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Номер 724, страница 207 - 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. 27. Формулы тригонометрических функций двойного и половинного углов - номер 724, страница 207.
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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 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=> 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" => 1107669 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "27. 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{#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "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 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "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" => "" 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=> 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. 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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 {#1125 …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" => 1107669 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "27. Формулы тригонометрических функций двойного и половинного углов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "203" "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 {#1126 #items: array:3 [ 0 => App\Models\Element {#1127 #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" => 1277079 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Edition {#1129 #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 {#1130 …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 {#1132 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1134 …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 {#1130 …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 {#1132 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1134 …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" => "1108465" "type" => "task" ] "text" => "<p><strong>724.</strong> Найдите $ \sin \frac{\alpha}{2} $, $ \cos \frac{\alpha}{2} $, $ \operatorname{tg} \frac{\alpha}{2} $ и $ \operatorname{ctg} \frac{\alpha}{2} $, если:</p><p><strong>а)</strong> $ \cos \alpha = \frac{3}{4} $ и $ \frac{3\pi}{2} < \alpha < 2\pi $;</p><p><strong>Б)</strong> $ \sin \alpha = -\frac{4}{5} $ и $ \pi < \alpha < \frac{3\pi}{2} $;</p><p><strong>В)</strong> $ \operatorname{tg} \alpha = -\frac{7}{24} $ и $ \frac{\pi}{2} < \alpha < \pi $.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "724-1.jpg" "alt" => null "width" => "1359" "height" => 408 "path" => "/media/algebra_09/soltan-u19/0-1/724-1.webp?ts=1752834710" ] ] ] #original: array:7 [ "id" => 1277079 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128} "task" => array:2 [ "refs" => "1108465" "type" => "task" ] "text" => "<p><strong>724.</strong> Найдите $ \sin \frac{\alpha}{2} $, $ \cos \frac{\alpha}{2} $, $ \operatorname{tg} \frac{\alpha}{2} $ и $ \operatorname{ctg} \frac{\alpha}{2} $, если:</p><p><strong>а)</strong> $ \cos \alpha = \frac{3}{4} $ и $ \frac{3\pi}{2} < \alpha < 2\pi $;</p><p><strong>Б)</strong> $ \sin \alpha = -\frac{4}{5} $ и $ \pi < \alpha < \frac{3\pi}{2} $;</p><p><strong>В)</strong> $ \operatorname{tg} \alpha = -\frac{7}{24} $ и $ \frac{\pi}{2} < \alpha < \pi $.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "724-1.jpg" "alt" => null "width" => "1359" "height" => 408 "path" => "/media/algebra_09/soltan-u19/0-1/724-1.webp?ts=1752834710" ] ] ] #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 {#1135 #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" => 1278370 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136 #items: array:1 [ 0 => App\Models\Edition {#1137 #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 {#1138 …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 {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …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 {#1138 …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 {#1140 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 …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" => "1108465" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "724-1.jpg" "alt" => null "width" => "1264" "height" => 767 "path" => "/media/algebra_09/soltan-u19/1-1/724-1.webp?ts=1752831506" ] 