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Номер 662, страница 192 - 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 662, страница 192.
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" => 1108399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "192" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-662" "field_display_title" => "662" "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 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …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 {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …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 {#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 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …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 {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …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 {#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 …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 {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 …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 {#1095 …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 {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 …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" => 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" => 1277017 "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 …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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …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 {#1102 …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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …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" => "1108399" "type" => "task" ] "text" => "<p><strong>662.</strong> Найдите значение выражения:</p><p><strong>a)</strong> $ \operatorname{sin}^2 120^\circ + \operatorname{cos}^2 150^\circ + \operatorname{tg}^2 225^\circ - \operatorname{ctg}^2 210^\circ $</p><p><strong>б)</strong> $ \operatorname{sin}^2 (-330^\circ) - \operatorname{cos}^2 (-120^\circ) - \operatorname{tg}^2 (-240^\circ) - \operatorname{ctg}^2 (-330^\circ) $</p><p><strong>в)</strong> $ 6 \operatorname{sin} (-240^\circ) \cdot \operatorname{cos} 315^\circ - 4 \operatorname{cos} 135^\circ \cdot \operatorname{tg} (-225^\circ) \cdot \operatorname{tg} 120^\circ $</p><p><strong>г)</strong> $ \frac{\operatorname{sin} (-120^\circ)}{\operatorname{sin} 300^\circ} - \frac{\operatorname{ctg} 120^\circ \cdot \operatorname{cos} 330^\circ}{\operatorname{cos} 360^\circ} + \operatorname{sin} (-210^\circ) \cdot \operatorname{tg} 150^\circ $</p>" "img" => array:1 [ 0 => array:5 [ "name" => "662-1.jpg" "alt" => null "width" => "1357" "height" => 444 "path" => "/media/algebra_09/soltan-u19/0-1/662-1.webp?ts=1752834668" ] ] ] #original: array:7 [ "id" => 1277017 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1108399" "type" => "task" ] "text" => "<p><strong>662.</strong> Найдите значение выражения:</p><p><strong>a)</strong> $ \operatorname{sin}^2 120^\circ + \operatorname{cos}^2 150^\circ + \operatorname{tg}^2 225^\circ - \operatorname{ctg}^2 210^\circ $</p><p><strong>б)</strong> $ \operatorname{sin}^2 (-330^\circ) - \operatorname{cos}^2 (-120^\circ) - \operatorname{tg}^2 (-240^\circ) - \operatorname{ctg}^2 (-330^\circ) $</p><p><strong>в)</strong> $ 6 \operatorname{sin} (-240^\circ) \cdot \operatorname{cos} 315^\circ - 4 \operatorname{cos} 135^\circ \cdot \operatorname{tg} (-225^\circ) \cdot \operatorname{tg} 120^\circ $</p><p><strong>г)</strong> $ \frac{\operatorname{sin} (-120^\circ)}{\operatorname{sin} 300^\circ} - \frac{\operatorname{ctg} 120^\circ \cdot \operatorname{cos} 330^\circ}{\operatorname{cos} 360^\circ} + \operatorname{sin} (-210^\circ) \cdot \operatorname{tg} 150^\circ $</p>" "img" => array:1 [ 0 => array:5 [ "name" => "662-1.jpg" "alt" => null "width" => "1357" "height" => 444 "path" => "/media/algebra_09/soltan-u19/0-1/662-1.