995 "created_at" => "2026-02-28 22:16:36" "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" => "2026-02-28 22:16:36" "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 {#1053 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общеобразовательных школ естественно-математического направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "816" "field_priority" => "8" "field_default_folder" => "/algebra_10/shynybekov-u19/" "field_isbn" => "978-601-331-522-5" "field_cover" => array:1 [ 0 => "/media/algebra_10/shynybekov-u19/covers/cover1.webp" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/shynybekov-u19/covers/cover1.webp" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1062 #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" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 3423 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1037} "field_class" => Illuminate\Database\Eloquent\Collection {#1038} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1040} "field_author" => Illuminate\Database\Eloquent\Collection {#1042} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1046} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1047} "field_country" => Illuminate\Database\Eloquent\Collection {#1049} "field_city" => Illuminate\Database\Eloquent\Collection {#1051} "field_series" => Illuminate\Database\Eloquent\Collection {#1053} "field_umk" => Illuminate\Database\Eloquent\Collection {#1054} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1055} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1056} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1057} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1058} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общеобразовательных школ естественно-математического направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "816" "field_priority" => "8" "field_default_folder" => "/algebra_10/shynybekov-u19/" "field_isbn" => "978-601-331-522-5" "field_cover" => array:1 [ 0 => "/media/algebra_10/shynybekov-u19/covers/cover1.webp" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/shynybekov-u19/covers/cover1.webp" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1059} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1060} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1061} "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1062} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #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 "content" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:2 [ 0 => App\Models\Element {#1074 #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" => 1261381 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1083 #items: array:1 [ 0 => App\Models\Edition {#1075 #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" => 4955 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "title" => "Учебник рус" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1077 #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" => "rus0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1079 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4955 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "title" => "Учебник рус" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1077} "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" => "rus0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1078} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1079} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080} "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 } "book_page" => array:2 [ "refs" => "1098154" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "92-1.jpg" "alt" => null "width" => "1587" "height" => 2515 "path" => "/media/algebra_10/shynybekov-u19/rus0-1/92-1.webp" ] ] ] #original: array:6 [ "id" => 1261381 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1083} "book_page" => array:2 [ "refs" => "1098154" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "92-1.jpg" "alt" => null "width" => "1587" "height" => 2515 "path" => "/media/algebra_10/shynybekov-u19/rus0-1/92-1.webp" ] ] ] #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 {#1081 #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" => 1261406 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091 #items: array:1 [ 0 => App\Models\Edition {#1082 #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" => 4956 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "title" => "Учебник кз" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ …1] #escapeWhenCastingToString: false } "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" => "kz0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4956 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "title" => "Учебник кз" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1085} "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" => "kz0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1086} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1087} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088} "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 } "book_page" => array:2 [ "refs" => "1098154" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "92-1.jpg" "alt" => null "width" => "1645" "height" => 2535 "path" => "/media/algebra_10/shynybekov-u19/kz0-1/92-1.webp" ] ] ] #original: array:6 [ "id" => 1261406 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1091} "book_page" => array:2 [ "refs" => "1098154" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "92-1.jpg" "alt" => null "width" => "1645" "height" => 2535 "path" => "/media/algebra_10/shynybekov-u19/kz0-1/92-1.webp" ] ] ] #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" => Illuminate\Database\Eloquent\Collection {#3619 #items: array:1 [ 0 => App\Models\BookPage {#3612 #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:20 [ "id" => 1098155 