1 => array:5 [ "name" => "724-2.jpg" "alt" => null "width" => "1073" "height" => 2355 "path" => "/media/algebra_09/soltan-u19/1-1/724-2.webp?ts=1752831506" ] 2 => array:5 [ "name" => "724-3.jpg" "alt" => null "width" => "758" "height" => 214 "path" => "/media/algebra_09/soltan-u19/1-1/724-3.webp?ts=1752831506" ] ] ] #original: array:6 [ "id" => 1278370 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136} "task" => array:2 [ "refs" => "1108465" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "724-1.jpg" "alt" => null "width" => "1264" "height" => 767 "path" => "/media/algebra_09/soltan-u19/1-1/724-1.webp?ts=1752831506" ] 1 => array:5 [ "name" => "724-2.jpg" "alt" => null "width" => "1073" "height" => 2355 "path" => "/media/algebra_09/soltan-u19/1-1/724-2.webp?ts=1752831506" ] 2 => array:5 [ "name" => "724-3.jpg" "alt" => null "width" => "758" "height" => 214 "path" => "/media/algebra_09/soltan-u19/1-1/724-3.webp?ts=1752831506" ] ] ] #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 {#1143 #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" => 1554325 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:1 [ 0 => App\Models\Edition {#1145 #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 {#1146 …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 {#1148 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1150 …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 {#1146 …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 {#1148 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1150 …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" => "1108465" "type" => "task" ] "text" => "<p><strong>а)</strong> Дано: $ \cos \alpha = \frac{3}{4} $ и $ \frac{3\pi}{2} < \alpha < 2\pi $.</p><p>Сначала определим, в какой четверти находится угол $ \frac{\alpha}{2} $. Разделив неравенство для $ \alpha $ на 2, получаем: $ \frac{3\pi}{4} < \frac{\alpha}{2} < \pi $. Это означает, что угол $ \frac{\alpha}{2} $ находится во второй четверти. Во второй четверти синус положителен ($ \sin\frac{\alpha}{2} > 0 $), а косинус, тангенс и котангенс отрицательны ($ \cos\frac{\alpha}{2} < 0 $, $ \tg\frac{\alpha}{2} < 0 $, $ \ctg\frac{\alpha}{2} < 0 $).</p><p>Воспользуемся формулами половинного угла:</p><p>$ \sin^2\frac{\alpha}{2} = \frac{1 - \cos\alpha}{2} = \frac{1 - \frac{3}{4}}{2} = \frac{\frac{1}{4}}{2} = \frac{1}{8} $. Так как $ \sin\frac{\alpha}{2} > 0 $, то $ \sin\frac{\alpha}{2} = \sqrt{\frac{1}{8}} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4} $.</p><p>$ \cos^2\frac{\alpha}{2} = \frac{1 + \cos\alpha}{2} = \frac{1 + \frac{3}{4}}{2} = \frac{\frac{7}{4}}{2} = \frac{7}{8} $. Так как $ \cos\frac{\alpha}{2} < 0 $, то $ \cos\frac{\alpha}{2} = -\sqrt{\frac{7}{8}} = -\frac{\sqrt{7}}{2\sqrt{2}} = -\frac{\sqrt{14}}{4} $.</p><p>$ \tg\frac{\alpha}{2} = \frac{\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}} = \frac{\frac{\sqrt{2}}{4}}{-\frac{\sqrt{14}}{4}} = -\frac{\sqrt{2}}{\sqrt{14}} = -\frac{1}{\sqrt{7}} = -\frac{\sqrt{7}}{7} $.</p><p>$ \ctg\frac{\alpha}{2} = \frac{1}{\tg\frac{\alpha}{2}} = \frac{1}{-\frac{\sqrt{7}}{7}} = -\sqrt{7} $.</p><p><strong>Ответ:</strong> $ \sin\frac{\alpha}{2} = \frac{\sqrt{2}}{4} $, $ \cos\frac{\alpha}{2} = -\frac{\sqrt{14}}{4} $, $ \tg\frac{\alpha}{2} = -\frac{\sqrt{7}}{7} $, $ \ctg\frac{\alpha}{2} = -\sqrt{7} $.</p><p><strong>б)</strong> Дано: $ \sin \alpha = -\frac{4}{5} $ и $ \pi < \alpha < \frac{3\pi}{2} $.</p><p>Определим четверть для угла $ \frac{\alpha}{2} $. Разделив неравенство для $ \alpha $ на 2, получаем: $ \frac{\pi}{2} < \frac{\alpha}{2} < \frac{3\pi}{4} $. Угол $ \frac{\alpha}{2} $ находится во второй четверти. Следовательно, $ \sin\frac{\alpha}{2} > 0 $, $ \cos\frac{\alpha}{2} < 0 $, $ \tg\frac{\alpha}{2} < 0 $, $ \ctg\frac{\alpha}{2} < 0 $.</p><p>Для использования формул половинного угла нам нужен $ \cos\alpha $. Найдем его из основного тригонометрического тождества $ \sin^2\alpha + \cos^2\alpha = 1 $.</p><p>$ \cos^2\alpha = 1 - \sin^2\alpha = 1 - (-\frac{4}{5})^2 = 1 - \frac{16}{25} = \frac{9}{25} $.</p><p>Поскольку $ \alpha $ находится в третьей четверти ($ \pi < \alpha < \frac{3\pi}{2} $), $ \cos\alpha $ отрицателен. Значит, $ \cos\alpha = -\frac{3}{5} $.