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Номер 575, страница 168 - 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. 22. Синус, косинус, тангенс и котангенс произвольного угла - номер 575, страница 168.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108308 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "168" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-575" "field_display_title" => "575" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Branch {#1034 #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 {#1039 #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 {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #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 {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …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 {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …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 {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …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 {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. 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Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072 #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 {#1070 #items: array:1 [ 0 => App\Models\Book {#1073 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false 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"field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …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 {#1074 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 …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 {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1101 #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 {#1072} "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 {#1070} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1101} ] #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 {#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:24 [ "id" => 1107664 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "22. Синус, косинус, тангенс и котангенс произвольного угла" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "158" "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 {#1104 #items: array:1 [ 0 => App\Models\Book {#1105 #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 {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …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 {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …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 {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => 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"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 {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1133 #items: array:1 [ 0 => App\Models\Branch {#1134 #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 {#1135 …2} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137 …2} ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1135 …2} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107664 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "22. Синус, косинус, тангенс и котангенс произвольного угла" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103} "field_page_start" => "158" "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 {#1104} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1133} ] #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 {#1156 #items: array:3 [ 0 => App\Models\Element {#1166 #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" => 1276930 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:1 [ 0 => App\Models\Edition {#1167 #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 {#1169 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1169 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1170 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108308" "type" => "task" ] "text" => "<p><strong>575.</strong> Вычислите:</p><p><strong>a)</strong> $ \cos \pi + \tg^2(3\pi - \frac{\pi}{4}) + 2\sin(-\frac{\pi}{6}); $</p><p><strong>б)</strong> $ 1,5\tg^2(-\frac{\pi}{6}) - 0,5\ctg(4\pi + \frac{\pi}{4}) + \sin(\pi - \frac{\pi}{2}); $</p><p><strong>в)</strong> $ 3\ctg(5\pi - \frac{\pi}{3}) - 4\sin^2(-\frac{\pi}{6}) - 2\cos(\pi - \frac{\pi}{6}); $</p><p><strong>г)</strong> $ 0,5\tg^2(-\frac{\pi}{3}) - 3\tg 7\pi + 2\sin^2(3\pi - \frac{\pi}{3}). $</p>" "img" => array:1 [ 0 => array:5 [ "name" => "575-1.jpg" "alt" => null "width" => "1135" "height" => 648 "path" => "/media/algebra_09/soltan-u19/0-1/575-1.webp?ts=1752834575" ] ] ] #original: array:7 [ "id" => 1276930 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175} "task" => array:2 [ "refs" => "1108308" "type" => "task" ] "text" => "<p><strong>575.