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "93" "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/page-93" "field_display_title" => "93" "field_folder" => "1" "field_image_name" => "93" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#3629 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#3628 #items: array:1 [ 0 => App\Models\Book {#1034} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "content" => Illuminate\Database\Eloquent\Collection {#3616 #items: array:2 [ 0 => App\Models\Element {#3734 #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" => 1261383 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#3736 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1098155" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1261383 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#3736} "book_page" => array:2 [ "refs" => "1098155" "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 => "*" ] } 1 => App\Models\Element {#3732 #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" => 1261409 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#3715 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1098155" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1261409 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#3715} "book_page" => array:2 [ "refs" => "1098155" "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" => "1098156" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098154" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#3717 #items: [] #escapeWhenCastingToString: false } ] #original: array:20 [ "id" => 1098155 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "93" "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/page-93" "field_display_title" => "93" "field_folder" => "1" "field_image_name" => "93" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#3629} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#3628} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "content" => Illuminate\Database\Eloquent\Collection {#3616} "next" => array:2 [ "refs" => "1098156" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098154" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#3717} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "previous" => Illuminate\Database\Eloquent\Collection {#3609 #items: array:1 [ 0 => App\Models\BookPage {#3608 #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:20 [ "id" => 1098153 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "91" "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/page-91" "field_display_title" => "91" "field_folder" => "1" "field_image_name" => "91" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#3610 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#3611 #items: array:1 [ 0 => App\Models\Book {#1034} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "content" => Illuminate\Database\Eloquent\Collection {#3613 #items: array:2 [ 0 => App\Models\Element {#3622 #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" => 1261379 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#3623 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1098153" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1261379 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#3623} "book_page" => array:2 [ "refs" => "1098153" "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 => "*" ] } 1 => App\Models\Element {#3624 #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" => 1261404 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#3625 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1098153" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1261404 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#3625} "book_page" => array:2 [ "refs" => "1098153" "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" => "1098154" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098152" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#3614 #items: array:6 [ 0 => App\Models\Task {#3632 #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" => 1110099 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/3-46" "field_display_title" => "3.46" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#3636 #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 {#3635 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#3634 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#3672 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#3637 #items: array:4 [ …4] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1110100" "type" => "task" ] "previous" => array:2 [ "refs" => "1110098" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#3663 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098153" "type" => "book_page" ] ] #original: array:24 [ "id" => 1110099 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/3-46" "field_display_title" => "3.46" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#3636} "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 {#3635} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#3634} "parent_branches" => Illuminate\Database\Eloquent\Collection {#3672} "content" => Illuminate\Database\Eloquent\Collection {#3637} "next" => array:2 [ "refs" => "1110100" "type" => "task" ] "previous" => array:2 [ "refs" => "1110098" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#3663} "page" => array:2 [ "refs" => "1098153" "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 {#3669 #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" => 1110100 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/3-47" "field_display_title" => "3.47" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#3677 #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 {#3665 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#3666 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#3664 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#3640 #items: array:4 [ …4] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1110101" "type" => "task" ] "previous" => array:2 [ "refs" => "1110099" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#3646 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098153" "type" => "book_page" ] ] #original: array:24 [ "id" => 1110100 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/3-47" "field_display_title" => "3.47" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#3677} "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 {#3665} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#3666} "parent_branches" => Illuminate\Database\Eloquent\Collection {#3664} "content" => Illuminate\Database\Eloquent\Collection {#3640} "next" => array:2 [ "refs" => "1110101" "type" => "task" ] "previous" => array:2 [ "refs" => "1110099" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#3646} "page" => array:2 [ "refs" => "1098153" "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 {#3667 #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" => 1110101 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/3-48" "field_display_title" => "3.48" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#3668 #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 {#3650 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#3651 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#3649 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#3641 #items: array:4 [ …4] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1110102" "type" => "task" ] "previous" => array:2 [ "refs" => "1110100" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#3689 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1110101 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/3-48" "field_display_title" => "3.48" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#3668} "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 {#3650} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#3651} "parent_branches" => Illuminate\Database\Eloquent\Collection {#3649} "content" => Illuminate\Database\Eloquent\Collection {#3641} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#3689} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#3648 #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" => 1110102 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/3-49" "field_display_title" => "3.49" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#3652 …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 {#3674 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#3676 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#3688 …2} "content" => Illuminate\Database\Eloquent\Collection {#3654 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#3705 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1110102 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/3-49" "field_display_title" => "3.49" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#3652 …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 {#3674 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#3676 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#3688 …2} "content" => Illuminate\Database\Eloquent\Collection {#3654 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#3705 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#3673 #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" => 1110103 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/3-50" "field_display_title" => "3.50" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#3675 …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 {#3703 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#3702 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#3704 …2} "content" => Illuminate\Database\Eloquent\Collection {#3679 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#3721 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1110103 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/3-50" "field_display_title" => "3.50" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#3675 …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 {#3703 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#3702 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#3704 …2} "content" => Illuminate\Database\Eloquent\Collection {#3679 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#3721 …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 {#3701 #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" => 1110104 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/3-51" "field_display_title" => "3.51" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#3700 …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 {#3719 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#3718 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#3720 …2} "content" => Illuminate\Database\Eloquent\Collection {#3699 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#3737 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1110104 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "91" "field_page_end" => null "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/3-51" "field_display_title" => "3.51" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#3700 …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 {#3719 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#3718 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#3720 …2} "content" => Illuminate\Database\Eloquent\Collection {#3699 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#3737 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:20 [ "id" => 1098153 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "91" "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/page-91" "field_display_title" => "91" "field_folder" => "1" "field_image_name" => "91" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#3610} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#3611} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "content" => Illuminate\Database\Eloquent\Collection {#3613} "next" => array:2 [ "refs" => "1098154" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098152" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#3614} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "tasks" => Illuminate\Database\Eloquent\Collection {#1090 #items: array:1 [ 0 => App\Models\Task {#1089 #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" => 1110105 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "92" "field_page_end" => null "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/dokazhi-92" "field_display_title" => "Докажите самостоятельно" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1092 #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 {#1093 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\Branch {#1102 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:23 [ "id" => 1109815 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_display_title" => "Тригонометрические уравнения и их системы" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_page_start" => "70" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1106 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1105 …2} ] #original: array:23 [ "id" => 1109815 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_display_title" => "Тригонометрические уравнения и их системы" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1107 …2} "field_page_start" => "70" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1106 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1105 …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 } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1136 #items: array:2 [ 0 => App\Models\Branch {#1144 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:23 [ "id" => 1109815 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_display_title" => "Тригонометрические уравнения и их системы" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1145 …2} "field_page_start" => "70" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1146 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1147 …2} ] #original: array:23 [ "id" => 1109815 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_display_title" => "Тригонометрические уравнения и их системы" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1145 …2} "field_page_start" => "70" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1146 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1147 …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\Branch {#1148 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:23 [ "id" => 1109818 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_display_title" => "3.3. Обратные тригонометрические уравнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_page_start" => "92" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1150 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1151 …2} ] #original: array:23 [ "id" => 1109818 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_display_title" => "3.3. Обратные тригонометрические уравнения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_page_start" => "92" "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" => [] "book" => Illuminate\Database\Eloquent\Collection {#1150 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1151 …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 } "content" => Illuminate\Database\Eloquent\Collection {#1095 #items: array:4 [ 0 => 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:7 [ "id" => 1284380 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1124 …2} "task" => array:2 [ …2] "text" => "<p><strong>Докажите самостоятельно</strong></p><p>Докажите формулы 2, 3, 4.</p>" "img" => array:1 [ …1] ] #original: array:7 [ "id" => 1284380 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1124 …2} "task" => array:2 [ …2] "text" => "<p><strong>Докажите самостоятельно</strong></p><p>Докажите формулы 2, 3, 4.</p>" "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 {#1123 #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" => 1284912 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1121 …2} "task" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1284912 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1121 …2} "task" => 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 => "*" ] } 2 => App\Models\Element {#1122 #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" => 1285727 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1113 …2} "task" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1285727 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1113 …2} "task" => 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 => "*" ] } 3 => App\Models\Element {#1115 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1543813 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1096 …2} "task" => array:2 [ …2] "text" => "<p>Поскольку в условии задачи не указаны сами формулы, будем доказывать наиболее вероятный набор стандартных тригонометрических тождеств, которые часто нумеруются и доказываются в такой последовательности. Будем считать, что основная формула косинуса разности уже известна или доказана ранее (как формула 1): $cos(α - β) = cosα cosβ + sinα sinβ$.</p><p><strong>2.</strong> Доказательство формулы косинуса суммы: $cos(α + β) = cosα cosβ - sinα sinβ$.<br>Представим сумму $α + β$ в виде разности $α - (-β)$.<br>Применим формулу косинуса разности, подставив $x = α$ и $y = -β$:<br>$cos(α + β) = cos(α - (-β)) = cosα cos(-β) + sinα sin(-β)$.<br>Воспользуемся свойствами четности и нечетности тригонометрических функций: $cos(-β) = cosβ$ (косинус — четная функция) и $sin(-β) = -sinβ$ (синус — нечетная функция).<br>Подставим эти выражения в нашу формулу:<br>$cosα cosβ + sinα (-sinβ) = cosα cosβ - sinα sinβ$.<br>Таким образом, доказано, что $cos(α + β) = cosα cosβ - sinα sinβ$.<br>Ответ: Формула $cos(α + β) = cosα cosβ - sinα sinβ$ доказана.</p><p><strong>3.