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" => 1278308 "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 …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 {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …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 {#1110 …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 {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …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" => "1108399" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "662-1.jpg" "alt" => null "width" => "1403" "height" => 2438 "path" => "/media/algebra_09/soltan-u19/1-1/662-1.webp?ts=1752831451" ] ] ] #original: array:6 [ "id" => 1278308 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1108399" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "662-1.jpg" "alt" => null "width" => "1403" "height" => 2438 "path" => "/media/algebra_09/soltan-u19/1-1/662-1.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" => 1554263 "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 …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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …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 {#1118 …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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …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" => "1108399" "type" => "task" ] "text" => "<p><strong>а)</strong> Для вычисления значения выражения $ \sin^2 120^\circ + \cos^2 150^\circ + \tg^2 225^\circ - \ctg^2 210^\circ $ воспользуемся формулами приведения и значениями тригонометрических функций для стандартных углов. Вычислим значение каждого слагаемого: $ \sin 120^\circ = \sin(180^\circ - 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2} $, откуда $ \sin^2 120^\circ = (\frac{\sqrt{3}}{2})^2 = \frac{3}{4} $. Далее, $ \cos 150^\circ = \cos(180^\circ - 30^\circ) = -\cos 30^\circ = -\frac{\sqrt{3}}{2} $, откуда $ \cos^2 150^\circ = (-\frac{\sqrt{3}}{2})^2 = \frac{3}{4} $. Затем, $ \tg 225^\circ = \tg(180^\circ + 45^\circ) = \tg 45^\circ = 1 $, откуда $ \tg^2 225^\circ = 1^2 = 1 $. И, наконец, $ \ctg 210^\circ = \ctg(180^\circ + 30^\circ) = \ctg 30^\circ = \sqrt{3} $, откуда $ \ctg^2 210^\circ = (\sqrt{3})^2 = 3 $. Подставляем найденные значения в исходное выражение: $ \frac{3}{4} + \frac{3}{4} + 1 - 3 = \frac{6}{4} - 2 = \frac{3}{2} - 2 = \frac{3-4}{2} = -\frac{1}{2} $. Ответ: $ -\frac{1}{2} $.</p><p><strong>б)</strong> Рассмотрим выражение $ \sin^2(-330^\circ) - \cos^2(-120^\circ) - \tg^2(-240^\circ) - \ctg^2(-330^\circ) $. Используем свойства четности и нечетности тригонометрических функций: $ \sin(-\alpha) = -\sin(\alpha) $, $ \cos(-\alpha) = \cos(\alpha) $, $ \tg(-\alpha) = -\tg(\alpha) $, $ \ctg(-\alpha) = -\ctg(\alpha) $. Так как все функции в выражении возведены в квадрат, знаки минуса при вынесении из-под функции исчезают, и выражение принимает вид: $ \sin^2(330^\circ) - \cos^2(120^\circ) - \tg^2(240^\circ) - \ctg^2(330^\circ) $. Вычислим значения, используя формулы приведения: $ \sin 330^\circ = \sin(360^\circ - 30^\circ) = -\sin 30^\circ = -\frac{1}{2} $, поэтому $ \sin^2 330^\circ = \frac{1}{4} $. $ \cos 120^\circ = \cos(180^\circ - 60^\circ) = -\cos 60^\circ = -\frac{1}{2} $, поэтому $ \cos^2 120^\circ = \frac{1}{4} $. $ \tg 240^\circ = \tg(180^\circ + 60^\circ) = \tg 60^\circ = \sqrt{3} $, поэтому $ \tg^2 240^\circ = 3 $. $ \ctg 330^\circ = \ctg(360^\circ - 30^\circ) = -\ctg 30^\circ = -\sqrt{3} $, поэтому $ \ctg^2 330^\circ = 3 $. Подставляем значения в выражение: $ \frac{1}{4} - \frac{1}{4} - 3 - 3 = 0 - 6 = -6 $. Ответ: $ -6 $.