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Номер 674, страница 194 - гдз по алгебре 9 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2019 - 2026
ISBN: 978-601-317-424-2
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 9 классе
IV. Тригонометрия. 25. Формулы приведения - номер 674, страница 194.
App\Models\Task {#1024 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "194" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-674" "field_display_title" => "674" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1035 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1050 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037} "field_author" => Illuminate\Database\Eloquent\Collection {#1039} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044} "field_country" => Illuminate\Database\Eloquent\Collection {#1046} "field_city" => Illuminate\Database\Eloquent\Collection {#1048} "field_series" => Illuminate\Database\Eloquent\Collection {#1050} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107667 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "25. Формулы приведения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "189" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Book {#1065 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1066 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1068 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => 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 "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1066} "field_class" => Illuminate\Database\Eloquent\Collection {#1068} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070} "field_author" => Illuminate\Database\Eloquent\Collection {#1072} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077} "field_country" => Illuminate\Database\Eloquent\Collection {#1079} "field_city" => Illuminate\Database\Eloquent\Collection {#1081} "field_series" => Illuminate\Database\Eloquent\Collection {#1083} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Branch {#1094 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1096 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1096} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107667 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "25. Формулы приведения" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "189" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1064} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:3 [ 0 => App\Models\Element {#1099 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1277029 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1101 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 #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 {#1104 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1104} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106} "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" => "1108411" "type" => "task" ] "text" => "<p><strong>674.</strong> Упростите выражение:</p><p><strong>а)</strong> $ \frac{\text{tg}(5\pi - x) + \text{tg}(10\pi - x) - \text{tg}(21\pi + x)}{\text{sin } 6,5\pi - \text{cos } 15\pi + \text{cos } 20\pi} $</p><p><strong>б)</strong> $ \frac{\text{cos}(5\pi - x) + \text{sin}(11\pi + x)}{\text{tg}(\pi - x) \cdot \text{tg}(0,5\pi + x) + \text{tg}(20\pi + x)} $</p><p><strong>в)</strong> $ \frac{\text{tg}^2(9\pi - x) \cdot \text{sin}^2(x - 4,5\pi)}{\text{sin}(19\pi - x) \cdot \text{sin}(-x - 15\pi)} $</p><p><strong>г)</strong> $ \frac{\text{cos}^2(5,5\pi - x)}{\text{tg}^2(x - 10\pi)} + \frac{\text{cos}^2(-x)}{\text{tg}^2(x - 7,5\pi)} $</p>" "img" => array:1 [ 0 => array:5 [ "name" => "674-1.jpg" "alt" => null "width" => "1001" "height" => 700 "path" => "/media/algebra_09/soltan-u19/0-1/674-1.webp?ts=1752834668" ] ] ] #original: array:7 [ "id" => 1277029 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1108411" "type" => "task" ] "text" => "<p><strong>674.