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Номер 682, страница 198 - 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. 26. Формулы косинуса, синуса, тангенса и котангенса суммы и разности двух углов - номер 682, страница 198.
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" => 1108420 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "198" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-682" "field_display_title" => "682" "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 #items: array:1 [ 0 => App\Models\Term {#1042 …30} ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1043 #items: array:1 [ 0 => App\Models\Term {#1044 …30} ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 #items: array:1 [ 0 => App\Models\Term {#1046 …30} ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1047 #items: array:3 [ 0 => App\Models\Term {#1048 …30} …2 ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 #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 {#1064 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 #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 {#1041} "field_class" => Illuminate\Database\Eloquent\Collection {#1043} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045} "field_author" => Illuminate\Database\Eloquent\Collection {#1047} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052} "field_country" => Illuminate\Database\Eloquent\Collection {#1054} "field_city" => Illuminate\Database\Eloquent\Collection {#1056} "field_series" => Illuminate\Database\Eloquent\Collection {#1058} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063} "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 {#1064} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067} "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 {#1068 #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 {#1039} "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} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068} ] #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 {#1138 #items: array:2 [ 0 => App\Models\Branch {#1069 #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 {#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 #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 {#1074 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1076 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1080 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1091 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1092 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096 #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 {#1097 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 #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 {#1074} "field_class" => Illuminate\Database\Eloquent\Collection {#1076} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078} "field_author" => Illuminate\Database\Eloquent\Collection {#1080} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1084} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1085} "field_country" => Illuminate\Database\Eloquent\Collection {#1087} "field_city" => Illuminate\Database\Eloquent\Collection {#1089} "field_series" => Illuminate\Database\Eloquent\Collection {#1091} "field_umk" => Illuminate\Database\Eloquent\Collection {#1092} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1093} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1094} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1095} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1096} "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 {#1097} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100} "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 {#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" => 1107668 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "26. Формулы косинуса, синуса, тангенса и котангенса суммы и разности двух углов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "196" "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 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1112 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1123 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 #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 {#1129 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 #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 {#1106} "field_class" => Illuminate\Database\Eloquent\Collection {#1108} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110} "field_author" => Illuminate\Database\Eloquent\Collection {#1112} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117} "field_country" => Illuminate\Database\Eloquent\Collection {#1119} "field_city" => Illuminate\Database\Eloquent\Collection {#1121} "field_series" => Illuminate\Database\Eloquent\Collection {#1123} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128} "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 {#1129} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132} "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 {#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 #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 {#1136 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137 #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 {#1135} "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} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1137} ] #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" => 1107668 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "26. Формулы косинуса, синуса, тангенса и котангенса суммы и разности двух углов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103} "field_page_start" => "196" "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 {#1141 #items: array:3 [ 0 => App\Models\Element {#1150 #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" => 1277037 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1159 #items: array:1 [ 0 => App\Models\Edition {#1151 #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 {#1153 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1154 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1155 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1156 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4953 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153} "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 {#1154} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1155} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1156} "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" => "1108420" "type" => "task" ] "text" => "<p><strong>682.</strong> Преобразуйте выражение с помощью формул суммы и разности двух углов:</p><p><strong>a)</strong> $\sin(30^\circ + \alpha)$;</p><p><strong>б)</strong> $\sin(60^\circ - \alpha)$;</p><p><strong>в)</strong> $\cos(\alpha + 45^\circ)$;</p><p><strong>г)</strong> $\cos(\alpha - 60^\circ)$;</p><p><strong>д)</strong> $\operatorname{tg}(45^\circ + \alpha)$;</p><p><strong>е)</strong> $\operatorname{tg}(60^\circ - \alpha)$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "682-1.jpg" "alt" => null "width" => "1571" "height" => 348 "path" => "/media/algebra_09/soltan-u19/0-1/682-1.webp?ts=1752834689" ] ] ] #original: array:7 [ "id" => 1277037 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1159} "task" => array:2 [ "refs" => "1108420" "type" => "task" ] "text" => "<p><strong>682.</strong> Преобразуйте выражение с помощью формул суммы и разности двух углов:</p><p><strong>a)</strong> $\sin(30^\circ + \alpha)$;</p><p><strong>б)</strong> $\sin(60^\circ - \alpha)$;</p><p><strong>в)</strong> $\cos(\alpha + 45^\circ)$;</p><p><strong>г)</strong> $\cos(\alpha - 60^\circ)$;</p><p><strong>д)</strong> $\operatorname{tg}(45^\circ + \alpha)$;</p><p><strong>е)</strong> $\operatorname{tg}(60^\circ - \alpha)$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "682-1.jpg" "alt" => null "width" => "1571" "height" => 348 "path" => "/media/algebra_09/soltan-u19/0-1/682-1.webp?ts=1752834689" ] ] ] #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 {#1157 #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" => 1278328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1167 #items: array:1 [ 0 => App\Models\Edition {#1158 #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 {#1161 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1162 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1163 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1164 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4954 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1161} "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 {#1162} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1163} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1164} "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" => "1108420" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "682-1.jpg" "alt" => null "width" => "1258" "height" => 1784 "path" => "/media/algebra_09/soltan-u19/1-1/682-1.webp?ts=1752831477" ] ] ] #original: array:6 [ "id" => 1278328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1167} "task" => array:2 [ "refs" => "1108420" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "682-1.jpg" "alt" => null "width" => "1258" "height" => 1784 "path" => "/media/algebra_09/soltan-u19/1-1/682-1.webp?ts=1752831477" ] ] ] #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 {#1165 #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" => 1554283 