</strong> Доказательство формулы синуса разности: $sin(α - β) = sinα cosβ - cosα sinβ$.<br>Воспользуемся формулой приведения $sin(x) = cos(\frac{π}{2} - x)$.<br>Применим ее к нашему выражению, взяв $x = α - β$:<br>$sin(α - β) = cos(\frac{π}{2} - (α - β)) = cos((\frac{π}{2} - α) + β)$.<br>Теперь используем доказанную в пункте 2 формулу косинуса суммы $cos(x + y) = cosx cosy - sinx siny$, где $x = \frac{π}{2} - α$ и $y = β$:<br>$cos(\frac{π}{2} - α)cosβ - sin(\frac{π}{2} - α)sinβ$.<br>Снова применим формулы приведения: $cos(\frac{π}{2} - α) = sinα$ и $sin(\frac{π}{2} - α) = cosα$.<br>Подставим их в выражение:<br>$sinα cosβ - cosα sinβ$.<br>Таким образом, доказано, что $sin(α - β) = sinα cosβ - cosα sinβ$.<br>Ответ: Формула $sin(α - β) = sinα cosβ - cosα sinβ$ доказана.</p><p><strong>4.</strong> Доказательство формулы синуса суммы: $sin(α + β) = sinα cosβ + cosα sinβ$.<br>Представим сумму $α + β$ в виде разности $α - (-β)$.<br>Применим доказанную в пункте 3 формулу синуса разности $sin(x - y) = sinx cosy - cosx siny$, подставив $x = α$ и $y = -β$:<br>$sin(α + β) = sin(α - (-β)) = sinα cos(-β) - cosα sin(-β)$.<br>Используем свойства четности косинуса ($cos(-β) = cosβ$) и нечетности синуса ($sin(-β) = -sinβ$):<br>$sinα cosβ - cosα (-sinβ) = sinα cosβ + cosα sinβ$.<br>Таким образом, доказано, что $sin(α + β) = sinα cosβ + cosα sinβ$.<br>Ответ: Формула $sin(α + β) = sinα cosβ + cosα sinβ$ доказана.</p>" ] #original: array:6 [ "id" => 1543813 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1096 …2} "task" => array:2 [ …2] "text" => "<p>Поскольку в условии задачи не указаны сами формулы, будем доказывать наиболее вероятный набор стандартных тригонометрических тождеств, которые часто нумеруются и доказываются в такой последовательности. Будем считать, что основная формула косинуса разности уже известна или доказана ранее (как формула 1): $cos(α - β) = cosα cosβ + sinα sinβ$.</p><p><strong>2.</strong> Доказательство формулы косинуса суммы: $cos(α + β) = cosα cosβ - sinα sinβ$.<br>Представим сумму $α + β$ в виде разности $α - (-β)$.<br>Применим формулу косинуса разности, подставив $x = α$ и $y = -β$:<br>$cos(α + β) = cos(α - (-β)) = cosα cos(-β) + sinα sin(-β)$.<br>Воспользуемся свойствами четности и нечетности тригонометрических функций: $cos(-β) = cosβ$ (косинус — четная функция) и $sin(-β) = -sinβ$ (синус — нечетная функция).<br>Подставим эти выражения в нашу формулу:<br>$cosα cosβ + sinα (-sinβ) = cosα cosβ - sinα sinβ$.<br>Таким образом, доказано, что $cos(α + β) = cosα cosβ - sinα sinβ$.<br>Ответ: Формула $cos(α + β) = cosα cosβ - sinα sinβ$ доказана.</p><p><strong>3.</strong> Доказательство формулы синуса разности: $sin(α - β) = sinα cosβ - cosα sinβ$.<br>Воспользуемся формулой приведения $sin(x) = cos(\frac{π}{2} - x)$.<br>Применим ее к нашему выражению, взяв $x = α - β$:<br>$sin(α - β) = cos(\frac{π}{2} - (α - β)) = cos((\frac{π}{2} - α) + β)$.<br>Теперь используем доказанную в пункте 2 формулу косинуса суммы $cos(x + y) = cosx cosy - sinx siny$, где $x = \frac{π}{2} - α$ и $y = β$:<br>$cos(\frac{π}{2} - α)cosβ - sin(\frac{π}{2} - α)sinβ$.<br>Снова применим формулы приведения: $cos(\frac{π}{2} - α) = sinα$ и $sin(\frac{π}{2} - α) = cosα$.<br>Подставим их в выражение:<br>$sinα cosβ - cosα sinβ$.<br>Таким образом, доказано, что $sin(α - β) = sinα cosβ - cosα sinβ$.<br>Ответ: Формула $sin(α - β) = sinα cosβ - cosα sinβ$ доказана.</p><p><strong>4.</strong> Доказательство формулы синуса суммы: $sin(α + β) = sinα cosβ + cosα sinβ$.<br>Представим сумму $α + β$ в виде разности $α - (-β)$.<br>Применим доказанную в пункте 3 формулу синуса разности $sin(x - y) = sinx cosy - cosx siny$, подставив $x = α$ и $y = -β$:<br>$sin(α + β) = sin(α - (-β)) = sinα cos(-β) - cosα sin(-β)$.<br>Используем свойства четности косинуса ($cos(-β) = cosβ$) и нечетности синуса ($sin(-β) = -sinβ$):<br>$sinα cosβ - cosα (-sinβ) = sinα cosβ + cosα sinβ$.<br>Таким образом, доказано, что $sin(α + β) = sinα cosβ + cosα sinβ$.<br>Ответ: Формула $sin(α + β) = sinα cosβ + cosα sinβ$ доказана.</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" => "1110106" "type" => "task" ] "previous" => array:2 [ "refs" => "1110104" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1127 #items: array:1 [ 0 => App\Models\Book {#1034} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098154" "type" => "book_page" ] ] #original: array:24 [ "id" => 1110105 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "92" "field_page_end" => null "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/dokazhi-92" "field_display_title" => "Докажите самостоятельно" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1092} "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 {#1093} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1103} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1136} "content" => Illuminate\Database\Eloquent\Collection {#1095} "next" => array:2 [ "refs" => "1110106" "type" => "task" ] "previous" => array:2 [ "refs" => "1110104" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1127} "page" => array:2 [ "refs" => "1098154" "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 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:20 [ "id" => 1098154 "created_at" => "2026-02-28 22:16:36" "updated_at" => null "field_page_start" => "92" "field_url" => "/10-klass/algebra/atamyra-shynybekov-uchebnik/page-92" "field_display_title" => "92" "field_folder" => "1" "field_image_name" => "92" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1036} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1065} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "content" => Illuminate\Database\Eloquent\Collection {#1064} "next" => array:2 [ "refs" => "1098155" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098153" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1090} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
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