</strong> Упростите выражение:</p><p><strong>а)</strong> $ \frac{\text{tg}(5\pi - x) + \text{tg}(10\pi - x) - \text{tg}(21\pi + x)}{\text{sin } 6,5\pi - \text{cos } 15\pi + \text{cos } 20\pi} $</p><p><strong>б)</strong> $ \frac{\text{cos}(5\pi - x) + \text{sin}(11\pi + x)}{\text{tg}(\pi - x) \cdot \text{tg}(0,5\pi + x) + \text{tg}(20\pi + x)} $</p><p><strong>в)</strong> $ \frac{\text{tg}^2(9\pi - x) \cdot \text{sin}^2(x - 4,5\pi)}{\text{sin}(19\pi - x) \cdot \text{sin}(-x - 15\pi)} $</p><p><strong>г)</strong> $ \frac{\text{cos}^2(5,5\pi - x)}{\text{tg}^2(x - 10\pi)} + \frac{\text{cos}^2(-x)}{\text{tg}^2(x - 7,5\pi)} $</p>" "img" => array:1 [ 0 => array:5 [ "name" => "674-1.jpg" "alt" => null "width" => "1001" "height" => 700 "path" => "/media/algebra_09/soltan-u19/0-1/674-1.webp?ts=1752834668" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1107 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1278320 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1109 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 #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 {#1112 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1112} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114} "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" => "1108411" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "674-1.jpg" "alt" => null "width" => "1439" "height" => 1498 "path" => "/media/algebra_09/soltan-u19/1-1/674-1.webp?ts=1752831451" ] 1 => array:5 [ "name" => "674-2.jpg" "alt" => null "width" => "1252" "height" => 419 "path" => "/media/algebra_09/soltan-u19/1-1/674-2.webp?ts=1752831451" ] ] ] #original: array:6 [ "id" => 1278320 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1108411" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "674-1.jpg" "alt" => null "width" => "1439" "height" => 1498 "path" => "/media/algebra_09/soltan-u19/1-1/674-1.webp?ts=1752831451" ] 1 => array:5 [ "name" => "674-2.jpg" "alt" => null "width" => "1252" "height" => 419 "path" => "/media/algebra_09/soltan-u19/1-1/674-2.webp?ts=1752831451" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1115 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1554275 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1117 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1118 #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 {#1120 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1118} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1120} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122} "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" => "1108411" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Для упрощения выражения воспользуемся формулами приведения и свойством периодичности тригонометрических функций. Сначала упростим числитель и знаменатель по отдельности.</p><p>Упростим числитель:</p><p>Используя периодичность тангенса ($T = \pi$) и формулы приведения:</p><p>$ \text{tg}(5\pi - x) = \text{tg}(\pi - x) = -\text{tg}(x) $</p><p>$ \text{tg}(10\pi - x) = \text{tg}(-x) = -\text{tg}(x) $</p><p>$ \text{tg}(21\pi + x) = \text{tg}(\pi + x) = \text{tg}(x) $</p><p>Тогда числитель равен: $ -\text{tg}(x) + (-\text{tg}(x)) - \text{tg}(x) = -3\text{tg}(x) $.</p><p>Упростим знаменатель, вычислив значения тригонометрических функций:</p><p>$ \sin(6,5\pi) = \sin(6\pi + \frac{\pi}{2}) = \sin(\frac{\pi}{2}) = 1 $</p><p>$ \cos(15\pi) = \cos(14\pi + \pi) = \cos(\pi) = -1 $</p><p>$ \cos(20\pi) = \cos(0) = 1 $</p><p>Тогда знаменатель равен: $ 1 - (-1) + 1 = 3 $.</p><p>Подставим упрощенные части в исходное выражение:</p><p>$ \frac{-3\text{tg}(x)}{3} = -\text{tg}(x) $</p><p><strong>Ответ:</strong> $-\text{tg}(x)$</p><p><strong>б)</strong></p><p>Упростим числитель и знаменатель дроби, используя формулы приведения и периодичность.</p><p>Числитель:</p><p>$ \cos(5\pi - x) = \cos(4\pi + \pi - x) = \cos(\pi - x) = -\cos(x) $</p><p>$ \sin(11\pi + x) = \sin(10\pi + \pi + x) = \sin(\pi + x) = -\sin(x) $</p><p>Следовательно, числитель равен $ -\cos(x) - \sin(x) = -(\cos(x) + \sin(x)) $.</p><p>Знаменатель:</p><p>$ \text{tg}(\pi - x) = -\text{tg}(x) $</p><p>$ \text{tg}(0,5\pi + x) = \text{tg}(\frac{\pi}{2} + x) = -\text{ctg}(x) $</p><p>$ \text{tg}(20\pi + x) = \text{tg}(x) $</p><p>Произведение в знаменателе: $ \text{tg}(\pi - x) \cdot \text{tg}(0,5\pi + x) = (-\text{tg}(x)) \cdot (-\text{ctg}(x)) = \text{tg}(x)\text{ctg}(x) = 1 $.</p><p>Следовательно, знаменатель равен $ 1 + \text{tg}(x) $.