</strong> Вычислите:</p><p><strong>a)</strong> $ \cos \pi + \tg^2(3\pi - \frac{\pi}{4}) + 2\sin(-\frac{\pi}{6}); $</p><p><strong>б)</strong> $ 1,5\tg^2(-\frac{\pi}{6}) - 0,5\ctg(4\pi + \frac{\pi}{4}) + \sin(\pi - \frac{\pi}{2}); $</p><p><strong>в)</strong> $ 3\ctg(5\pi - \frac{\pi}{3}) - 4\sin^2(-\frac{\pi}{6}) - 2\cos(\pi - \frac{\pi}{6}); $</p><p><strong>г)</strong> $ 0,5\tg^2(-\frac{\pi}{3}) - 3\tg 7\pi + 2\sin^2(3\pi - \frac{\pi}{3}). $</p>" "img" => array:1 [ 0 => array:5 [ "name" => "575-1.jpg" "alt" => null "width" => "1135" "height" => 648 "path" => "/media/algebra_09/soltan-u19/0-1/575-1.webp?ts=1752834575" ] ] ] #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 {#1173 #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" => 1278221 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1183 #items: array:1 [ 0 => App\Models\Edition {#1174 #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 {#1177 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1177 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108308" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "575-1.jpg" "alt" => null "width" => "1391" "height" => 2420 "path" => "/media/algebra_09/soltan-u19/1-1/575-1.webp?ts=1752831341" ] ] ] #original: array:6 [ "id" => 1278221 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1183} "task" => array:2 [ "refs" => "1108308" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "575-1.jpg" "alt" => null "width" => "1391" "height" => 2420 "path" => "/media/algebra_09/soltan-u19/1-1/575-1.webp?ts=1752831341" ] ] ] #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 {#1181 #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" => 1554176 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191 #items: array:1 [ 0 => App\Models\Edition {#1182 #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 {#1185 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1185 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1108308" "type" => "task" ] "text" => "<p><strong>а)</strong> $cos \pi + tg^2(3\pi - \frac{\pi}{4}) + 2sin(-\frac{\pi}{6})$<br>Для решения данного выражения, вычислим значение каждого слагаемого по отдельности, используя тригонометрические тождества и формулы приведения.<br>1. Значение косинуса в точке $\pi$ равно: $cos \pi = -1$.<br>2. Упростим $tg^2(3\pi - \frac{\pi}{4})$. Используем периодичность тангенса ($T=\pi$), что позволяет отбросить $2\pi$: $tg(3\pi - \frac{\pi}{4}) = tg(\pi - \frac{\pi}{4})$. По формуле приведения $tg(\pi - \alpha) = -tg(\alpha)$, поэтому $tg(\pi - \frac{\pi}{4}) = -tg(\frac{\pi}{4}) = -1$. Возводим в квадрат: $tg^2(3\pi - \frac{\pi}{4}) = (-1)^2 = 1$.<br>3. Упростим $2sin(-\frac{\pi}{6})$. Синус является нечетной функцией, поэтому $sin(-\alpha) = -sin(\alpha)$. Следовательно, $sin(-\frac{\pi}{6}) = -sin(\frac{\pi}{6}) = -\frac{1}{2}$. Тогда $2sin(-\frac{\pi}{6}) = 2 \cdot (-\frac{1}{2}) = -1$.<br>4. Сложим полученные значения: $-1 + 1 + (-1) = -1$.<br>Ответ: -1</p><p><strong>б)</strong> $1,5tg^2(-\frac{\pi}{6}) - 0,5ctg(4\pi + \frac{\pi}{4}) + sin(\pi - \frac{\pi}{2})$<br>Вычислим значение каждого члена выражения.<br>1. Для $1,5tg^2(-\frac{\pi}{6})$ используем свойство нечетности тангенса $tg(-\alpha) = -tg(\alpha)$. $tg(-\frac{\pi}{6}) = -tg(\frac{\pi}{6}) = -\frac{1}{\sqrt{3}}$. Тогда $tg^2(-\frac{\pi}{6}) = (-\frac{1}{\sqrt{3}})^2 = \frac{1}{3}$. В итоге получаем $1,5 \cdot \frac{1}{3} = \frac{3}{2} \cdot \frac{1}{3} = \frac{1}{2} = 0,5$.<br>2. Для $0,5ctg(4\pi + \frac{\pi}{4})$ используем периодичность котангенса ($T=\pi$), что позволяет отбросить $4\pi$: $ctg(4\pi + \frac{\pi}{4}) = ctg(\frac{\pi}{4}) = 1$. Тогда $0,5ctg(4\pi + \frac{\pi}{4}) = 0,5 \cdot 1 = 0,5$.<br>3. Для $sin(\pi - \frac{\pi}{2})$ используем формулу приведения $sin(\pi - \alpha) = sin(\alpha)$. Получаем $sin(\pi - \frac{\pi}{2}) = sin(\frac{\pi}{2}) = 1$.<br>4. Подставим найденные значения в исходное выражение: $0,5 - 0,5 + 1 = 1$.<br>Ответ: 1</p><p><strong>в)</strong> $3ctg(5\pi - \frac{\pi}{3}) - 4sin^2(-\frac{\pi}{6}) - 2cos(\pi - \frac{\pi}{6})$<br>Вычислим значение каждого члена выражения.