</p><p>Теперь применяем формулы половинного угла:</p><p>$ \sin^2\frac{\alpha}{2} = \frac{1 - \cos\alpha}{2} = \frac{1 - (-\frac{3}{5})}{2} = \frac{1 + \frac{3}{5}}{2} = \frac{\frac{8}{5}}{2} = \frac{4}{5} $. Так как $ \sin\frac{\alpha}{2} > 0 $, то $ \sin\frac{\alpha}{2} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5} $.</p><p>$ \cos^2\frac{\alpha}{2} = \frac{1 + \cos\alpha}{2} = \frac{1 + (-\frac{3}{5})}{2} = \frac{1 - \frac{3}{5}}{2} = \frac{\frac{2}{5}}{2} = \frac{1}{5} $. Так как $ \cos\frac{\alpha}{2} < 0 $, то $ \cos\frac{\alpha}{2} = -\sqrt{\frac{1}{5}} = -\frac{1}{\sqrt{5}} = -\frac{\sqrt{5}}{5} $.</p><p>$ \tg\frac{\alpha}{2} = \frac{\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}} = \frac{\frac{2\sqrt{5}}{5}}{-\frac{\sqrt{5}}{5}} = -2 $.</p><p>$ \ctg\frac{\alpha}{2} = \frac{1}{\tg\frac{\alpha}{2}} = -\frac{1}{2} $.</p><p><strong>Ответ:</strong> $ \sin\frac{\alpha}{2} = \frac{2\sqrt{5}}{5} $, $ \cos\frac{\alpha}{2} = -\frac{\sqrt{5}}{5} $, $ \tg\frac{\alpha}{2} = -2 $, $ \ctg\frac{\alpha}{2} = -\frac{1}{2} $.</p><p><strong>в)</strong> Дано: $ \tg \alpha = -\frac{7}{24} $ и $ \frac{\pi}{2} < \alpha < \pi $.</p><p>Определим четверть для угла $ \frac{\alpha}{2} $. Разделив неравенство для $ \alpha $ на 2, получаем: $ \frac{\pi}{4} < \frac{\alpha}{2} < \frac{\pi}{2} $. Угол $ \frac{\alpha}{2} $ находится в первой четверти. В первой четверти все тригонометрические функции положительны: $ \sin\frac{\alpha}{2} > 0 $, $ \cos\frac{\alpha}{2} > 0 $, $ \tg\frac{\alpha}{2} > 0 $, $ \ctg\frac{\alpha}{2} > 0 $.</p><p>Найдем $ \cos\alpha $, используя тождество $ 1 + \tg^2\alpha = \frac{1}{\cos^2\alpha} $.</p><p>$ \cos^2\alpha = \frac{1}{1 + \tg^2\alpha} = \frac{1}{1 + (-\frac{7}{24})^2} = \frac{1}{1 + \frac{49}{576}} = \frac{1}{\frac{625}{576}} = \frac{576}{625} $.</p><p>Поскольку $ \alpha $ находится во второй четверти ($ \frac{\pi}{2} < \alpha < \pi $), $ \cos\alpha $ отрицателен. Значит, $ \cos\alpha = -\sqrt{\frac{576}{625}} = -\frac{24}{25} $.</p><p>Теперь применяем формулы половинного угла:</p><p>$ \sin^2\frac{\alpha}{2} = \frac{1 - \cos\alpha}{2} = \frac{1 - (-\frac{24}{25})}{2} = \frac{1 + \frac{24}{25}}{2} = \frac{\frac{49}{25}}{2} = \frac{49}{50} $. Так как $ \sin\frac{\alpha}{2} > 0 $, то $ \sin\frac{\alpha}{2} = \sqrt{\frac{49}{50}} = \frac{7}{5\sqrt{2}} = \frac{7\sqrt{2}}{10} $.</p><p>$ \cos^2\frac{\alpha}{2} = \frac{1 + \cos\alpha}{2} = \frac{1 + (-\frac{24}{25})}{2} = \frac{1 - \frac{24}{25}}{2} = \frac{\frac{1}{25}}{2} = \frac{1}{50} $. Так как $ \cos\frac{\alpha}{2} > 0 $, то $ \cos\frac{\alpha}{2} = \sqrt{\frac{1}{50}} = \frac{1}{5\sqrt{2}} = \frac{\sqrt{2}}{10} $.</p><p>$ \tg\frac{\alpha}{2} = \frac{\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}} = \frac{\frac{7\sqrt{2}}{10}}{\frac{\sqrt{2}}{10}} = 7 $.</p><p>$ \ctg\frac{\alpha}{2} = \frac{1}{\tg\frac{\alpha}{2}} = \frac{1}{7} $.</p><p><strong>Ответ:</strong> $ \sin\frac{\alpha}{2} = \frac{7\sqrt{2}}{10} $, $ \cos\frac{\alpha}{2} = \frac{\sqrt{2}}{10} $, $ \tg\frac{\alpha}{2} = 7 $, $ \ctg\frac{\alpha}{2} = \frac{1}{7} $.</p>" ] #original: array:6 [ "id" => 1554325 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144} "task" => array:2 [ "refs" => "1108465" "type" => "task" ] "text" => "<p><strong>а)</strong> Дано: $ \cos \alpha = \frac{3}{4} $ и $ \frac{3\pi}{2} < \alpha < 2\pi $.</p><p>Сначала определим, в какой четверти находится угол $ \frac{\alpha}{2} $. Разделив неравенство для $ \alpha $ на 2, получаем: $ \frac{3\pi}{4} < \frac{\alpha}{2} < \pi $. Это означает, что угол $ \frac{\alpha}{2} $ находится во второй четверти. Во второй четверти синус положителен ($ \sin\frac{\alpha}{2} > 0 $), а косинус, тангенс и котангенс отрицательны ($ \cos\frac{\alpha}{2} < 0 $, $ \tg\frac{\alpha}{2} < 0 $, $ \ctg\frac{\alpha}{2} < 0 $).</p><p>Воспользуемся формулами половинного угла:</p><p>$ \sin^2\frac{\alpha}{2} = \frac{1 - \cos\alpha}{2} = \frac{1 - \frac{3}{4}}{2} = \frac{\frac{1}{4}}{2} = \frac{1}{8} $. Так как $ \sin\frac{\alpha}{2} > 0 $, то $ \sin\frac{\alpha}{2} = \sqrt{\frac{1}{8}} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4} $.</p><p>$ \cos^2\frac{\alpha}{2} = \frac{1 + \cos\alpha}{2} = \frac{1 + \frac{3}{4}}{2} = \frac{\frac{7}{4}}{2} = \frac{7}{8} $. Так как $ \cos\frac{\alpha}{2} < 0 $, то $ \cos\frac{\alpha}{2} = -\sqrt{\frac{7}{8}} = -\frac{\sqrt{7}}{2\sqrt{2}} = -\frac{\sqrt{14}}{4} $.