</p><p><strong>в)</strong> Найдем значение выражения $ 6\sin(-240^\circ) \cdot \cos 315^\circ - 4\cos 135^\circ \cdot \tg(-225^\circ) \cdot \tg 120^\circ $. Сначала упростим каждый тригонометрический множитель: $ \sin(-240^\circ) = -\sin(240^\circ) = -\sin(180^\circ + 60^\circ) = -(-\sin 60^\circ) = \frac{\sqrt{3}}{2} $. $ \cos 315^\circ = \cos(360^\circ - 45^\circ) = \cos 45^\circ = \frac{\sqrt{2}}{2} $. $ \cos 135^\circ = \cos(180^\circ - 45^\circ) = -\cos 45^\circ = -\frac{\sqrt{2}}{2} $. $ \tg(-225^\circ) = -\tg(225^\circ) = -\tg(180^\circ + 45^\circ) = -\tg 45^\circ = -1 $. $ \tg 120^\circ = \tg(180^\circ - 60^\circ) = -\tg 60^\circ = -\sqrt{3} $. Теперь подставим эти значения в исходное выражение. Первое слагаемое: $ 6 \cdot \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = \frac{6\sqrt{6}}{4} = \frac{3\sqrt{6}}{2} $. Второе слагаемое (вычитаемое): $ 4\cos 135^\circ \cdot \tg(-225^\circ) \cdot \tg 120^\circ = 4 \cdot (-\frac{\sqrt{2}}{2}) \cdot (-1) \cdot (-\sqrt{3}) = -2\sqrt{6} $. Вычисляем разность: $ \frac{3\sqrt{6}}{2} - (-2\sqrt{6}) = \frac{3\sqrt{6}}{2} + 2\sqrt{6} = \frac{3\sqrt{6} + 4\sqrt{6}}{2} = \frac{7\sqrt{6}}{2} $. Ответ: $ \frac{7\sqrt{6}}{2} $.</p><p><strong>г)</strong> Рассмотрим выражение $ \frac{\sin(-120^\circ)}{\sin 300^\circ} - \frac{\ctg 120^\circ \cdot \cos 330^\circ}{\cos 360^\circ} + \sin(-210^\circ) \cdot \tg 150^\circ $. Вычислим значение каждого слагаемого по отдельности. Первое слагаемое: $ \frac{\sin(-120^\circ)}{\sin 300^\circ} $. Так как $ \sin(-120^\circ) = -\sin(120^\circ) = -\frac{\sqrt{3}}{2} $ и $ \sin 300^\circ = \sin(360^\circ-60^\circ) = -\sin 60^\circ = -\frac{\sqrt{3}}{2} $, то значение дроби равно $ \frac{-\frac{\sqrt{3}}{2}}{-\frac{\sqrt{3}}{2}} = 1 $. Второе слагаемое (вычитаемое): $ \frac{\ctg 120^\circ \cdot \cos 330^\circ}{\cos 360^\circ} $. Имеем: $ \ctg 120^\circ = \ctg(180^\circ-60^\circ) = -\ctg 60^\circ = -\frac{1}{\sqrt{3}} $, $ \cos 330^\circ = \cos(360^\circ-30^\circ) = \cos 30^\circ = \frac{\sqrt{3}}{2} $, $ \cos 360^\circ = 1 $. Тогда значение дроби равно $ \frac{-\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{2}}{1} = -\frac{1}{2} $. Третье слагаемое: $ \sin(-210^\circ) \cdot \tg 150^\circ $. Имеем: $ \sin(-210^\circ) = -\sin(210^\circ) = -\sin(180^\circ+30^\circ) = -(-\sin 30^\circ) = \frac{1}{2} $, $ \tg 150^\circ = \tg(180^\circ-30^\circ) = -\tg 30^\circ = -\frac{1}{\sqrt{3}} $. Произведение равно $ \frac{1}{2} \cdot (-\frac{1}{\sqrt{3}}) = -\frac{1}{2\sqrt{3}} = -\frac{\sqrt{3}}{6} $. Собираем все части вместе: $ 1 - (-\frac{1}{2}) + (-\frac{\sqrt{3}}{6}) = 1 + \frac{1}{2} - \frac{\sqrt{3}}{6} = \frac{3}{2} - \frac{\sqrt{3}}{6} = \frac{9}{6} - \frac{\sqrt{3}}{6} = \frac{9 - \sqrt{3}}{6} $. Ответ: $ \frac{9 - \sqrt{3}}{6} $.</p>" ] #original: array:6 [ "id" => 1554263 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1108399" "type" => "task" ] "text" => "<p><strong>а)</strong> Для вычисления значения выражения $ \sin^2 120^\circ + \cos^2 150^\circ + \tg^2 225^\circ - \ctg^2 210^\circ $ воспользуемся формулами приведения и значениями тригонометрических функций для стандартных углов. Вычислим значение каждого слагаемого: $ \sin 120^\circ = \sin(180^\circ - 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2} $, откуда $ \sin^2 120^\circ = (\frac{\sqrt{3}}{2})^2 = \frac{3}{4} $. Далее, $ \cos 150^\circ = \cos(180^\circ - 30^\circ) = -\cos 30^\circ = -\frac{\sqrt{3}}{2} $, откуда $ \cos^2 150^\circ = (-\frac{\sqrt{3}}{2})^2 = \frac{3}{4} $. Затем, $ \tg 225^\circ = \tg(180^\circ + 45^\circ) = \tg 45^\circ = 1 $, откуда $ \tg^2 225^\circ = 1^2 = 1 $. И, наконец, $ \ctg 210^\circ = \ctg(180^\circ + 30^\circ) = \ctg 30^\circ = \sqrt{3} $, откуда $ \ctg^2 210^\circ = (\sqrt{3})^2 = 3 $. Подставляем найденные значения в исходное выражение: $ \frac{3}{4} + \frac{3}{4} + 1 - 3 = \frac{6}{4} - 2 = \frac{3}{2} - 2 = \frac{3-4}{2} = -\frac{1}{2} $. Ответ: $ -\frac{1}{2} $.</p><p><strong>б)</strong> Рассмотрим выражение $ \sin^2(-330^\circ) - \cos^2(-120^\circ) - \tg^2(-240^\circ) - \ctg^2(-330^\circ) $. Используем свойства четности и нечетности тригонометрических функций: $ \sin(-\alpha) = -\sin(\alpha) $, $ \cos(-\alpha) = \cos(\alpha) $, $ \tg(-\alpha) = -\tg(\alpha) $, $ \ctg(-\alpha) = -\ctg(\alpha) $. Так как все функции в выражении возведены в квадрат, знаки минуса при вынесении из-под функции исчезают, и выражение принимает вид: $ \sin^2(330^\circ) - \cos^2(120^\circ) - \tg^2(240^\circ) - \ctg^2(330^\circ) $. Вычислим значения, используя формулы приведения: $ \sin 330^\circ = \sin(360^\circ - 30^\circ) = -\sin 30^\circ = -\frac{1}{2} $, поэтому $ \sin^2 330^\circ = \frac{1}{4} $. $ \cos 120^\circ = \cos(180^\circ - 60^\circ) = -\cos 60^\circ = -\frac{1}{2} $, поэтому $ \cos^2 120^\circ = \frac{1}{4} $. $ \tg 240^\circ = \tg(180^\circ + 60^\circ) = \tg 60^\circ = \sqrt{3} $, поэтому $ \tg^2 240^\circ = 3 $. $ \ctg 330^\circ = \ctg(360^\circ - 30^\circ) = -\ctg 30^\circ = -\sqrt{3} $, поэтому $ \ctg^2 330^\circ = 3 $. Подставляем значения в выражение: $ \frac{1}{4} - \frac{1}{4} - 3 - 3 = 0 - 6 = -6 $. Ответ: $ -6 $.</p><p><strong>в)</strong> Найдем значение выражения $ 6\sin(-240^\circ) \cdot \cos 315^\circ - 4\cos 135^\circ \cdot \tg(-225^\circ) \cdot \tg 120^\circ $. Сначала упростим каждый тригонометрический множитель: $ \sin(-240^\circ) = -\sin(240^\circ) = -\sin(180^\circ + 60^\circ) = -(-\sin 60^\circ) = \frac{\sqrt{3}}{2} $. $ \cos 315^\circ = \cos(360^\circ - 45^\circ) = \cos 45^\circ = \frac{\sqrt{2}}{2} $. $ \cos 135^\circ = \cos(180^\circ - 45^\circ) = -\cos 45^\circ = -\frac{\sqrt{2}}{2} $. $ \tg(-225^\circ) = -\tg(225^\circ) = -\tg(180^\circ + 45^\circ) = -\tg 45^\circ = -1 $. $ \tg 120^\circ = \tg(180^\circ - 60^\circ) = -\tg 60^\circ = -\sqrt{3} $. Теперь подставим эти значения в исходное выражение. Первое слагаемое: $ 6 \cdot \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = \frac{6\sqrt{6}}{4} = \frac{3\sqrt{6}}{2} $. Второе слагаемое (вычитаемое): $ 4\cos 135^\circ \cdot \tg(-225^\circ) \cdot \tg 120^\circ = 4 \cdot (-\frac{\sqrt{2}}{2}) \cdot (-1) \cdot (-\sqrt{3}) = -2\sqrt{6} $. Вычисляем разность: $ \frac{3\sqrt{6}}{2} - (-2\sqrt{6}) = \frac{3\sqrt{6}}{2} + 2\sqrt{6} = \frac{3\sqrt{6} + 4\sqrt{6}}{2} = \frac{7\sqrt{6}}{2} $. Ответ: $ \frac{7\sqrt{6}}{2} $.</p><p><strong>г)</strong> Рассмотрим выражение $ \frac{\sin(-120^\circ)}{\sin 300^\circ} - \frac{\ctg 120^\circ \cdot \cos 330^\circ}{\cos 360^\circ} + \sin(-210^\circ) \cdot \tg 150^\circ $. Вычислим значение каждого слагаемого по отдельности. Первое слагаемое: $ \frac{\sin(-120^\circ)}{\sin 300^\circ} $. Так как $ \sin(-120^\circ) = -\sin(120^\circ) = -\frac{\sqrt{3}}{2} $ и $ \sin 300^\circ = \sin(360^\circ-60^\circ) = -\sin 60^\circ = -\frac{\sqrt{3}}{2} $, то значение дроби равно $ \frac{-\frac{\sqrt{3}}{2}}{-\frac{\sqrt{3}}{2}} = 1 $. Второе слагаемое (вычитаемое): $ \frac{\ctg 120^\circ \cdot \cos 330^\circ}{\cos 360^\circ} $. Имеем: $ \ctg 120^\circ = \ctg(180^\circ-60^\circ) = -\ctg 