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Номер 735, страница 209 - 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 735, страница 209.
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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 } 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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. 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Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1087 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } 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Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1096 #items: array:1 [ …1] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1096} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125} ] #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" => 1277090 "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 #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 {#1132 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1133 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1134 #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 {#1130} "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} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1133} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1134} "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" => "1108476" "type" => "task" ] "text" => "<p><strong>735.</strong> Найдите угол $ \alpha $, если $ \alpha \in [0; 2\pi] $ и выполняется условие:</p><p><strong>a)</strong> $ \sin \frac{\alpha}{2} \cdot \cos \frac{\alpha}{2} = \frac{1}{4}; $</p><p><strong>б)</strong> $ \sin \alpha \cdot \cos \alpha = \frac{1}{2}; $</p><p><strong>в)</strong> $ \sin^2 \frac{\alpha}{4} - \cos^2 \frac{\alpha}{4} = \frac{1}{2}; $</p><p><strong>г)</strong> $ \cos^2 2\alpha - \sin^2 2\alpha = 1. $</p>" "img" => array:1 [ 0 => array:5 [ "name" => "735-1.jpg" "alt" => null "width" => "1371" "height" => 395 "path" => "/media/algebra_09/soltan-u19/0-1/735-1.webp?ts=1752834734" ] ] ] #original: array:7 [ "id" => 1277090 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128} "task" => array:2 [ "refs" => "1108476" "type" => "task" ] "text" => "<p><strong>735.</strong> Найдите угол $ \alpha $, если $ \alpha \in [0; 2\pi] $ и выполняется условие:</p><p><strong>a)</strong> $ \sin \frac{\alpha}{2} \cdot \cos \frac{\alpha}{2} = \frac{1}{4}; $</p><p><strong>б)</strong> $ \sin \alpha \cdot \cos \alpha = \frac{1}{2}; $</p><p><strong>в)</strong> $ \sin^2 \frac{\alpha}{4} - \cos^2 \frac{\alpha}{4} = \frac{1}{2}; $</p><p><strong>г)</strong> $ \cos^2 2\alpha - \sin^2 2\alpha = 1. $</p>" "img" => array:1 [ 0 => array:5 [ "name" => "735-1.jpg" "alt" => null "width" => "1371" "height" => 395 "path" => "/media/algebra_09/soltan-u19/0-1/735-1.webp?ts=1752834734" ] ] ] #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" => 1278381 "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 #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 {#1140 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142 #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 {#1138} "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} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1141} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1142} "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" => "1108476" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "735-1.jpg" "alt" => null "width" => "879" "height" => 836 "path" => "/media/algebra_09/soltan-u19/1-1/735-1.webp?ts=1752831534" ] 1 => array:5 [ "name" => "735-2.jpg" "alt" => null "width" => "785" "height" => 2350 "path" => "/media/algebra_09/soltan-u19/1-1/735-2.webp?ts=1752831534" ] 2 => array:5 [ "name" => "735-3.jpg" "alt" => null "width" => "921" "height" => 396 "path" => "/media/algebra_09/soltan-u19/1-1/735-3.webp?ts=1752831534" ] ] ] #original: array:6 [ "id" => 1278381 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136} "task" => array:2 [ "refs" => "1108476" "type" => "task" ] "img" => array:3 [ 0 => array:5 [ "name" => "735-1.jpg" "alt" => null "width" => "879" "height" => 836 "path" => "/media/algebra_09/soltan-u19/1-1/735-1.webp?ts=1752831534" ] 1 => array:5 [ "name" => "735-2.jpg" "alt" => null "width" => "785" "height" => 2350 "path" => "/media/algebra_09/soltan-u19/1-1/735-2.webp?ts=1752831534" ] 2 => array:5 [ "name" => "735-3.jpg" "alt" => null "width" => "921" "height" => 396 "path" => "/media/algebra_09/soltan-u19/1-1/735-3.webp?ts=1752831534" ] ] ] #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" => 1554336 "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 #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 {#1148 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1149 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1150 #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 {#1146} "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} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1149} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1150} "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" => "1108476" "type" => "task" ] "text" => "<p><strong>а)</strong> Дано уравнение $ \sin\frac{\alpha}{2} \cdot \cos\frac{\alpha}{2} = \frac{1}{4} $. Применим формулу синуса двойного угла $ \sin(2x) = 2\sin x \cos x $. Если взять $ x = \frac{\alpha}{2} $, то получим $ \sin\alpha = 2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2} $. Из этой формулы следует, что $ \sin\frac{\alpha}{2}\cos\frac{\alpha}{2} = \frac{1}{2}\sin\alpha $. Подставим это выражение в исходное уравнение:<br>$ \frac{1}{2}\sin\alpha = \frac{1}{4} $<br>$ \sin\alpha = \frac{1}{2} $<br>Теперь найдем все значения $ \alpha $ из интервала $ [0; 2\pi] $, которые удовлетворяют этому уравнению. Такими значениями являются $ \alpha_1 = \frac{\pi}{6} $ и $ \alpha_2 = \pi - \frac{\pi}{6} = \frac{5\pi}{6} $.<br>Ответ: $ \frac{\pi}{6}; \frac{5\pi}{6} $.</p><p><strong>б)</strong> Дано уравнение $ \sin\alpha \cdot \cos\alpha = \frac{1}{2} $. Используем ту же формулу синуса двойного угла $ \sin(2\alpha) = 2\sin\alpha\cos\alpha $, из которой $ \sin\alpha\cos\alpha = \frac{1}{2}\sin(2\alpha) $. Подставим в уравнение:<br>$ \frac{1}{2}\sin(2\alpha) = \frac{1}{2} $<br>$ \sin(2\alpha) = 1 $<br>Пусть $ t = 2\alpha $. Так как по условию $ \alpha \in [0; 2\pi] $, то $ t \in [0; 4\pi] $. Решим уравнение $ \sin t = 1 $ на интервале $ [0; 4\pi] $. Решениями являются $ t_1 = \frac{\pi}{2} $ и $ t_2 = \frac{\pi}{2} + 2\pi = \frac{5\pi}{2} $. Теперь вернемся к $ \alpha $:<br>$ 2\alpha_1 = \frac{\pi}{2} \Rightarrow \alpha_1 = \frac{\pi}{4} $<br>$ 2\alpha_2 = \frac{5\pi}{2} \Rightarrow \alpha_2 = \frac{5\pi}{4} $<br>Оба найденных значения принадлежат заданному интервалу $ [0; 2\pi] $.<br>Ответ: $ \frac{\pi}{4}; \frac{5\pi}{4} $.</p><p><strong>в)</strong> Дано уравнение $ \sin^2\frac{\alpha}{4} - \cos^2\frac{\alpha}{4} = \frac{1}{2} $. Воспользуемся формулой косинуса двойного угла $ \cos(2x) = \cos^2 x - \sin^2 x $. Левая часть уравнения представляет собой $ -(\cos^2\frac{\alpha}{4} - \sin^2\frac{\alpha}{4}) $, что равно $ -\cos(2 \cdot \frac{\alpha}{4}) = -\cos\frac{\alpha}{2} $. Таким образом, уравнение принимает вид:<br>$ -\cos\frac{\alpha}{2} = \frac{1}{2} $<br>$ \cos\frac{\alpha}{2} = -\frac{1}{2} $<br>Пусть $ t = \frac{\alpha}{2} $. Поскольку $ \alpha \in [0; 2\pi] $, то $ t \in [0; \pi] $. Нам нужно решить уравнение $ \cos t = -\frac{1}{2} $ на отрезке $ [0; \pi] $. Единственным решением на этом отрезке является $ t = \frac{2\pi}{3} $. Выполним обратную замену:<br>$ \frac{\alpha}{2} = \frac{2\pi}{3} \Rightarrow \alpha = \frac{4\pi}{3} $.<br>Данное значение находится в пределах интервала $ [0; 2\pi] $.<br>Ответ: $ \frac{4\pi}{3} $.</p><p><strong>г)</strong> Дано уравнение $ \cos^2(2\alpha) - \sin^2(2\alpha) = 1 $. Применим формулу косинуса двойного угла $ \cos(2x) = \cos^2 x - \sin^2 x $. При $ x = 2\alpha $ левая часть уравнения становится равной $ \cos(2 \cdot 2\alpha) = \cos(4\alpha) $. Уравнение сводится к следующему:<br>$ \cos(4\alpha) = 1 $<br>Пусть $ t = 4\alpha $. Учитывая, что $ \alpha \in [0; 2\pi] $, соответствующий интервал для $ t $ будет $ [0; 8\pi] $. Решим уравнение $ \cos t = 1 $ на интервале $ [0; 8\pi] $. Решениями являются значения $ t = 2\pi k $, где $ k $ — целое число. Выберем значения $ t $, попадающие в интервал $ [0; 8\pi] $:<br>$ t_0 = 0 $ (при k=0)<br>$ t_1 = 2\pi $ (при k=1)<br>$ t_2 = 4\pi $ (при k=2)<br>$ t_3 = 6\pi $ (при k=3)<br>$ t_4 = 8\pi $ (при k=4)<br>Теперь найдем соответствующие значения $ \alpha $:<br>$ 4\alpha_0 = 0 \Rightarrow \alpha_0 = 0 $<br>$ 4\alpha_1 = 2\pi \Rightarrow \alpha_1 = \frac{\pi}{2} $<br>$ 4\alpha_2 = 4\pi \Rightarrow \alpha_2 = \pi $<br>$ 4\alpha_3 = 6\pi \Rightarrow \alpha_3 = \frac{3\pi}{2} $<br>$ 4\alpha_4 = 8\pi \Rightarrow \alpha_4 = 2\pi $<br>Все пять значений входят в заданный интервал $ [0; 2\pi] $.