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:1 [ 0 => App\Models\Edition {#1166 #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 {#1169 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1170 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5665 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1169} "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 {#1170} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1171} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1172} "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" => "1108420" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Для преобразования выражения $sin(30° + α)$ воспользуемся формулой синуса суммы двух углов: $sin(x+y) = sin(x)cos(y) + cos(x)sin(y)$.</p><p>В данном выражении $x = 30°$ и $y = α$. Применим формулу:</p><p>$sin(30° + α) = sin(30°)cos(α) + cos(30°)sin(α)$</p><p>Подставим табличные значения тригонометрических функций $sin(30°) = \frac{1}{2}$ и $cos(30°) = \frac{\sqrt{3}}{2}$:</p><p>$sin(30° + α) = \frac{1}{2} \cdot cos(α) + \frac{\sqrt{3}}{2} \cdot sin(α) = \frac{cos(α) + \sqrt{3}sin(α)}{2}$</p><p><strong>Ответ:</strong> $\frac{cos(α) + \sqrt{3}sin(α)}{2}$.</p><p><strong>б)</strong></p><p>Для преобразования выражения $sin(60° - α)$ воспользуемся формулой синуса разности двух углов: $sin(x-y) = sin(x)cos(y) - cos(x)sin(y)$.</p><p>Здесь $x = 60°$ и $y = α$. Применим формулу:</p><p>$sin(60° - α) = sin(60°)cos(α) - cos(60°)sin(α)$</p><p>Подставим табличные значения $sin(60°) = \frac{\sqrt{3}}{2}$ и $cos(60°) = \frac{1}{2}$:</p><p>$sin(60° - α) = \frac{\sqrt{3}}{2} \cdot cos(α) - \frac{1}{2} \cdot sin(α) = \frac{\sqrt{3}cos(α) - sin(α)}{2}$</p><p><strong>Ответ:</strong> $\frac{\sqrt{3}cos(α) - sin(α)}{2}$.</p><p><strong>в)</strong></p><p>Для преобразования выражения $cos(α + 45°)$ воспользуемся формулой косинуса суммы двух углов: $cos(x+y) = cos(x)cos(y) - sin(x)sin(y)$.</p><p>Здесь $x = α$ и $y = 45°$. Применим формулу:</p><p>$cos(α + 45°) = cos(α)cos(45°) - sin(α)sin(45°)$</p><p>Подставим табличные значения $cos(45°) = \frac{\sqrt{2}}{2}$ и $sin(45°) = \frac{\sqrt{2}}{2}$:</p><p>$cos(α + 45°) = cos(α) \cdot \frac{\sqrt{2}}{2} - sin(α) \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}(cos(α) - sin(α))$</p><p><strong>Ответ:</strong> $\frac{\sqrt{2}}{2}(cos(α) - sin(α))$.</p><p><strong>г)</strong></p><p>Для преобразования выражения $cos(α - 60°)$ воспользуемся формулой косинуса разности двух углов: $cos(x-y) = cos(x)cos(y) + sin(x)sin(y)$.</p><p>Здесь $x = α$ и $y = 60°$. Применим формулу:</p><p>$cos(α - 60°) = cos(α)cos(60°) + sin(α)sin(60°)$</p><p>Подставим табличные значения $cos(60°) = \frac{1}{2}$ и $sin(60°) = \frac{\sqrt{3}}{2}$:</p><p>$cos(α - 60°) = cos(α) \cdot \frac{1}{2} + sin(α) \cdot \frac{\sqrt{3}}{2} = \frac{cos(α) + \sqrt{3}sin(α)}{2}$</p><p><strong>Ответ:</strong> $\frac{cos(α) + \sqrt{3}sin(α)}{2}$.</p><p><strong>д)</strong></p><p>Для преобразования выражения $tg(45° + α)$ воспользуемся формулой тангенса суммы двух углов: $tg(x+y) = \frac{tg(x) + tg(y)}{1 - tg(x)tg(y)}$.</p><p>Здесь $x = 45°$ и $y = α$. Применим формулу:</p><p>$tg(45° + α) = \frac{tg(45°) + tg(α)}{1 - tg(45°)tg(α)}$</p><p>Подставим табличное значение $tg(45°) = 1$:</p><p>$tg(45° + α) = \frac{1 + tg(α)}{1 - 1 \cdot tg(α)} = \frac{1 + tg(α)}{1 - tg(α)}$</p><p><strong>Ответ:</strong> $\frac{1 + tg(α)}{1 - tg(α)}$.</p><p><strong>е)</strong></p><p>Для преобразования выражения $tg(60° - α)$ воспользуемся формулой тангенса разности двух углов: $tg(x-y) = \frac{tg(x) - tg(y)}{1 + tg(x)tg(y)}$.</p><p>Здесь $x = 60°$ и $y = α$. Применим формулу:</p><p>$tg(60° - α) = \frac{tg(60°) - tg(α)}{1 + tg(60°)tg(α)}$</p><p>Подставим табличное значение $tg(60°) = \sqrt{3}$:</p><p>$tg(60° - α) = \frac{\sqrt{3} - tg(α)}{1 + \sqrt{3}tg(α)}$</p><p><strong>Ответ:</strong> $\frac{\sqrt{3} - tg(α)}{1 + \sqrt{3}tg(α)}$.</p>" ] #original: array:6 [ "id" => 1554283 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1175} "task" => array:2 [ "refs" => "1108420" "type" => "task" ] "text" => "<p><strong>а)</strong></p><p>Для преобразования выражения $sin(30° + α)$ воспользуемся формулой синуса суммы двух углов: $sin(x+y) = sin(x)cos(y) + cos(x)sin(y)$.</p><p>В данном выражении $x = 30°$ и $y = α$. Применим формулу:</p><p>$sin(30° + α) = sin(30°)cos(α) + cos(30°)sin(α)$</p><p>Подставим табличные значения тригонометрических функций $sin(30°) = \frac{1}{2}$ и $cos(30°) = \frac{\sqrt{3}}{2}$:</p><p>$sin(30° + α) = \frac{1}{2} \cdot cos(α) + \frac{\sqrt{3}}{2} \cdot sin(α) = \frac{cos(α) + \sqrt{3}sin(α)}{2}$</p><p><strong>Ответ:</strong> $\frac{cos(α) + \sqrt{3}sin(α)}{2}$.</p><p><strong>б)</strong></p><p>Для преобразования выражения $sin(60° - α)$ воспользуемся формулой синуса разности двух углов: $sin(x-y) = sin(x)cos(y) - cos(x)sin(y)$.