</p><p>Соберем дробь и упростим, выразив тангенс через синус и косинус:</p><p>$ \frac{-(\cos(x) + \sin(x))}{1 + \text{tg}(x)} = \frac{-(\cos(x) + \sin(x))}{1 + \frac{\sin(x)}{\cos(x)}} = \frac{-(\cos(x) + \sin(x))}{\frac{\cos(x) + \sin(x)}{\cos(x)}} = -(\cos(x) + \sin(x)) \cdot \frac{\cos(x)}{\cos(x) + \sin(x)} = -\cos(x) $</p><p><strong>Ответ:</strong> $-\cos(x)$</p><p><strong>в)</strong></p><p>Упростим числитель и знаменатель по отдельности.</p><p>Упростим числитель:</p><p>$ \text{tg}^2(9\pi - x) = (\text{tg}(9\pi - x))^2 = (\text{tg}(\pi - x))^2 = (-\text{tg}(x))^2 = \text{tg}^2(x) $.</p><p>$ \sin^2(x - 4,5\pi) = \sin^2(x - \frac{9\pi}{2}) = \sin^2(-(\frac{9\pi}{2} - x)) = \sin^2(\frac{9\pi}{2} - x) = \sin^2(4\pi + \frac{\pi}{2} - x) = \sin^2(\frac{\pi}{2} - x) = \cos^2(x) $.</p><p>Числитель равен: $ \text{tg}^2(x) \cdot \cos^2(x) = \frac{\sin^2(x)}{\cos^2(x)} \cdot \cos^2(x) = \sin^2(x) $.</p><p>Упростим знаменатель:</p><p>$ \sin(19\pi - x) = \sin(18\pi + \pi - x) = \sin(\pi - x) = \sin(x) $.</p><p>$ \sin(-x - 15\pi) = -\sin(x + 15\pi) = -\sin(x + 14\pi + \pi) = -\sin(x + \pi) = -(-\sin(x)) = \sin(x) $.</p><p>Знаменатель равен: $ \sin(x) \cdot \sin(x) = \sin^2(x) $.</p><p>Таким образом, выражение равно:</p><p>$ \frac{\sin^2(x)}{\sin^2(x)} = 1 $</p><p><strong>Ответ:</strong> $1$</p><p><strong>г)</strong></p><p>Упростим каждое слагаемое отдельно.</p><p>Первое слагаемое: $ \frac{\cos^2(5,5\pi - x)}{\text{tg}^2(x - 10\pi)} $.</p><p>Числитель: $ \cos^2(5,5\pi - x) = \cos^2(\frac{11\pi}{2} - x) $. По формуле приведения $ \cos(\frac{11\pi}{2} - x) = \cos(4\pi + \frac{3\pi}{2} - x) = \cos(\frac{3\pi}{2} - x) = -\sin(x) $. Тогда числитель равен $ (-\sin(x))^2 = \sin^2(x) $.</p><p>Знаменатель: $ \text{tg}^2(x - 10\pi) = \text{tg}^2(x) $ из-за периодичности тангенса ($T=\pi$).</p><p>Первое слагаемое равно: $ \frac{\sin^2(x)}{\text{tg}^2(x)} = \frac{\sin^2(x)}{\frac{\sin^2(x)}{\cos^2(x)}} = \cos^2(x) $.</p><p>Второе слагаемое: $ \frac{\cos^2(-x)}{\text{tg}^2(x - 7,5\pi)} $.</p><p>Числитель: $ \cos^2(-x) = \cos^2(x) $, так как косинус - четная функция.</p><p>Знаменатель: $ \text{tg}^2(x - 7,5\pi) = \text{tg}^2(x - \frac{15\pi}{2}) $. Используя периодичность, $ \text{tg}(x - \frac{15\pi}{2}) = \text{tg}(x - \frac{15\pi}{2} + 8\pi) = \text{tg}(x + \frac{\pi}{2}) = -\text{ctg}(x) $. Тогда знаменатель равен $ (-\text{ctg}(x))^2 = \text{ctg}^2(x) $.</p><p>Второе слагаемое равно: $ \frac{\cos^2(x)}{\text{ctg}^2(x)} = \frac{\cos^2(x)}{\frac{\cos^2(x)}{\sin^2(x)}} = \sin^2(x) $.</p><p>Суммируем оба слагаемых, используя основное тригонометрическое тождество:</p><p>$ \cos^2(x) + \sin^2(x) = 1 $.</p><p><strong>Ответ:</strong> $1$</p>" ] #original: array:6 [ "id" => 1554275 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1108411" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Для упрощения выражения воспользуемся формулами приведения и свойством периодичности тригонометрических функций. Сначала упростим числитель и знаменатель по отдельности.</p><p>Упростим числитель:</p><p>Используя периодичность тангенса ($T = \pi$) и формулы приведения:</p><p>$ \text{tg}(5\pi - x) = \text{tg}(\pi - x) = -\text{tg}(x) $</p><p>$ \text{tg}(10\pi - x) = \text{tg}(-x) = -\text{tg}(x) $</p><p>$ \text{tg}(21\pi + x) = \text{tg}(\pi + x) = \text{tg}(x) $</p><p>Тогда числитель равен: $ -\text{tg}(x) + (-\text{tg}(x)) - \text{tg}(x) = -3\text{tg}(x) $.</p><p>Упростим знаменатель, вычислив значения тригонометрических функций:</p><p>$ \sin(6,5\pi) = \sin(6\pi + \frac{\pi}{2}) = \sin(\frac{\pi}{2}) = 1 $</p><p>$ \cos(15\pi) = \cos(14\pi + \pi) = \cos(\pi) = -1 $</p><p>$ \cos(20\pi) = \cos(0) = 1 $</p><p>Тогда знаменатель равен: $ 1 - (-1) + 1 = 3 $.</p><p>Подставим упрощенные части в исходное выражение:</p><p>$ \frac{-3\text{tg}(x)}{3} = -\text{tg}(x) $</p><p><strong>Ответ:</strong> $-\text{tg}(x)$</p><p><strong>б)</strong></p><p>Упростим числитель и знаменатель дроби, используя формулы приведения и периодичность.</p><p>Числитель:</p><p>$ \cos(5\pi - x) = \cos(4\pi + \pi - x) = \cos(\pi - x) = -\cos(x) $</p><p>$ \sin(11\pi + x) = \sin(10\pi + \pi + x) = \sin(\pi + x) = -\sin(x) $</p><p>Следовательно, числитель равен $ -\cos(x) - \sin(x) = -(\cos(x) + \sin(x)) $.