</p><p>$ \tg\frac{\alpha}{2} = \frac{\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}} = \frac{\frac{\sqrt{2}}{4}}{-\frac{\sqrt{14}}{4}} = -\frac{\sqrt{2}}{\sqrt{14}} = -\frac{1}{\sqrt{7}} = -\frac{\sqrt{7}}{7} $.</p><p>$ \ctg\frac{\alpha}{2} = \frac{1}{\tg\frac{\alpha}{2}} = \frac{1}{-\frac{\sqrt{7}}{7}} = -\sqrt{7} $.</p><p><strong>Ответ:</strong> $ \sin\frac{\alpha}{2} = \frac{\sqrt{2}}{4} $, $ \cos\frac{\alpha}{2} = -\frac{\sqrt{14}}{4} $, $ \tg\frac{\alpha}{2} = -\frac{\sqrt{7}}{7} $, $ \ctg\frac{\alpha}{2} = -\sqrt{7} $.</p><p><strong>б)</strong> Дано: $ \sin \alpha = -\frac{4}{5} $ и $ \pi < \alpha < \frac{3\pi}{2} $.</p><p>Определим четверть для угла $ \frac{\alpha}{2} $. Разделив неравенство для $ \alpha $ на 2, получаем: $ \frac{\pi}{2} < \frac{\alpha}{2} < \frac{3\pi}{4} $. Угол $ \frac{\alpha}{2} $ находится во второй четверти. Следовательно, $ \sin\frac{\alpha}{2} > 0 $, $ \cos\frac{\alpha}{2} < 0 $, $ \tg\frac{\alpha}{2} < 0 $, $ \ctg\frac{\alpha}{2} < 0 $.</p><p>Для использования формул половинного угла нам нужен $ \cos\alpha $. Найдем его из основного тригонометрического тождества $ \sin^2\alpha + \cos^2\alpha = 1 $.</p><p>$ \cos^2\alpha = 1 - \sin^2\alpha = 1 - (-\frac{4}{5})^2 = 1 - \frac{16}{25} = \frac{9}{25} $.</p><p>Поскольку $ \alpha $ находится в третьей четверти ($ \pi < \alpha < \frac{3\pi}{2} $), $ \cos\alpha $ отрицателен. Значит, $ \cos\alpha = -\frac{3}{5} $.</p><p>Теперь применяем формулы половинного угла:</p><p>$ \sin^2\frac{\alpha}{2} = \frac{1 - \cos\alpha}{2} = \frac{1 - (-\frac{3}{5})}{2} = \frac{1 + \frac{3}{5}}{2} = \frac{\frac{8}{5}}{2} = \frac{4}{5} $. Так как $ \sin\frac{\alpha}{2} > 0 $, то $ \sin\frac{\alpha}{2} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5} $.</p><p>$ \cos^2\frac{\alpha}{2} = \frac{1 + \cos\alpha}{2} = \frac{1 + (-\frac{3}{5})}{2} = \frac{1 - \frac{3}{5}}{2} = \frac{\frac{2}{5}}{2} = \frac{1}{5} $. Так как $ \cos\frac{\alpha}{2} < 0 $, то $ \cos\frac{\alpha}{2} = -\sqrt{\frac{1}{5}} = -\frac{1}{\sqrt{5}} = -\frac{\sqrt{5}}{5} $.</p><p>$ \tg\frac{\alpha}{2} = \frac{\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}} = \frac{\frac{2\sqrt{5}}{5}}{-\frac{\sqrt{5}}{5}} = -2 $.</p><p>$ \ctg\frac{\alpha}{2} = \frac{1}{\tg\frac{\alpha}{2}} = -\frac{1}{2} $.</p><p><strong>Ответ:</strong> $ \sin\frac{\alpha}{2} = \frac{2\sqrt{5}}{5} $, $ \cos\frac{\alpha}{2} = -\frac{\sqrt{5}}{5} $, $ \tg\frac{\alpha}{2} = -2 $, $ \ctg\frac{\alpha}{2} = -\frac{1}{2} $.</p><p><strong>в)</strong> Дано: $ \tg \alpha = -\frac{7}{24} $ и $ \frac{\pi}{2} < \alpha < \pi $.</p><p>Определим четверть для угла $ \frac{\alpha}{2} $. Разделив неравенство для $ \alpha $ на 2, получаем: $ \frac{\pi}{4} < \frac{\alpha}{2} < \frac{\pi}{2} $. Угол $ \frac{\alpha}{2} $ находится в первой четверти. В первой четверти все тригонометрические функции положительны: $ \sin\frac{\alpha}{2} > 0 $, $ \cos\frac{\alpha}{2} > 0 $, $ \tg\frac{\alpha}{2} > 0 $, $ \ctg\frac{\alpha}{2} > 0 $.</p><p>Найдем $ \cos\alpha $, используя тождество $ 1 + \tg^2\alpha = \frac{1}{\cos^2\alpha} $.</p><p>$ \cos^2\alpha = \frac{1}{1 + \tg^2\alpha} = \frac{1}{1 + (-\frac{7}{24})^2} = \frac{1}{1 + \frac{49}{576}} = \frac{1}{\frac{625}{576}} = \frac{576}{625} $.</p><p>Поскольку $ \alpha $ находится во второй четверти ($ \frac{\pi}{2} < \alpha < \pi $), $ \cos\alpha $ отрицателен. Значит, $ \cos\alpha = -\sqrt{\frac{576}{625}} = -\frac{24}{25} $.</p><p>Теперь применяем формулы половинного угла:</p><p>$ \sin^2\frac{\alpha}{2} = \frac{1 - \cos\alpha}{2} = \frac{1 - (-\frac{24}{25})}{2} = \frac{1 + \frac{24}{25}}{2} = \frac{\frac{49}{25}}{2} = \frac{49}{50} $. Так как $ \sin\frac{\alpha}{2} > 0 $, то $ \sin\frac{\alpha}{2} = \sqrt{\frac{49}{50}} = \frac{7}{5\sqrt{2}} = \frac{7\sqrt{2}}{10} $.</p><p>$ \cos^2\frac{\alpha}{2} = \frac{1 + \cos\alpha}{2} = \frac{1 + (-\frac{24}{25})}{2} = \frac{1 - \frac{24}{25}}{2} = \frac{\frac{1}{25}}{2} = \frac{1}{50} $. Так как $ \cos\frac{\alpha}{2} > 0 $, то $ \cos\frac{\alpha}{2} = \sqrt{\frac{1}{50}} = \frac{1}{5\sqrt{2}} = \frac{\sqrt{2}}{10} $.</p><p>$ \tg\frac{\alpha}{2} = \frac{\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}} = \frac{\frac{7\sqrt{2}}{10}}{\frac{\sqrt{2}}{10}} = 7 $.