<br>1. Упростим $3ctg(5\pi - \frac{\pi}{3})$. Используя периодичность котангенса ($T=\pi$): $ctg(5\pi - \frac{\pi}{3}) = ctg(\pi - \frac{\pi}{3})$. По формуле приведения $ctg(\pi - \alpha) = -ctg(\alpha)$, получаем $ctg(\pi - \frac{\pi}{3}) = -ctg(\frac{\pi}{3}) = -\frac{1}{\sqrt{3}}$. Тогда $3ctg(5\pi - \frac{\pi}{3}) = 3 \cdot (-\frac{1}{\sqrt{3}}) = -\frac{3}{\sqrt{3}} = -\sqrt{3}$.<br>2. Упростим $4sin^2(-\frac{\pi}{6})$. Синус — нечетная функция: $sin(-\frac{\pi}{6}) = -sin(\frac{\pi}{6}) = -\frac{1}{2}$. Тогда $sin^2(-\frac{\pi}{6}) = (-\frac{1}{2})^2 = \frac{1}{4}$. В итоге получаем $4 \cdot \frac{1}{4} = 1$.<br>3. Упростим $2cos(\pi - \frac{\pi}{6})$. По формуле приведения $cos(\pi - \alpha) = -cos(\alpha)$, получаем $cos(\pi - \frac{\pi}{6}) = -cos(\frac{\pi}{6}) = -\frac{\sqrt{3}}{2}$. Тогда $2cos(\pi - \frac{\pi}{6}) = 2 \cdot (-\frac{\sqrt{3}}{2}) = -\sqrt{3}$.<br>4. Подставим найденные значения в исходное выражение: $-\sqrt{3} - 1 - (-\sqrt{3}) = -\sqrt{3} - 1 + \sqrt{3} = -1$.<br>Ответ: -1</p><p><strong>г)</strong> $0,5tg^2(-\frac{\pi}{3}) - 3tg(7\pi) + 2sin^2(3\pi - \frac{\pi}{3})$<br>Вычислим значение каждого члена выражения.<br>1. Упростим $0,5tg^2(-\frac{\pi}{3})$. Тангенс — нечетная функция: $tg(-\frac{\pi}{3}) = -tg(\frac{\pi}{3}) = -\sqrt{3}$. Тогда $tg^2(-\frac{\pi}{3}) = (-\sqrt{3})^2 = 3$. В итоге получаем $0,5 \cdot 3 = 1,5$.<br>2. Упростим $3tg(7\pi)$. Используя периодичность тангенса ($T=\pi$): $tg(7\pi) = tg(0) = 0$. Тогда $3 \cdot 0 = 0$.<br>3. Упростим $2sin^2(3\pi - \frac{\pi}{3})$. Используя периодичность синуса ($T=2\pi$): $sin(3\pi - \frac{\pi}{3}) = sin(\pi - \frac{\pi}{3})$. По формуле приведения $sin(\pi - \alpha) = sin(\alpha)$, получаем $sin(\pi - \frac{\pi}{3}) = sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$. Тогда $sin^2(3\pi - \frac{\pi}{3}) = (\frac{\sqrt{3}}{2})^2 = \frac{3}{4}$. В итоге получаем $2 \cdot \frac{3}{4} = \frac{3}{2} = 1,5$.<br>4. Подставим найденные значения в исходное выражение: $1,5 - 0 + 1,5 = 3$.<br>Ответ: 3</p>" ] #original: array:6 [ "id" => 1554176 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1191} "task" => array:2 [ "refs" => "1108308" "type" => "task" ] "text" => "<p><strong>а)</strong> $cos \pi + tg^2(3\pi - \frac{\pi}{4}) + 2sin(-\frac{\pi}{6})$<br>Для решения данного выражения, вычислим значение каждого слагаемого по отдельности, используя тригонометрические тождества и формулы приведения.<br>1. Значение косинуса в точке $\pi$ равно: $cos \pi = -1$.<br>2. Упростим $tg^2(3\pi - \frac{\pi}{4})$. Используем периодичность тангенса ($T=\pi$), что позволяет отбросить $2\pi$: $tg(3\pi - \frac{\pi}{4}) = tg(\pi - \frac{\pi}{4})$. По формуле приведения $tg(\pi - \alpha) = -tg(\alpha)$, поэтому $tg(\pi - \frac{\pi}{4}) = -tg(\frac{\pi}{4}) = -1$. Возводим в квадрат: $tg^2(3\pi - \frac{\pi}{4}) = (-1)^2 = 1$.<br>3. Упростим $2sin(-\frac{\pi}{6})$. Синус является нечетной функцией, поэтому $sin(-\alpha) = -sin(\alpha)$. Следовательно, $sin(-\frac{\pi}{6}) = -sin(\frac{\pi}{6}) = -\frac{1}{2}$. Тогда $2sin(-\frac{\pi}{6}) = 2 \cdot (-\frac{1}{2}) = -1$.<br>4. Сложим полученные значения: $-1 + 1 + (-1) = -1$.<br>Ответ: -1</p><p><strong>б)</strong> $1,5tg^2(-\frac{\pi}{6}) - 0,5ctg(4\pi + \frac{\pi}{4}) + sin(\pi - \frac{\pi}{2})$<br>Вычислим значение каждого члена выражения.<br>1. Для $1,5tg^2(-\frac{\pi}{6})$ используем свойство нечетности тангенса $tg(-\alpha) = -tg(\alpha)$. $tg(-\frac{\pi}{6}) = -tg(\frac{\pi}{6}) = -\frac{1}{\sqrt{3}}$. Тогда $tg^2(-\frac{\pi}{6}) = (-\frac{1}{\sqrt{3}})^2 = \frac{1}{3}$. В итоге получаем $1,5 \cdot \frac{1}{3} = \frac{3}{2} \cdot \frac{1}{3} = \frac{1}{2} = 0,5$.<br>2. Для $0,5ctg(4\pi + \frac{\pi}{4})$ используем периодичность котангенса ($T=\pi$), что позволяет отбросить $4\pi$: $ctg(4\pi + \frac{\pi}{4}) = ctg(\frac{\pi}{4}) = 1$. Тогда $0,5ctg(4\pi + \frac{\pi}{4}) = 0,5 \cdot 1 = 0,5$.<br>3. Для $sin(\pi - \frac{\pi}{2})$ используем формулу приведения $sin(\pi - \alpha) = sin(\alpha)$. Получаем $sin(\pi - \frac{\pi}{2}) = sin(\frac{\pi}{2}) = 1$.<br>4. Подставим найденные значения в исходное выражение: $0,5 - 0,5 + 1 = 1$.<br>Ответ: 1</p><p><strong>в)</strong> $3ctg(5\pi - \frac{\pi}{3}) - 4sin^2(-\frac{\pi}{6}) - 2cos(\pi - \frac{\pi}{6})$<br>Вычислим значение каждого члена выражения.<br>1. Упростим $3ctg(5\pi - \frac{\pi}{3})$. Используя периодичность котангенса ($T=\pi$): $ctg(5\pi - \frac{\pi}{3}) = ctg(\pi - \frac{\pi}{3})$. По формуле приведения $ctg(\pi - \alpha) = -ctg(\alpha)$, получаем $ctg(\pi - \frac{\pi}{3}) = -ctg(\frac{\pi}{3}) = -\frac{1}{\sqrt{3}}$. Тогда $3ctg(5\pi - \frac{\pi}{3}) = 3 \cdot (-\frac{1}{\sqrt{3}}) = -\frac{3}{\sqrt{3}} = -\sqrt{3}$.