</p><p>Здесь $x = 60°$ и $y = α$. Применим формулу:</p><p>$sin(60° - α) = sin(60°)cos(α) - cos(60°)sin(α)$</p><p>Подставим табличные значения $sin(60°) = \frac{\sqrt{3}}{2}$ и $cos(60°) = \frac{1}{2}$:</p><p>$sin(60° - α) = \frac{\sqrt{3}}{2} \cdot cos(α) - \frac{1}{2} \cdot sin(α) = \frac{\sqrt{3}cos(α) - sin(α)}{2}$</p><p><strong>Ответ:</strong> $\frac{\sqrt{3}cos(α) - sin(α)}{2}$.</p><p><strong>в)</strong></p><p>Для преобразования выражения $cos(α + 45°)$ воспользуемся формулой косинуса суммы двух углов: $cos(x+y) = cos(x)cos(y) - sin(x)sin(y)$.</p><p>Здесь $x = α$ и $y = 45°$. Применим формулу:</p><p>$cos(α + 45°) = cos(α)cos(45°) - sin(α)sin(45°)$</p><p>Подставим табличные значения $cos(45°) = \frac{\sqrt{2}}{2}$ и $sin(45°) = \frac{\sqrt{2}}{2}$:</p><p>$cos(α + 45°) = cos(α) \cdot \frac{\sqrt{2}}{2} - sin(α) \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}(cos(α) - sin(α))$</p><p><strong>Ответ:</strong> $\frac{\sqrt{2}}{2}(cos(α) - sin(α))$.</p><p><strong>г)</strong></p><p>Для преобразования выражения $cos(α - 60°)$ воспользуемся формулой косинуса разности двух углов: $cos(x-y) = cos(x)cos(y) + sin(x)sin(y)$.</p><p>Здесь $x = α$ и $y = 60°$. Применим формулу:</p><p>$cos(α - 60°) = cos(α)cos(60°) + sin(α)sin(60°)$</p><p>Подставим табличные значения $cos(60°) = \frac{1}{2}$ и $sin(60°) = \frac{\sqrt{3}}{2}$:</p><p>$cos(α - 60°) = cos(α) \cdot \frac{1}{2} + sin(α) \cdot \frac{\sqrt{3}}{2} = \frac{cos(α) + \sqrt{3}sin(α)}{2}$</p><p><strong>Ответ:</strong> $\frac{cos(α) + \sqrt{3}sin(α)}{2}$.</p><p><strong>д)</strong></p><p>Для преобразования выражения $tg(45° + α)$ воспользуемся формулой тангенса суммы двух углов: $tg(x+y) = \frac{tg(x) + tg(y)}{1 - tg(x)tg(y)}$.</p><p>Здесь $x = 45°$ и $y = α$. Применим формулу:</p><p>$tg(45° + α) = \frac{tg(45°) + tg(α)}{1 - tg(45°)tg(α)}$</p><p>Подставим табличное значение $tg(45°) = 1$:</p><p>$tg(45° + α) = \frac{1 + tg(α)}{1 - 1 \cdot tg(α)} = \frac{1 + tg(α)}{1 - tg(α)}$</p><p><strong>Ответ:</strong> $\frac{1 + tg(α)}{1 - tg(α)}$.</p><p><strong>е)</strong></p><p>Для преобразования выражения $tg(60° - α)$ воспользуемся формулой тангенса разности двух углов: $tg(x-y) = \frac{tg(x) - tg(y)}{1 + tg(x)tg(y)}$.</p><p>Здесь $x = 60°$ и $y = α$. Применим формулу:</p><p>$tg(60° - α) = \frac{tg(60°) - tg(α)}{1 + tg(60°)tg(α)}$</p><p>Подставим табличное значение $tg(60°) = \sqrt{3}$:</p><p>$tg(60° - α) = \frac{\sqrt{3} - tg(α)}{1 + \sqrt{3}tg(α)}$</p><p><strong>Ответ:</strong> $\frac{\sqrt{3} - tg(α)}{1 + \sqrt{3}tg(α)}$.</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" => "1108421" "type" => "task" ] "previous" => array:2 [ "refs" => "1108419" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1195 #items: array:1 [ 0 => App\Models\Book {#1147 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true 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#dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1173 #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 {#1174 #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 {#1176 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1177 #items: array:1 [ 0 => App\Models\Term {#1178 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1179 #items: array:1 [ 0 => App\Models\Term {#1180 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1181 #items: array:1 [ 0 => App\Models\Term {#1182 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1183 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1184 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1185 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1186 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1187 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1188 #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 {#1189 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1190 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1191 #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 {#1140} "field_class" => Illuminate\Database\Eloquent\Collection {#1146} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1144} "field_author" => Illuminate\Database\Eloquent\Collection {#1145} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1176} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1177} "field_country" => Illuminate\Database\Eloquent\Collection {#1179} "field_city" => Illuminate\Database\Eloquent\Collection {#1181} "field_series" => Illuminate\Database\Eloquent\Collection {#1183} "field_umk" => Illuminate\Database\Eloquent\Collection {#1184} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1185} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1186} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1187} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1188} "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 {#1189} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1190} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1191} "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" => 1097940 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "198" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-198" "field_display_title" => "198" "field_folder" => "folder1" "field_image_name" => "198" "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 {#1147} ] #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" => 1261091 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214 #items: array:1 [ …1] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1097940" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #original: array:6 [ "id" => 1261091 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1214} "book_page" => array:2 [ "refs" => "1097940" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ …5] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1097941" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097939" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1370 #items: array:2 [ 0 => App\Models\Task {#1378 #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" => 1108419 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "198" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/vopr-198" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1379 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1380 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1381 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1382 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1385 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108420" "type" => "task" ] "previous" => array:2 [ "refs" => "1108418" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1391 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097940" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108419 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "198" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/vopr-198" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1379} "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 {#1380} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1381} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1382} "content" => Illuminate\Database\Eloquent\Collection {#1385} "next" => array:2 [ "refs" => "1108420" "type" => "task" ] "previous" => array:2 [ "refs" => "1108418" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1391} "page" => array:2 [ "refs" => "1097940" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1384 #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" => 1108420 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "198" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-682" "field_display_title" => "682" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1383 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1390 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1392 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1388 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1386 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108421" "type" => "task" ] "previous" => array:2 [ "refs" => "1108419" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1389 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1097940" "type" => "book_page" ] ] #original: array:24 [ "id" => 1108420 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "198" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-682" "field_display_title" => "682" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1383} "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 {#1390} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1392} "parent_branches" => 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№682 (с. 198)
Условие. №682 (с. 198)
скриншот условия
682. Преобразуйте выражение с помощью формул суммы и разности двух углов:
a) $\sin(30^\circ + \alpha)$;
б) $\sin(60^\circ - \alpha)$;
в) $\cos(\alpha + 45^\circ)$;
г) $\cos(\alpha - 60^\circ)$;
д) $\operatorname{tg}(45^\circ + \alpha)$;
е) $\operatorname{tg}(60^\circ - \alpha)$.
Решение. №682 (с. 198)
Решение 2 (rus). №682 (с. 198)
а)
Для преобразования выражения $sin(30° + α)$ воспользуемся формулой синуса суммы двух углов: $sin(x+y) = sin(x)cos(y) + cos(x)sin(y)$.
В данном выражении $x = 30°$ и $y = α$. Применим формулу:
$sin(30° + α) = sin(30°)cos(α) + cos(30°)sin(α)$
Подставим табличные значения тригонометрических функций $sin(30°) = \frac{1}{2}$ и $cos(30°) = \frac{\sqrt{3}}{2}$:
$sin(30° + α) = \frac{1}{2} \cdot cos(α) + \frac{\sqrt{3}}{2} \cdot sin(α) = \frac{cos(α) + \sqrt{3}sin(α)}{2}$
Ответ: $\frac{cos(α) + \sqrt{3}sin(α)}{2}$.