60^\circ = -\frac{1}{\sqrt{3}} $, $ \cos 330^\circ = \cos(360^\circ-30^\circ) = \cos 30^\circ = \frac{\sqrt{3}}{2} $, $ \cos 360^\circ = 1 $. Тогда значение дроби равно $ \frac{-\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{2}}{1} = -\frac{1}{2} $. Третье слагаемое: $ \sin(-210^\circ) \cdot \tg 150^\circ $. Имеем: $ \sin(-210^\circ) = -\sin(210^\circ) = -\sin(180^\circ+30^\circ) = -(-\sin 30^\circ) = \frac{1}{2} $, $ \tg 150^\circ = \tg(180^\circ-30^\circ) = -\tg 30^\circ = -\frac{1}{\sqrt{3}} $. Произведение равно $ \frac{1}{2} \cdot (-\frac{1}{\sqrt{3}}) = -\frac{1}{2\sqrt{3}} = -\frac{\sqrt{3}}{6} $. Собираем все части вместе: $ 1 - (-\frac{1}{2}) + (-\frac{\sqrt{3}}{6}) = 1 + \frac{1}{2} - \frac{\sqrt{3}}{6} = \frac{3}{2} - \frac{\sqrt{3}}{6} = \frac{9}{6} - \frac{\sqrt{3}}{6} = \frac{9 - \sqrt{3}}{6} $. Ответ: $ \frac{9 - \sqrt{3}}{6} $.</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" => "1108400" "type" => "task" ] "previous" => array:2 [ "refs" => "1108398" "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: [] #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" => 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{#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 {#1166 #items: array:1 [ 0 => App\Models\BookPage {#1165 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1097934 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "192" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-192" "field_display_title" => "192" "field_folder" => "folder1" "field_image_name" => "192" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1167 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Book {#1168 #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 {#1170 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1191 …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 {#1192 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1195 …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 {#1170 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1191 …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 {#1192 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1195 …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 {#1196 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1208 #items: array:1 [ 0 => App\Models\Element {#1197 #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" => 1261085 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1200 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261085 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1200 …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" => "1097935" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097933" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1975 #items: array:6 [ 0 => App\Models\Task {#1206 #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" => 1108398 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "192" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-661" "field_display_title" => "661" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1209 …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 {#1210 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1211 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1244 …2} "content" => Illuminate\Database\Eloquent\Collection {#1281 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1306 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108398 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "192" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-661" "field_display_title" => "661" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1209 …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 {#1210 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1211 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1244 …2} "content" => Illuminate\Database\Eloquent\Collection {#1281 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1306 …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 {#1335 #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" => 1108399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "192" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-662" "field_display_title" => "662" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1336 …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 {#1338 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1339 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1372 …2} "content" => Illuminate\Database\Eloquent\Collection {#1409 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1434 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108399 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "192" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-662" "field_display_title" => "662" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1336 …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 {#1338 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1339 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1372 …2} "content" => Illuminate\Database\Eloquent\Collection {#1409 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1434 …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 {#1463 #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" => 1108400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "192" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-663" "field_display_title" => "663" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1464 …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 {#1466 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1467 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1500 …2} "content" => Illuminate\Database\Eloquent\Collection {#1537 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1562 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "192" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-663" "field_display_title" => "663" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1464 …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 {#1466 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1467 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1500 …2} "content" => Illuminate\Database\Eloquent\Collection {#1537 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1562 …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 {#1591 #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" => 1108401 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "192" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-664" "field_display_title" => "664" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1592 …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 {#1594 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1595 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1628 …2} "content" => Illuminate\Database\Eloquent\Collection {#1665 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1690 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108401 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "192" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-664" "field_display_title" => "664" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1592 …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 {#1594 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1595 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1628 …2} "content" => Illuminate\Database\Eloquent\Collection {#1665 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1690 …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 {#1719 #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" => 1108402 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "192" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-665" "field_display_title" => "665" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1720 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null 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№662 (с. 192)
Условие. №662 (с. 192)
скриншот условия
662. Найдите значение выражения:
a) $ \operatorname{sin}^2 120^\circ + \operatorname{cos}^2 150^\circ + \operatorname{tg}^2 225^\circ - \operatorname{ctg}^2 210^\circ $
б) $ \operatorname{sin}^2 (-330^\circ) - \operatorname{cos}^2 (-120^\circ) - \operatorname{tg}^2 (-240^\circ) - \operatorname{ctg}^2 (-330^\circ) $
в) $ 6 \operatorname{sin} (-240^\circ) \cdot \operatorname{cos} 315^\circ - 4 \operatorname{cos} 135^\circ \cdot \operatorname{tg} (-225^\circ) \cdot \operatorname{tg} 120^\circ $
г) $ \frac{\operatorname{sin} (-120^\circ)}{\operatorname{sin} 300^\circ} - \frac{\operatorname{ctg} 120^\circ \cdot \operatorname{cos} 330^\circ}{\operatorname{cos} 360^\circ} + \operatorname{sin} (-210^\circ) \cdot \operatorname{tg} 150^\circ $
Решение. №662 (с. 192)
Решение 2 (rus). №662 (с. 192)
а) Для вычисления значения выражения $ \sin^2 120^\circ + \cos^2 150^\circ + \tg^2 225^\circ - \ctg^2 210^\circ $ воспользуемся формулами приведения и значениями тригонометрических функций для стандартных углов. Вычислим значение каждого слагаемого: $ \sin 120^\circ = \sin(180^\circ - 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2} $, откуда $ \sin^2 120^\circ = (\frac{\sqrt{3}}{2})^2 = \frac{3}{4} $. Далее, $ \cos 150^\circ = \cos(180^\circ - 30^\circ) = -\cos 30^\circ = -\frac{\sqrt{3}}{2} $, откуда $ \cos^2 150^\circ = (-\frac{\sqrt{3}}{2})^2 = \frac{3}{4} $. Затем, $ \tg 225^\circ = \tg(180^\circ + 45^\circ) = \tg 45^\circ = 1 $, откуда $ \tg^2 225^\circ = 1^2 = 1 $. И, наконец, $ \ctg 210^\circ = \ctg(180^\circ + 30^\circ) = \ctg 30^\circ = \sqrt{3} $, откуда $ \ctg^2 210^\circ = (\sqrt{3})^2 = 3 $. Подставляем найденные значения в исходное выражение: $ \frac{3}{4} + \frac{3}{4} + 1 - 3 = \frac{6}{4} - 2 = \frac{3}{2} - 2 = \frac{3-4}{2} = -\frac{1}{2} $. Ответ: $ -\frac{1}{2} $.
б) Рассмотрим выражение $ \sin^2(-330^\circ) - \cos^2(-120^\circ) - \tg^2(-240^\circ) - \ctg^2(-330^\circ) $. Используем свойства четности и нечетности тригонометрических функций: $ \sin(-\alpha) = -\sin(\alpha) $, $ \cos(-\alpha) = \cos(\alpha) $, $ \tg(-\alpha) = -\tg(\alpha) $, $ \ctg(-\alpha) = -\ctg(\alpha) $. Так как все функции в выражении возведены в квадрат, знаки минуса при вынесении из-под функции исчезают, и выражение принимает вид: $ \sin^2(330^\circ) - \cos^2(120^\circ) - \tg^2(240^\circ) - \ctg^2(330^\circ) $. Вычислим значения, используя формулы приведения: $ \sin 330^\circ = \sin(360^\circ - 30^\circ) = -\sin 30^\circ = -\frac{1}{2} $, поэтому $ \sin^2 330^\circ = \frac{1}{4} $. $ \cos 120^\circ = \cos(180^\circ - 60^\circ) = -\cos 60^\circ = -\frac{1}{2} $, поэтому $ \cos^2 120^\circ = \frac{1}{4} $. $ \tg 240^\circ = \tg(180^\circ + 60^\circ) = \tg 60^\circ = \sqrt{3} $, поэтому $ \tg^2 240^\circ = 3 $. $ \ctg 330^\circ = \ctg(360^\circ - 30^\circ) = -\ctg 30^\circ = -\sqrt{3} $, поэтому $ \ctg^2 330^\circ = 3 $. Подставляем значения в выражение: $ \frac{1}{4} - \frac{1}{4} - 3 - 3 = 0 - 6 = -6 $. Ответ: $ -6 $.