<br>Ответ: $ 0; \frac{\pi}{2}; \pi; \frac{3\pi}{2}; 2\pi $.</p>" ] #original: array:6 [ "id" => 1554336 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144} "task" => array:2 [ "refs" => "1108476" "type" => "task" ] "text" => "<p><strong>а)</strong> Дано уравнение $ \sin\frac{\alpha}{2} \cdot \cos\frac{\alpha}{2} = \frac{1}{4} $. Применим формулу синуса двойного угла $ \sin(2x) = 2\sin x \cos x $. Если взять $ x = \frac{\alpha}{2} $, то получим $ \sin\alpha = 2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2} $. Из этой формулы следует, что $ \sin\frac{\alpha}{2}\cos\frac{\alpha}{2} = \frac{1}{2}\sin\alpha $. Подставим это выражение в исходное уравнение:<br>$ \frac{1}{2}\sin\alpha = \frac{1}{4} $<br>$ \sin\alpha = \frac{1}{2} $<br>Теперь найдем все значения $ \alpha $ из интервала $ [0; 2\pi] $, которые удовлетворяют этому уравнению. Такими значениями являются $ \alpha_1 = \frac{\pi}{6} $ и $ \alpha_2 = \pi - \frac{\pi}{6} = \frac{5\pi}{6} $.<br>Ответ: $ \frac{\pi}{6}; \frac{5\pi}{6} $.</p><p><strong>б)</strong> Дано уравнение $ \sin\alpha \cdot \cos\alpha = \frac{1}{2} $. Используем ту же формулу синуса двойного угла $ \sin(2\alpha) = 2\sin\alpha\cos\alpha $, из которой $ \sin\alpha\cos\alpha = \frac{1}{2}\sin(2\alpha) $. Подставим в уравнение:<br>$ \frac{1}{2}\sin(2\alpha) = \frac{1}{2} $<br>$ \sin(2\alpha) = 1 $<br>Пусть $ t = 2\alpha $. Так как по условию $ \alpha \in [0; 2\pi] $, то $ t \in [0; 4\pi] $. Решим уравнение $ \sin t = 1 $ на интервале $ [0; 4\pi] $. Решениями являются $ t_1 = \frac{\pi}{2} $ и $ t_2 = \frac{\pi}{2} + 2\pi = \frac{5\pi}{2} $. Теперь вернемся к $ \alpha $:<br>$ 2\alpha_1 = \frac{\pi}{2} \Rightarrow \alpha_1 = \frac{\pi}{4} $<br>$ 2\alpha_2 = \frac{5\pi}{2} \Rightarrow \alpha_2 = \frac{5\pi}{4} $<br>Оба найденных значения принадлежат заданному интервалу $ [0; 2\pi] $.<br>Ответ: $ \frac{\pi}{4}; \frac{5\pi}{4} $.</p><p><strong>в)</strong> Дано уравнение $ \sin^2\frac{\alpha}{4} - \cos^2\frac{\alpha}{4} = \frac{1}{2} $. Воспользуемся формулой косинуса двойного угла $ \cos(2x) = \cos^2 x - \sin^2 x $. Левая часть уравнения представляет собой $ -(\cos^2\frac{\alpha}{4} - \sin^2\frac{\alpha}{4}) $, что равно $ -\cos(2 \cdot \frac{\alpha}{4}) = -\cos\frac{\alpha}{2} $. Таким образом, уравнение принимает вид:<br>$ -\cos\frac{\alpha}{2} = \frac{1}{2} $<br>$ \cos\frac{\alpha}{2} = -\frac{1}{2} $<br>Пусть $ t = \frac{\alpha}{2} $. Поскольку $ \alpha \in [0; 2\pi] $, то $ t \in [0; \pi] $. Нам нужно решить уравнение $ \cos t = -\frac{1}{2} $ на отрезке $ [0; \pi] $. Единственным решением на этом отрезке является $ t = \frac{2\pi}{3} $. Выполним обратную замену:<br>$ \frac{\alpha}{2} = \frac{2\pi}{3} \Rightarrow \alpha = \frac{4\pi}{3} $.<br>Данное значение находится в пределах интервала $ [0; 2\pi] $.<br>Ответ: $ \frac{4\pi}{3} $.</p><p><strong>г)</strong> Дано уравнение $ \cos^2(2\alpha) - \sin^2(2\alpha) = 1 $. Применим формулу косинуса двойного угла $ \cos(2x) = \cos^2 x - \sin^2 x $. При $ x = 2\alpha $ левая часть уравнения становится равной $ \cos(2 \cdot 2\alpha) = \cos(4\alpha) $. Уравнение сводится к следующему:<br>$ \cos(4\alpha) = 1 $<br>Пусть $ t = 4\alpha $. Учитывая, что $ \alpha \in [0; 2\pi] $, соответствующий интервал для $ t $ будет $ [0; 8\pi] $. Решим уравнение $ \cos t = 1 $ на интервале $ [0; 8\pi] $. Решениями являются значения $ t = 2\pi k $, где $ k $ — целое число. Выберем значения $ t $, попадающие в интервал $ [0; 8\pi] $:<br>$ t_0 = 0 $ (при k=0)<br>$ t_1 = 2\pi $ (при k=1)<br>$ t_2 = 4\pi $ (при k=2)<br>$ t_3 = 6\pi $ (при k=3)<br>$ t_4 = 8\pi $ (при k=4)<br>Теперь найдем соответствующие значения $ \alpha $:<br>$ 4\alpha_0 = 0 \Rightarrow \alpha_0 = 0 $<br>$ 4\alpha_1 = 2\pi \Rightarrow \alpha_1 = \frac{\pi}{2} $<br>$ 4\alpha_2 = 4\pi \Rightarrow \alpha_2 = \pi $<br>$ 4\alpha_3 = 6\pi \Rightarrow \alpha_3 = \frac{3\pi}{2} $<br>$ 4\alpha_4 = 8\pi \Rightarrow \alpha_4 = 2\pi $<br>Все пять значений входят в заданный интервал $ [0; 2\pi] $.<br>Ответ: $ 0; \frac{\pi}{2}; \pi; \frac{3\pi}{2}; 2\pi $.