<br>2. Упростим $4sin^2(-\frac{\pi}{6})$. Синус — нечетная функция: $sin(-\frac{\pi}{6}) = -sin(\frac{\pi}{6}) = -\frac{1}{2}$. Тогда $sin^2(-\frac{\pi}{6}) = (-\frac{1}{2})^2 = \frac{1}{4}$. В итоге получаем $4 \cdot \frac{1}{4} = 1$.<br>3. Упростим $2cos(\pi - \frac{\pi}{6})$. По формуле приведения $cos(\pi - \alpha) = -cos(\alpha)$, получаем $cos(\pi - \frac{\pi}{6}) = -cos(\frac{\pi}{6}) = -\frac{\sqrt{3}}{2}$. Тогда $2cos(\pi - \frac{\pi}{6}) = 2 \cdot (-\frac{\sqrt{3}}{2}) = -\sqrt{3}$.<br>4. Подставим найденные значения в исходное выражение: $-\sqrt{3} - 1 - (-\sqrt{3}) = -\sqrt{3} - 1 + \sqrt{3} = -1$.<br>Ответ: -1</p><p><strong>г)</strong> $0,5tg^2(-\frac{\pi}{3}) - 3tg(7\pi) + 2sin^2(3\pi - \frac{\pi}{3})$<br>Вычислим значение каждого члена выражения.<br>1. Упростим $0,5tg^2(-\frac{\pi}{3})$. Тангенс — нечетная функция: $tg(-\frac{\pi}{3}) = -tg(\frac{\pi}{3}) = -\sqrt{3}$. Тогда $tg^2(-\frac{\pi}{3}) = (-\sqrt{3})^2 = 3$. В итоге получаем $0,5 \cdot 3 = 1,5$.<br>2. Упростим $3tg(7\pi)$. Используя периодичность тангенса ($T=\pi$): $tg(7\pi) = tg(0) = 0$. Тогда $3 \cdot 0 = 0$.<br>3. Упростим $2sin^2(3\pi - \frac{\pi}{3})$. Используя периодичность синуса ($T=2\pi$): $sin(3\pi - \frac{\pi}{3}) = sin(\pi - \frac{\pi}{3})$. По формуле приведения $sin(\pi - \alpha) = sin(\alpha)$, получаем $sin(\pi - \frac{\pi}{3}) = sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$. Тогда $sin^2(3\pi - \frac{\pi}{3}) = (\frac{\sqrt{3}}{2})^2 = \frac{3}{4}$. В итоге получаем $2 \cdot \frac{3}{4} = \frac{3}{2} = 1,5$.<br>4. Подставим найденные значения в исходное выражение: $1,5 - 0 + 1,5 = 3$.<br>Ответ: 3</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108309" "type" => "task" ] "previous" => array:2 [ "refs" => "1108307" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Book {#1163 #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 {#1148 #items: array:1 [ 0 => App\Models\Term {#1147 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "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 [ …6] "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 {#1149 #items: array:1 [ 0 => App\Models\Term {#1154 #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" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #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 {#1153 #items: array:1 [ 0 => App\Models\Term {#1152 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1150 #items: array:3 [ 0 => App\Models\Term {#1151 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "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 {#1140 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1141 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Term {#1145 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:1 [ 0 => App\Models\Term {#1142 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Term {#1157 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1155 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1162 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1164 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1160 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1158 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1161 #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 {#1159 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1189 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1190 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192 #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 {#1148} "field_class" => Illuminate\Database\Eloquent\Collection {#1149} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153} "field_author" => Illuminate\Database\Eloquent\Collection {#1150} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1139} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_country" => Illuminate\Database\Eloquent\Collection {#1144} "field_city" => Illuminate\Database\Eloquent\Collection {#1143} "field_series" => Illuminate\Database\Eloquent\Collection {#1155} "field_umk" => Illuminate\Database\Eloquent\Collection {#1162} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1164} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1160} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1158} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1161} "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 {#1159} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1189} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1190} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1192} "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 {#1200 #items: array:1 [ 0 => App\Models\BookPage {#1199 #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" => 1097910 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "168" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-168" "field_display_title" => "168" "field_folder" => "folder1" "field_image_name" => "168" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1201 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1202 #items: array:1 [ 0 => App\Models\Book {#1163} ] #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 {#1203 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1213 #items: array:1 [ 0 => App\Models\Element {#1212 #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" => 1261061 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261061 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214 …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" => "1097911" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097909" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1881 #items: array:7 [ 0 => App\Models\Task {#1895 #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" => 1108304 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "168" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-571" "field_display_title" => "571" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1896 …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 {#1897 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1898 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1899 …2} "content" => Illuminate\Database\Eloquent\Collection {#1925 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1926 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108304 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "168" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-571" "field_display_title" => "571" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1896 …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 {#1897 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1898 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1899 …2} "content" => Illuminate\Database\Eloquent\Collection {#1925 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1926 …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 {#1924 #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" => 1108305 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "168" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-572" "field_display_title" => "572" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1927 …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 {#1928 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1929 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1930 …2} "content" => Illuminate\Database\Eloquent\Collection {#1956 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1957 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108305 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "168" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-572" "field_display_title" => "572" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1927 …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 {#1928 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1929 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1930 …2} "content" => Illuminate\Database\Eloquent\Collection {#1956 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1957 …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 {#1955 #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" => 1108306 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "168" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-573" "field_display_title" => "573" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1958 …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 {#1959 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1960 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1961 …2} "content" => Illuminate\Database\Eloquent\Collection {#1987 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1988 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108306 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "168" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-573" "field_display_title" => "573" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1958 …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 {#1959 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1960 