в) Найдем значение выражения $ 6\sin(-240^\circ) \cdot \cos 315^\circ - 4\cos 135^\circ \cdot \tg(-225^\circ) \cdot \tg 120^\circ $. Сначала упростим каждый тригонометрический множитель: $ \sin(-240^\circ) = -\sin(240^\circ) = -\sin(180^\circ + 60^\circ) = -(-\sin 60^\circ) = \frac{\sqrt{3}}{2} $. $ \cos 315^\circ = \cos(360^\circ - 45^\circ) = \cos 45^\circ = \frac{\sqrt{2}}{2} $. $ \cos 135^\circ = \cos(180^\circ - 45^\circ) = -\cos 45^\circ = -\frac{\sqrt{2}}{2} $. $ \tg(-225^\circ) = -\tg(225^\circ) = -\tg(180^\circ + 45^\circ) = -\tg 45^\circ = -1 $. $ \tg 120^\circ = \tg(180^\circ - 60^\circ) = -\tg 60^\circ = -\sqrt{3} $. Теперь подставим эти значения в исходное выражение. Первое слагаемое: $ 6 \cdot \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = \frac{6\sqrt{6}}{4} = \frac{3\sqrt{6}}{2} $. Второе слагаемое (вычитаемое): $ 4\cos 135^\circ \cdot \tg(-225^\circ) \cdot \tg 120^\circ = 4 \cdot (-\frac{\sqrt{2}}{2}) \cdot (-1) \cdot (-\sqrt{3}) = -2\sqrt{6} $. Вычисляем разность: $ \frac{3\sqrt{6}}{2} - (-2\sqrt{6}) = \frac{3\sqrt{6}}{2} + 2\sqrt{6} = \frac{3\sqrt{6} + 4\sqrt{6}}{2} = \frac{7\sqrt{6}}{2} $. Ответ: $ \frac{7\sqrt{6}}{2} $.
г) Рассмотрим выражение $ \frac{\sin(-120^\circ)}{\sin 300^\circ} - \frac{\ctg 120^\circ \cdot \cos 330^\circ}{\cos 360^\circ} + \sin(-210^\circ) \cdot \tg 150^\circ $. Вычислим значение каждого слагаемого по отдельности. Первое слагаемое: $ \frac{\sin(-120^\circ)}{\sin 300^\circ} $. Так как $ \sin(-120^\circ) = -\sin(120^\circ) = -\frac{\sqrt{3}}{2} $ и $ \sin 300^\circ = \sin(360^\circ-60^\circ) = -\sin 60^\circ = -\frac{\sqrt{3}}{2} $, то значение дроби равно $ \frac{-\frac{\sqrt{3}}{2}}{-\frac{\sqrt{3}}{2}} = 1 $. Второе слагаемое (вычитаемое): $ \frac{\ctg 120^\circ \cdot \cos 330^\circ}{\cos 360^\circ} $. Имеем: $ \ctg 120^\circ = \ctg(180^\circ-60^\circ) = -\ctg 60^\circ = -\frac{1}{\sqrt{3}} $, $ \cos 330^\circ = \cos(360^\circ-30^\circ) = \cos 30^\circ = \frac{\sqrt{3}}{2} $, $ \cos 360^\circ = 1 $. Тогда значение дроби равно $ \frac{-\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{2}}{1} = -\frac{1}{2} $. Третье слагаемое: $ \sin(-210^\circ) \cdot \tg 150^\circ $. Имеем: $ \sin(-210^\circ) = -\sin(210^\circ) = -\sin(180^\circ+30^\circ) = -(-\sin 30^\circ) = \frac{1}{2} $, $ \tg 150^\circ = \tg(180^\circ-30^\circ) = -\tg 30^\circ = -\frac{1}{\sqrt{3}} $. Произведение равно $ \frac{1}{2} \cdot (-\frac{1}{\sqrt{3}}) = -\frac{1}{2\sqrt{3}} = -\frac{\sqrt{3}}{6} $. Собираем все части вместе: $ 1 - (-\frac{1}{2}) + (-\frac{\sqrt{3}}{6}) = 1 + \frac{1}{2} - \frac{\sqrt{3}}{6} = \frac{3}{2} - \frac{\sqrt{3}}{6} = \frac{9}{6} - \frac{\sqrt{3}}{6} = \frac{9 - \sqrt{3}}{6} $. Ответ: $ \frac{9 - \sqrt{3}}{6} $.
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