</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" => "1108477" "type" => "task" ] "previous" => array:2 [ "refs" => "1108475" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1151 #items: array:1 [ 0 => App\Models\Book {#1152 #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 {#1153 #items: array:1 [ 0 => App\Models\Term {#1154 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1155 #items: array:1 [ 0 => App\Models\Term {#1156 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ "field_accusative_case" => "девятый" "field_creative_case" => "девятым" "field_dative_case" => "девятому" "field_genitive_case" => "девятого" "field_nominative_case" => "девятый" "field_prepositional_case" => "девятом" ] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1157 #items: array:1 [ 0 => App\Models\Term {#1158 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ "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" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "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" => "Kokshetau" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1159 #items: array:3 [ 0 => App\Models\Term {#1160 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1161 #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 {#1162 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true 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"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 {#1168 #items: array:1 [ 0 => App\Models\Term {#1169 #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 {#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 {#1194 #items: array:1 [ 0 => App\Models\BookPage {#1193 #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" => 1097951 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-209" "field_display_title" => "209" "field_folder" => "folder1" "field_image_name" => "209" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1195 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:1 [ 0 => App\Models\Book {#1196 #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 {#1198 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1199 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1201 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1207 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1216 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1217 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1218 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1219 …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 {#1220 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1221 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1222 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1223 …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 {#1198} "field_class" => Illuminate\Database\Eloquent\Collection {#1199} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1201} "field_author" => Illuminate\Database\Eloquent\Collection {#1203} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1207} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1216 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1217 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1218 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1219 …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 {#1220 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1221 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1222 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1223 …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 {#1224 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:1 [ 0 => App\Models\Element {#1234 #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" => 1261102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1244 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261102 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1244 …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" => "1097952" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097950" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1738 #items: array:4 [ 0 => App\Models\Task {#1748 #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" => 1108475 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-734" "field_display_title" => "734" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1752 …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 {#1751 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1786 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1881 …2} "content" => Illuminate\Database\Eloquent\Collection {#1899 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1906 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108475 