</p><p>Знаменатель:</p><p>$ \text{tg}(\pi - x) = -\text{tg}(x) $</p><p>$ \text{tg}(0,5\pi + x) = \text{tg}(\frac{\pi}{2} + x) = -\text{ctg}(x) $</p><p>$ \text{tg}(20\pi + x) = \text{tg}(x) $</p><p>Произведение в знаменателе: $ \text{tg}(\pi - x) \cdot \text{tg}(0,5\pi + x) = (-\text{tg}(x)) \cdot (-\text{ctg}(x)) = \text{tg}(x)\text{ctg}(x) = 1 $.</p><p>Следовательно, знаменатель равен $ 1 + \text{tg}(x) $.</p><p>Соберем дробь и упростим, выразив тангенс через синус и косинус:</p><p>$ \frac{-(\cos(x) + \sin(x))}{1 + \text{tg}(x)} = \frac{-(\cos(x) + \sin(x))}{1 + \frac{\sin(x)}{\cos(x)}} = \frac{-(\cos(x) + \sin(x))}{\frac{\cos(x) + \sin(x)}{\cos(x)}} = -(\cos(x) + \sin(x)) \cdot \frac{\cos(x)}{\cos(x) + \sin(x)} = -\cos(x) $</p><p><strong>Ответ:</strong> $-\cos(x)$</p><p><strong>в)</strong></p><p>Упростим числитель и знаменатель по отдельности.</p><p>Упростим числитель:</p><p>$ \text{tg}^2(9\pi - x) = (\text{tg}(9\pi - x))^2 = (\text{tg}(\pi - x))^2 = (-\text{tg}(x))^2 = \text{tg}^2(x) $.</p><p>$ \sin^2(x - 4,5\pi) = \sin^2(x - \frac{9\pi}{2}) = \sin^2(-(\frac{9\pi}{2} - x)) = \sin^2(\frac{9\pi}{2} - x) = \sin^2(4\pi + \frac{\pi}{2} - x) = \sin^2(\frac{\pi}{2} - x) = \cos^2(x) $.</p><p>Числитель равен: $ \text{tg}^2(x) \cdot \cos^2(x) = \frac{\sin^2(x)}{\cos^2(x)} \cdot \cos^2(x) = \sin^2(x) $.</p><p>Упростим знаменатель:</p><p>$ \sin(19\pi - x) = \sin(18\pi + \pi - x) = \sin(\pi - x) = \sin(x) $.</p><p>$ \sin(-x - 15\pi) = -\sin(x + 15\pi) = -\sin(x + 14\pi + \pi) = -\sin(x + \pi) = -(-\sin(x)) = \sin(x) $.</p><p>Знаменатель равен: $ \sin(x) \cdot \sin(x) = \sin^2(x) $.</p><p>Таким образом, выражение равно:</p><p>$ \frac{\sin^2(x)}{\sin^2(x)} = 1 $</p><p><strong>Ответ:</strong> $1$</p><p><strong>г)</strong></p><p>Упростим каждое слагаемое отдельно.</p><p>Первое слагаемое: $ \frac{\cos^2(5,5\pi - x)}{\text{tg}^2(x - 10\pi)} $.</p><p>Числитель: $ \cos^2(5,5\pi - x) = \cos^2(\frac{11\pi}{2} - x) $. По формуле приведения $ \cos(\frac{11\pi}{2} - x) = \cos(4\pi + \frac{3\pi}{2} - x) = \cos(\frac{3\pi}{2} - x) = -\sin(x) $. Тогда числитель равен $ (-\sin(x))^2 = \sin^2(x) $.</p><p>Знаменатель: $ \text{tg}^2(x - 10\pi) = \text{tg}^2(x) $ из-за периодичности тангенса ($T=\pi$).</p><p>Первое слагаемое равно: $ \frac{\sin^2(x)}{\text{tg}^2(x)} = \frac{\sin^2(x)}{\frac{\sin^2(x)}{\cos^2(x)}} = \cos^2(x) $.</p><p>Второе слагаемое: $ \frac{\cos^2(-x)}{\text{tg}^2(x - 7,5\pi)} $.</p><p>Числитель: $ \cos^2(-x) = \cos^2(x) $, так как косинус - четная функция.</p><p>Знаменатель: $ \text{tg}^2(x - 7,5\pi) = \text{tg}^2(x - \frac{15\pi}{2}) $. Используя периодичность, $ \text{tg}(x - \frac{15\pi}{2}) = \text{tg}(x - \frac{15\pi}{2} + 8\pi) = \text{tg}(x + \frac{\pi}{2}) = -\text{ctg}(x) $. Тогда знаменатель равен $ (-\text{ctg}(x))^2 = \text{ctg}^2(x) $.</p><p>Второе слагаемое равно: $ \frac{\cos^2(x)}{\text{ctg}^2(x)} = \frac{\cos^2(x)}{\frac{\cos^2(x)}{\sin^2(x)}} = \sin^2(x) $.</p><p>Суммируем оба слагаемых, используя основное тригонометрическое тождество:</p><p>$ \cos^2(x) + \sin^2(x) = 1 $.</p><p><strong>Ответ:</strong> $1$</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" => "1108412" "type" => "task" ] "previous" => array:2 [ "refs" => "1108410" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Book {#1124 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] 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"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" => "2026-04-10 13:58:26" "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 => "*" ] } ] 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"field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1142 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1143 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1144 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1146 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1147 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1148 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1125} "field_class" => Illuminate\Database\Eloquent\Collection {#1127} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1129} "field_author" => Illuminate\Database\Eloquent\Collection {#1131} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1135} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1136} "field_country" => Illuminate\Database\Eloquent\Collection {#1138} "field_city" => Illuminate\Database\Eloquent\Collection {#1140} "field_series" => Illuminate\Database\Eloquent\Collection {#1142} "field_umk" => Illuminate\Database\Eloquent\Collection {#1143} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1144} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1145} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1147} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1148} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1347 #items: array:1 [ 0 => App\Models\BookPage {#1155 #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" => 1097936 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "194" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-194" "field_display_title" => "194" "field_folder" => "folder1" "field_image_name" => "194" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1159 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Book {#1160 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1161 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1161 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1163 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1165 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1174 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1188 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1189 #items: array:1 [ 0 => App\Models\Element {#1190 #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" => 1261087 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261087 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1097937" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097935" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:5 [ 0 => App\Models\Task {#1199 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108408 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "194" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-671" "field_display_title" => "671" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200 …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 {#1202 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236 …2} "content" => Illuminate\Database\Eloquent\Collection {#1273 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1292 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108408 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "194" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-671" "field_display_title" => "671" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1200 …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 {#1202 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1203 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1236 …2} "content" => Illuminate\Database\Eloquent\Collection {#1273 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1292 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1293 #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" => 1108409 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "194" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-672" "field_display_title" => "672" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1294 …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 {#1295 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1296 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1297 …2} "content" => Illuminate\Database\Eloquent\Collection {#1298 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1305 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108409 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "194" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-672" "field_display_title" => "672" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1294 …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 {#1295 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1296 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1297 …2} "content" => Illuminate\Database\Eloquent\Collection {#1298 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1305 