</p><p>$ \ctg\frac{\alpha}{2} = \frac{1}{\tg\frac{\alpha}{2}} = \frac{1}{7} $.</p><p><strong>Ответ:</strong> $ \sin\frac{\alpha}{2} = \frac{7\sqrt{2}}{10} $, $ \cos\frac{\alpha}{2} = \frac{\sqrt{2}}{10} $, $ \tg\frac{\alpha}{2} = 7 $, $ \ctg\frac{\alpha}{2} = \frac{1}{7} $.</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" => "1108466" "type" => "task" ] "previous" => array:2 [ "refs" => "1108464" "type" => "task" ] "book" => 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#dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1170 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1171 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1172 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1173 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1174 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1175 #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 {#1176 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1177 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1178 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1179 #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 {#1153} "field_class" => Illuminate\Database\Eloquent\Collection {#1155} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1157} "field_author" => Illuminate\Database\Eloquent\Collection {#1159} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1163} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1164} "field_country" => Illuminate\Database\Eloquent\Collection {#1166} "field_city" => Illuminate\Database\Eloquent\Collection {#1168} "field_series" => Illuminate\Database\Eloquent\Collection {#1170} "field_umk" => Illuminate\Database\Eloquent\Collection {#1171} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1172} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1173} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1174} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1175} "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 {#1176} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1177} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1178} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1179} "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 {#1829 #items: array:1 [ 0 => App\Models\BookPage {#1183 #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" => 1097949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-207" "field_display_title" => "207" "field_folder" => "folder1" "field_image_name" => "207" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1187 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1186 #items: array:1 [ 0 => App\Models\Book {#1188 #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 {#1189 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1202 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1204 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1209 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1211 …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 {#1212 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1215 …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 {#1189 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1202 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1204 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1209 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1211 …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 {#1212 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1215 …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 {#1216 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ 0 => App\Models\Element {#1218 #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" => 1261100 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261100 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219 …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" => "1097950" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097948" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:6 [ 0 => App\Models\Task {#1227 #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" => 1108462 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-721" "field_display_title" => "721" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228 …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 {#1230 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1231 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1264 …2} "content" => Illuminate\Database\Eloquent\Collection {#1301 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1326 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108462 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-721" "field_display_title" => "721" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228 …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 {#1230 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1231 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1264 …2} "content" => Illuminate\Database\Eloquent\Collection {#1301 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1326 …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 {#1327 #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" => 1108463 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-722" "field_display_title" => "722" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1328 …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 {#1330 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1331 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1364 …2} "content" => Illuminate\Database\Eloquent\Collection {#1401 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1426 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108463 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-722" "field_display_title" => "722" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1328 …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 {#1330 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1331 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1364 …2} "content" => Illuminate\Database\Eloquent\Collection {#1401 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1426 …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 {#1427 #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" => 1108464 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-723" "field_display_title" => "723" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1428 …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 {#1430 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1431 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1464 …2} "content" => Illuminate\Database\Eloquent\Collection {#1501 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1526 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108464 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-723" "field_display_title" => "723" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1428 …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 {#1430 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1431 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1464 …2} "content" => Illuminate\Database\Eloquent\Collection {#1501 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1526 …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 {#1527 #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" => 1108465 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-724" "field_display_title" => "724" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1528 …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 {#1530 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1531 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1564 …2} "content" => Illuminate\Database\Eloquent\Collection {#1601 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1626 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108465 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-724" "field_display_title" => "724" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1528 …2} "field_metatags_title" => null "field_metatags_description" => null 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№724 (с. 207)
Условие. №724 (с. 207)
скриншот условия
724. Найдите $ \sin \frac{\alpha}{2} $, $ \cos \frac{\alpha}{2} $, $ \operatorname{tg} \frac{\alpha}{2} $ и $ \operatorname{ctg} \frac{\alpha}{2} $, если:
а) $ \cos \alpha = \frac{3}{4} $ и $ \frac{3\pi}{2} < \alpha < 2\pi $;
Б) $ \sin \alpha = -\frac{4}{5} $ и $ \pi < \alpha < \frac{3\pi}{2} $;
В) $ \operatorname{tg} \alpha = -\frac{7}{24} $ и $ \frac{\pi}{2} < \alpha < \pi $.