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1961 …2} "content" => Illuminate\Database\Eloquent\Collection {#1987 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1988 …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 {#1986 #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" => 1108307 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "168" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-574" "field_display_title" => "574" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1989 …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 {#1990 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1991 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1992 …2} "content" => Illuminate\Database\Eloquent\Collection {#2018 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2019 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108307 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "168" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-574" "field_display_title" => "574" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1989 …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 {#1990 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1991 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1992 …2} "content" => Illuminate\Database\Eloquent\Collection {#2018 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2019 …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 {#2017 #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" => 1108308 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "168" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-575" "field_display_title" => "575" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2020 …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 {#2021 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2022 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2023 …2} "content" => Illuminate\Database\Eloquent\Collection {#2024 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2025 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108308 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "168" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-575" "field_display_title" => "575" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2020 …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 {#2021 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2022 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2023 …2} "content" => Illuminate\Database\Eloquent\Collection {#2024 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2025 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] 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№575 (с. 168)
Условие. №575 (с. 168)
скриншот условия
575. Вычислите:
a) $ \cos \pi + \tg^2(3\pi - \frac{\pi}{4}) + 2\sin(-\frac{\pi}{6}); $
б) $ 1,5\tg^2(-\frac{\pi}{6}) - 0,5\ctg(4\pi + \frac{\pi}{4}) + \sin(\pi - \frac{\pi}{2}); $
в) $ 3\ctg(5\pi - \frac{\pi}{3}) - 4\sin^2(-\frac{\pi}{6}) - 2\cos(\pi - \frac{\pi}{6}); $
г) $ 0,5\tg^2(-\frac{\pi}{3}) - 3\tg 7\pi + 2\sin^2(3\pi - \frac{\pi}{3}). $
Решение. №575 (с. 168)
Решение 2 (rus). №575 (с. 168)
а) $cos \pi + tg^2(3\pi - \frac{\pi}{4}) + 2sin(-\frac{\pi}{6})$
Для решения данного выражения, вычислим значение каждого слагаемого по отдельности, используя тригонометрические тождества и формулы приведения.
1. Значение косинуса в точке $\pi$ равно: $cos \pi = -1$.
2. Упростим $tg^2(3\pi - \frac{\pi}{4})$. Используем периодичность тангенса ($T=\pi$), что позволяет отбросить $2\pi$: $tg(3\pi - \frac{\pi}{4}) = tg(\pi - \frac{\pi}{4})$. По формуле приведения $tg(\pi - \alpha) = -tg(\alpha)$, поэтому $tg(\pi - \frac{\pi}{4}) = -tg(\frac{\pi}{4}) = -1$. Возводим в квадрат: $tg^2(3\pi - \frac{\pi}{4}) = (-1)^2 = 1$.
3. Упростим $2sin(-\frac{\pi}{6})$. Синус является нечетной функцией, поэтому $sin(-\alpha) = -sin(\alpha)$. Следовательно, $sin(-\frac{\pi}{6}) = -sin(\frac{\pi}{6}) = -\frac{1}{2}$. Тогда $2sin(-\frac{\pi}{6}) = 2 \cdot (-\frac{1}{2}) = -1$.
4. Сложим полученные значения: $-1 + 1 + (-1) = -1$.
Ответ: -1
б) $1,5tg^2(-\frac{\pi}{6}) - 0,5ctg(4\pi + \frac{\pi}{4}) + sin(\pi - \frac{\pi}{2})$
Вычислим значение каждого члена выражения.
1. Для $1,5tg^2(-\frac{\pi}{6})$ используем свойство нечетности тангенса $tg(-\alpha) = -tg(\alpha)$. $tg(-\frac{\pi}{6}) = -tg(\frac{\pi}{6}) = -\frac{1}{\sqrt{3}}$. Тогда $tg^2(-\frac{\pi}{6}) = (-\frac{1}{\sqrt{3}})^2 = \frac{1}{3}$. В итоге получаем $1,5 \cdot \frac{1}{3} = \frac{3}{2} \cdot \frac{1}{3} = \frac{1}{2} = 0,5$.