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1306 #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" => 1108410 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "194" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-673" "field_display_title" => "673" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1307 …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 {#1308 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1309 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1310 …2} "content" => Illuminate\Database\Eloquent\Collection {#1311 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1318 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108410 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "194" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-673" "field_display_title" => "673" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1307 …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 {#1308 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1309 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1310 …2} "content" => Illuminate\Database\Eloquent\Collection {#1311 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1318 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1319 #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" => 1108411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "194" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-674" "field_display_title" => "674" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1320 …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 {#1321 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1322 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1323 …2} "content" => Illuminate\Database\Eloquent\Collection {#1324 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1331 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "194" "field_page_end" => null "field_url" 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№674 (с. 194)
Условие. №674 (с. 194)
скриншот условия
674. Упростите выражение:
а) $ \frac{\text{tg}(5\pi - x) + \text{tg}(10\pi - x) - \text{tg}(21\pi + x)}{\text{sin } 6,5\pi - \text{cos } 15\pi + \text{cos } 20\pi} $
б) $ \frac{\text{cos}(5\pi - x) + \text{sin}(11\pi + x)}{\text{tg}(\pi - x) \cdot \text{tg}(0,5\pi + x) + \text{tg}(20\pi + x)} $
в) $ \frac{\text{tg}^2(9\pi - x) \cdot \text{sin}^2(x - 4,5\pi)}{\text{sin}(19\pi - x) \cdot \text{sin}(-x - 15\pi)} $
г) $ \frac{\text{cos}^2(5,5\pi - x)}{\text{tg}^2(x - 10\pi)} + \frac{\text{cos}^2(-x)}{\text{tg}^2(x - 7,5\pi)} $
Решение. №674 (с. 194)
Решение 2 (rus). №674 (с. 194)
а)
Для упрощения выражения воспользуемся формулами приведения и свойством периодичности тригонометрических функций. Сначала упростим числитель и знаменатель по отдельности.
Упростим числитель:
Используя периодичность тангенса ($T = \pi$) и формулы приведения:
$ \text{tg}(5\pi - x) = \text{tg}(\pi - x) = -\text{tg}(x) $
$ \text{tg}(10\pi - x) = \text{tg}(-x) = -\text{tg}(x) $
$ \text{tg}(21\pi + x) = \text{tg}(\pi + x) = \text{tg}(x) $
Тогда числитель равен: $ -\text{tg}(x) + (-\text{tg}(x)) - \text{tg}(x) = -3\text{tg}(x) $.
Упростим знаменатель, вычислив значения тригонометрических функций:
$ \sin(6,5\pi) = \sin(6\pi + \frac{\pi}{2}) = \sin(\frac{\pi}{2}) = 1 $
$ \cos(15\pi) = \cos(14\pi + \pi) = \cos(\pi) = -1 $
$ \cos(20\pi) = \cos(0) = 1 $
Тогда знаменатель равен: $ 1 - (-1) + 1 = 3 $.
Подставим упрощенные части в исходное выражение:
$ \frac{-3\text{tg}(x)}{3} = -\text{tg}(x) $
Ответ: $-\text{tg}(x)$
б)
Упростим числитель и знаменатель дроби, используя формулы приведения и периодичность.
Числитель:
$ \cos(5\pi - x) = \cos(4\pi + \pi - x) = \cos(\pi - x) = -\cos(x) $
$ \sin(11\pi + x) = \sin(10\pi + \pi + x) = \sin(\pi + x) = -\sin(x) $
Следовательно, числитель равен $ -\cos(x) - \sin(x) = -(\cos(x) + \sin(x)) $.
Знаменатель:
$ \text{tg}(\pi - x) = -\text{tg}(x) $
$ \text{tg}(0,5\pi + x) = \text{tg}(\frac{\pi}{2} + x) = -\text{ctg}(x) $
$ \text{tg}(20\pi + x) = \text{tg}(x) $
Произведение в знаменателе: $ \text{tg}(\pi - x) \cdot \text{tg}(0,5\pi + x) = (-\text{tg}(x)) \cdot (-\text{ctg}(x)) = \text{tg}(x)\text{ctg}(x) = 1 $.