Решение. №724 (с. 207)
Решение 2 (rus). №724 (с. 207)
а) Дано: $ \cos \alpha = \frac{3}{4} $ и $ \frac{3\pi}{2} < \alpha < 2\pi $.
Сначала определим, в какой четверти находится угол $ \frac{\alpha}{2} $. Разделив неравенство для $ \alpha $ на 2, получаем: $ \frac{3\pi}{4} < \frac{\alpha}{2} < \pi $. Это означает, что угол $ \frac{\alpha}{2} $ находится во второй четверти. Во второй четверти синус положителен ($ \sin\frac{\alpha}{2} > 0 $), а косинус, тангенс и котангенс отрицательны ($ \cos\frac{\alpha}{2} < 0 $, $ \tg\frac{\alpha}{2} < 0 $, $ \ctg\frac{\alpha}{2} < 0 $).
Воспользуемся формулами половинного угла:
$ \sin^2\frac{\alpha}{2} = \frac{1 - \cos\alpha}{2} = \frac{1 - \frac{3}{4}}{2} = \frac{\frac{1}{4}}{2} = \frac{1}{8} $. Так как $ \sin\frac{\alpha}{2} > 0 $, то $ \sin\frac{\alpha}{2} = \sqrt{\frac{1}{8}} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4} $.
$ \cos^2\frac{\alpha}{2} = \frac{1 + \cos\alpha}{2} = \frac{1 + \frac{3}{4}}{2} = \frac{\frac{7}{4}}{2} = \frac{7}{8} $. Так как $ \cos\frac{\alpha}{2} < 0 $, то $ \cos\frac{\alpha}{2} = -\sqrt{\frac{7}{8}} = -\frac{\sqrt{7}}{2\sqrt{2}} = -\frac{\sqrt{14}}{4} $.
$ \tg\frac{\alpha}{2} = \frac{\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}} = \frac{\frac{\sqrt{2}}{4}}{-\frac{\sqrt{14}}{4}} = -\frac{\sqrt{2}}{\sqrt{14}} = -\frac{1}{\sqrt{7}} = -\frac{\sqrt{7}}{7} $.
$ \ctg\frac{\alpha}{2} = \frac{1}{\tg\frac{\alpha}{2}} = \frac{1}{-\frac{\sqrt{7}}{7}} = -\sqrt{7} $.
Ответ: $ \sin\frac{\alpha}{2} = \frac{\sqrt{2}}{4} $, $ \cos\frac{\alpha}{2} = -\frac{\sqrt{14}}{4} $, $ \tg\frac{\alpha}{2} = -\frac{\sqrt{7}}{7} $, $ \ctg\frac{\alpha}{2} = -\sqrt{7} $.
б) Дано: $ \sin \alpha = -\frac{4}{5} $ и $ \pi < \alpha < \frac{3\pi}{2} $.
Определим четверть для угла $ \frac{\alpha}{2} $. Разделив неравенство для $ \alpha $ на 2, получаем: $ \frac{\pi}{2} < \frac{\alpha}{2} < \frac{3\pi}{4} $. Угол $ \frac{\alpha}{2} $ находится во второй четверти. Следовательно, $ \sin\frac{\alpha}{2} > 0 $, $ \cos\frac{\alpha}{2} < 0 $, $ \tg\frac{\alpha}{2} < 0 $, $ \ctg\frac{\alpha}{2} < 0 $.
Для использования формул половинного угла нам нужен $ \cos\alpha $. Найдем его из основного тригонометрического тождества $ \sin^2\alpha + \cos^2\alpha = 1 $.
$ \cos^2\alpha = 1 - \sin^2\alpha = 1 - (-\frac{4}{5})^2 = 1 - \frac{16}{25} = \frac{9}{25} $.
Поскольку $ \alpha $ находится в третьей четверти ($ \pi < \alpha < \frac{3\pi}{2} $), $ \cos\alpha $ отрицателен. Значит, $ \cos\alpha = -\frac{3}{5} $.
Теперь применяем формулы половинного угла:
$ \sin^2\frac{\alpha}{2} = \frac{1 - \cos\alpha}{2} = \frac{1 - (-\frac{3}{5})}{2} = \frac{1 + \frac{3}{5}}{2} = \frac{\frac{8}{5}}{2} = \frac{4}{5} $. Так как $ \sin\frac{\alpha}{2} > 0 $, то $ \sin\frac{\alpha}{2} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5} $.