б)
Для преобразования выражения $sin(60° - α)$ воспользуемся формулой синуса разности двух углов: $sin(x-y) = sin(x)cos(y) - cos(x)sin(y)$.
Здесь $x = 60°$ и $y = α$. Применим формулу:
$sin(60° - α) = sin(60°)cos(α) - cos(60°)sin(α)$
Подставим табличные значения $sin(60°) = \frac{\sqrt{3}}{2}$ и $cos(60°) = \frac{1}{2}$:
$sin(60° - α) = \frac{\sqrt{3}}{2} \cdot cos(α) - \frac{1}{2} \cdot sin(α) = \frac{\sqrt{3}cos(α) - sin(α)}{2}$
Ответ: $\frac{\sqrt{3}cos(α) - sin(α)}{2}$.
в)
Для преобразования выражения $cos(α + 45°)$ воспользуемся формулой косинуса суммы двух углов: $cos(x+y) = cos(x)cos(y) - sin(x)sin(y)$.
Здесь $x = α$ и $y = 45°$. Применим формулу:
$cos(α + 45°) = cos(α)cos(45°) - sin(α)sin(45°)$
Подставим табличные значения $cos(45°) = \frac{\sqrt{2}}{2}$ и $sin(45°) = \frac{\sqrt{2}}{2}$:
$cos(α + 45°) = cos(α) \cdot \frac{\sqrt{2}}{2} - sin(α) \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}(cos(α) - sin(α))$
Ответ: $\frac{\sqrt{2}}{2}(cos(α) - sin(α))$.
г)
Для преобразования выражения $cos(α - 60°)$ воспользуемся формулой косинуса разности двух углов: $cos(x-y) = cos(x)cos(y) + sin(x)sin(y)$.
Здесь $x = α$ и $y = 60°$. Применим формулу:
$cos(α - 60°) = cos(α)cos(60°) + sin(α)sin(60°)$
Подставим табличные значения $cos(60°) = \frac{1}{2}$ и $sin(60°) = \frac{\sqrt{3}}{2}$:
$cos(α - 60°) = cos(α) \cdot \frac{1}{2} + sin(α) \cdot \frac{\sqrt{3}}{2} = \frac{cos(α) + \sqrt{3}sin(α)}{2}$
Ответ: $\frac{cos(α) + \sqrt{3}sin(α)}{2}$.
д)
Для преобразования выражения $tg(45° + α)$ воспользуемся формулой тангенса суммы двух углов: $tg(x+y) = \frac{tg(x) + tg(y)}{1 - tg(x)tg(y)}$.
Здесь $x = 45°$ и $y = α$. Применим формулу:
$tg(45° + α) = \frac{tg(45°) + tg(α)}{1 - tg(45°)tg(α)}$
Подставим табличное значение $tg(45°) = 1$:
$tg(45° + α) = \frac{1 + tg(α)}{1 - 1 \cdot tg(α)} = \frac{1 + tg(α)}{1 - tg(α)}$
Ответ: $\frac{1 + tg(α)}{1 - tg(α)}$.
е)
Для преобразования выражения $tg(60° - α)$ воспользуемся формулой тангенса разности двух углов: $tg(x-y) = \frac{tg(x) - tg(y)}{1 + tg(x)tg(y)}$.
Здесь $x = 60°$ и $y = α$. Применим формулу:
$tg(60° - α) = \frac{tg(60°) - tg(α)}{1 + tg(60°)tg(α)}$
Подставим табличное значение $tg(60°) = \sqrt{3}$:
$tg(60° - α) = \frac{\sqrt{3} - tg(α)}{1 + \sqrt{3}tg(α)}$
Ответ: $\frac{\sqrt{3} - tg(α)}{1 + \sqrt{3}tg(α)}$.
Другие задания:
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