Следовательно, знаменатель равен $ 1 + \text{tg}(x) $.
Соберем дробь и упростим, выразив тангенс через синус и косинус:
$ \frac{-(\cos(x) + \sin(x))}{1 + \text{tg}(x)} = \frac{-(\cos(x) + \sin(x))}{1 + \frac{\sin(x)}{\cos(x)}} = \frac{-(\cos(x) + \sin(x))}{\frac{\cos(x) + \sin(x)}{\cos(x)}} = -(\cos(x) + \sin(x)) \cdot \frac{\cos(x)}{\cos(x) + \sin(x)} = -\cos(x) $
Ответ: $-\cos(x)$
в)
Упростим числитель и знаменатель по отдельности.
Упростим числитель:
$ \text{tg}^2(9\pi - x) = (\text{tg}(9\pi - x))^2 = (\text{tg}(\pi - x))^2 = (-\text{tg}(x))^2 = \text{tg}^2(x) $.
$ \sin^2(x - 4,5\pi) = \sin^2(x - \frac{9\pi}{2}) = \sin^2(-(\frac{9\pi}{2} - x)) = \sin^2(\frac{9\pi}{2} - x) = \sin^2(4\pi + \frac{\pi}{2} - x) = \sin^2(\frac{\pi}{2} - x) = \cos^2(x) $.
Числитель равен: $ \text{tg}^2(x) \cdot \cos^2(x) = \frac{\sin^2(x)}{\cos^2(x)} \cdot \cos^2(x) = \sin^2(x) $.
Упростим знаменатель:
$ \sin(19\pi - x) = \sin(18\pi + \pi - x) = \sin(\pi - x) = \sin(x) $.
$ \sin(-x - 15\pi) = -\sin(x + 15\pi) = -\sin(x + 14\pi + \pi) = -\sin(x + \pi) = -(-\sin(x)) = \sin(x) $.
Знаменатель равен: $ \sin(x) \cdot \sin(x) = \sin^2(x) $.
Таким образом, выражение равно:
$ \frac{\sin^2(x)}{\sin^2(x)} = 1 $
Ответ: $1$
г)
Упростим каждое слагаемое отдельно.
Первое слагаемое: $ \frac{\cos^2(5,5\pi - x)}{\text{tg}^2(x - 10\pi)} $.
Числитель: $ \cos^2(5,5\pi - x) = \cos^2(\frac{11\pi}{2} - x) $. По формуле приведения $ \cos(\frac{11\pi}{2} - x) = \cos(4\pi + \frac{3\pi}{2} - x) = \cos(\frac{3\pi}{2} - x) = -\sin(x) $. Тогда числитель равен $ (-\sin(x))^2 = \sin^2(x) $.
Знаменатель: $ \text{tg}^2(x - 10\pi) = \text{tg}^2(x) $ из-за периодичности тангенса ($T=\pi$).
Первое слагаемое равно: $ \frac{\sin^2(x)}{\text{tg}^2(x)} = \frac{\sin^2(x)}{\frac{\sin^2(x)}{\cos^2(x)}} = \cos^2(x) $.
Второе слагаемое: $ \frac{\cos^2(-x)}{\text{tg}^2(x - 7,5\pi)} $.
Числитель: $ \cos^2(-x) = \cos^2(x) $, так как косинус - четная функция.
Знаменатель: $ \text{tg}^2(x - 7,5\pi) = \text{tg}^2(x - \frac{15\pi}{2}) $. Используя периодичность, $ \text{tg}(x - \frac{15\pi}{2}) = \text{tg}(x - \frac{15\pi}{2} + 8\pi) = \text{tg}(x + \frac{\pi}{2}) = -\text{ctg}(x) $. Тогда знаменатель равен $ (-\text{ctg}(x))^2 = \text{ctg}^2(x) $.
Второе слагаемое равно: $ \frac{\cos^2(x)}{\text{ctg}^2(x)} = \frac{\cos^2(x)}{\frac{\cos^2(x)}{\sin^2(x)}} = \sin^2(x) $.
Суммируем оба слагаемых, используя основное тригонометрическое тождество:
$ \cos^2(x) + \sin^2(x) = 1 $.
Ответ: $1$
Другие задания:
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