2. Для $0,5ctg(4\pi + \frac{\pi}{4})$ используем периодичность котангенса ($T=\pi$), что позволяет отбросить $4\pi$: $ctg(4\pi + \frac{\pi}{4}) = ctg(\frac{\pi}{4}) = 1$. Тогда $0,5ctg(4\pi + \frac{\pi}{4}) = 0,5 \cdot 1 = 0,5$.
3. Для $sin(\pi - \frac{\pi}{2})$ используем формулу приведения $sin(\pi - \alpha) = sin(\alpha)$. Получаем $sin(\pi - \frac{\pi}{2}) = sin(\frac{\pi}{2}) = 1$.
4. Подставим найденные значения в исходное выражение: $0,5 - 0,5 + 1 = 1$.
Ответ: 1
в) $3ctg(5\pi - \frac{\pi}{3}) - 4sin^2(-\frac{\pi}{6}) - 2cos(\pi - \frac{\pi}{6})$
Вычислим значение каждого члена выражения.
1. Упростим $3ctg(5\pi - \frac{\pi}{3})$. Используя периодичность котангенса ($T=\pi$): $ctg(5\pi - \frac{\pi}{3}) = ctg(\pi - \frac{\pi}{3})$. По формуле приведения $ctg(\pi - \alpha) = -ctg(\alpha)$, получаем $ctg(\pi - \frac{\pi}{3}) = -ctg(\frac{\pi}{3}) = -\frac{1}{\sqrt{3}}$. Тогда $3ctg(5\pi - \frac{\pi}{3}) = 3 \cdot (-\frac{1}{\sqrt{3}}) = -\frac{3}{\sqrt{3}} = -\sqrt{3}$.
2. Упростим $4sin^2(-\frac{\pi}{6})$. Синус — нечетная функция: $sin(-\frac{\pi}{6}) = -sin(\frac{\pi}{6}) = -\frac{1}{2}$. Тогда $sin^2(-\frac{\pi}{6}) = (-\frac{1}{2})^2 = \frac{1}{4}$. В итоге получаем $4 \cdot \frac{1}{4} = 1$.
3. Упростим $2cos(\pi - \frac{\pi}{6})$. По формуле приведения $cos(\pi - \alpha) = -cos(\alpha)$, получаем $cos(\pi - \frac{\pi}{6}) = -cos(\frac{\pi}{6}) = -\frac{\sqrt{3}}{2}$. Тогда $2cos(\pi - \frac{\pi}{6}) = 2 \cdot (-\frac{\sqrt{3}}{2}) = -\sqrt{3}$.
4. Подставим найденные значения в исходное выражение: $-\sqrt{3} - 1 - (-\sqrt{3}) = -\sqrt{3} - 1 + \sqrt{3} = -1$.
Ответ: -1
г) $0,5tg^2(-\frac{\pi}{3}) - 3tg(7\pi) + 2sin^2(3\pi - \frac{\pi}{3})$
Вычислим значение каждого члена выражения.
1. Упростим $0,5tg^2(-\frac{\pi}{3})$. Тангенс — нечетная функция: $tg(-\frac{\pi}{3}) = -tg(\frac{\pi}{3}) = -\sqrt{3}$. Тогда $tg^2(-\frac{\pi}{3}) = (-\sqrt{3})^2 = 3$. В итоге получаем $0,5 \cdot 3 = 1,5$.
2. Упростим $3tg(7\pi)$. Используя периодичность тангенса ($T=\pi$): $tg(7\pi) = tg(0) = 0$. Тогда $3 \cdot 0 = 0$.
3. Упростим $2sin^2(3\pi - \frac{\pi}{3})$. Используя периодичность синуса ($T=2\pi$): $sin(3\pi - \frac{\pi}{3}) = sin(\pi - \frac{\pi}{3})$. По формуле приведения $sin(\pi - \alpha) = sin(\alpha)$, получаем $sin(\pi - \frac{\pi}{3}) = sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$. Тогда $sin^2(3\pi - \frac{\pi}{3}) = (\frac{\sqrt{3}}{2})^2 = \frac{3}{4}$. В итоге получаем $2 \cdot \frac{3}{4} = \frac{3}{2} = 1,5$.
4. Подставим найденные значения в исходное выражение: $1,5 - 0 + 1,5 = 3$.
Ответ: 3
Другие задания:
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