$ \cos^2\frac{\alpha}{2} = \frac{1 + \cos\alpha}{2} = \frac{1 + (-\frac{3}{5})}{2} = \frac{1 - \frac{3}{5}}{2} = \frac{\frac{2}{5}}{2} = \frac{1}{5} $. Так как $ \cos\frac{\alpha}{2} < 0 $, то $ \cos\frac{\alpha}{2} = -\sqrt{\frac{1}{5}} = -\frac{1}{\sqrt{5}} = -\frac{\sqrt{5}}{5} $.
$ \tg\frac{\alpha}{2} = \frac{\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}} = \frac{\frac{2\sqrt{5}}{5}}{-\frac{\sqrt{5}}{5}} = -2 $.
$ \ctg\frac{\alpha}{2} = \frac{1}{\tg\frac{\alpha}{2}} = -\frac{1}{2} $.
Ответ: $ \sin\frac{\alpha}{2} = \frac{2\sqrt{5}}{5} $, $ \cos\frac{\alpha}{2} = -\frac{\sqrt{5}}{5} $, $ \tg\frac{\alpha}{2} = -2 $, $ \ctg\frac{\alpha}{2} = -\frac{1}{2} $.
в) Дано: $ \tg \alpha = -\frac{7}{24} $ и $ \frac{\pi}{2} < \alpha < \pi $.
Определим четверть для угла $ \frac{\alpha}{2} $. Разделив неравенство для $ \alpha $ на 2, получаем: $ \frac{\pi}{4} < \frac{\alpha}{2} < \frac{\pi}{2} $. Угол $ \frac{\alpha}{2} $ находится в первой четверти. В первой четверти все тригонометрические функции положительны: $ \sin\frac{\alpha}{2} > 0 $, $ \cos\frac{\alpha}{2} > 0 $, $ \tg\frac{\alpha}{2} > 0 $, $ \ctg\frac{\alpha}{2} > 0 $.
Найдем $ \cos\alpha $, используя тождество $ 1 + \tg^2\alpha = \frac{1}{\cos^2\alpha} $.
$ \cos^2\alpha = \frac{1}{1 + \tg^2\alpha} = \frac{1}{1 + (-\frac{7}{24})^2} = \frac{1}{1 + \frac{49}{576}} = \frac{1}{\frac{625}{576}} = \frac{576}{625} $.
Поскольку $ \alpha $ находится во второй четверти ($ \frac{\pi}{2} < \alpha < \pi $), $ \cos\alpha $ отрицателен. Значит, $ \cos\alpha = -\sqrt{\frac{576}{625}} = -\frac{24}{25} $.
Теперь применяем формулы половинного угла:
$ \sin^2\frac{\alpha}{2} = \frac{1 - \cos\alpha}{2} = \frac{1 - (-\frac{24}{25})}{2} = \frac{1 + \frac{24}{25}}{2} = \frac{\frac{49}{25}}{2} = \frac{49}{50} $. Так как $ \sin\frac{\alpha}{2} > 0 $, то $ \sin\frac{\alpha}{2} = \sqrt{\frac{49}{50}} = \frac{7}{5\sqrt{2}} = \frac{7\sqrt{2}}{10} $.
$ \cos^2\frac{\alpha}{2} = \frac{1 + \cos\alpha}{2} = \frac{1 + (-\frac{24}{25})}{2} = \frac{1 - \frac{24}{25}}{2} = \frac{\frac{1}{25}}{2} = \frac{1}{50} $. Так как $ \cos\frac{\alpha}{2} > 0 $, то $ \cos\frac{\alpha}{2} = \sqrt{\frac{1}{50}} = \frac{1}{5\sqrt{2}} = \frac{\sqrt{2}}{10} $.
$ \tg\frac{\alpha}{2} = \frac{\sin\frac{\alpha}{2}}{\cos\frac{\alpha}{2}} = \frac{\frac{7\sqrt{2}}{10}}{\frac{\sqrt{2}}{10}} = 7 $.
$ \ctg\frac{\alpha}{2} = \frac{1}{\tg\frac{\alpha}{2}} = \frac{1}{7} $.
Ответ: $ \sin\frac{\alpha}{2} = \frac{7\sqrt{2}}{10} $, $ \cos\frac{\alpha}{2} = \frac{\sqrt{2}}{10} $, $ \tg\frac{\alpha}{2} = 7 $, $ \ctg\frac{\alpha}{2} = \frac{1}{7} $.
Другие задания:
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