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-734" "field_display_title" => "734" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1752 …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 {#1751 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1786 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1881 …2} "content" => Illuminate\Database\Eloquent\Collection {#1899 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1906 …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 {#1890 #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" => 1108476 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-735" "field_display_title" => "735" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1891 …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 {#1892 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1897 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1896 …2} "content" => Illuminate\Database\Eloquent\Collection {#1885 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1901 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108476 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-735" "field_display_title" => "735" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1891 …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 {#1892 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1897 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1896 …2} "content" => Illuminate\Database\Eloquent\Collection {#1885 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1901 …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 {#1895 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-736" "field_display_title" => "736" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1894 …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 {#1893 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1888 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1889 …2} "content" => Illuminate\Database\Eloquent\Collection {#1907 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1938 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108477 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-736" "field_display_title" => "736" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1894 …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 {#1893 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1888 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1889 …2} "content" => Illuminate\Database\Eloquent\Collection {#1907 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1938 …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 {#1882 #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" => 1108478 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-737" "field_display_title" => "737" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1884 …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 {#1883 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1905 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1904 …2} "content" => Illuminate\Database\Eloquent\Collection {#1953 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1960 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108478 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "209" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-737" "field_display_title" => "737" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1884 …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 {#1883 …2} "top_parent_branch" => 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№735 (с. 209)
Условие. №735 (с. 209)
скриншот условия
735. Найдите угол $ \alpha $, если $ \alpha \in [0; 2\pi] $ и выполняется условие:
a) $ \sin \frac{\alpha}{2} \cdot \cos \frac{\alpha}{2} = \frac{1}{4}; $
б) $ \sin \alpha \cdot \cos \alpha = \frac{1}{2}; $
в) $ \sin^2 \frac{\alpha}{4} - \cos^2 \frac{\alpha}{4} = \frac{1}{2}; $
г) $ \cos^2 2\alpha - \sin^2 2\alpha = 1. $
Решение. №735 (с. 209)
Решение 2 (rus). №735 (с. 209)
а) Дано уравнение $ \sin\frac{\alpha}{2} \cdot \cos\frac{\alpha}{2} = \frac{1}{4} $. Применим формулу синуса двойного угла $ \sin(2x) = 2\sin x \cos x $. Если взять $ x = \frac{\alpha}{2} $, то получим $ \sin\alpha = 2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2} $. Из этой формулы следует, что $ \sin\frac{\alpha}{2}\cos\frac{\alpha}{2} = \frac{1}{2}\sin\alpha $. Подставим это выражение в исходное уравнение:
$ \frac{1}{2}\sin\alpha = \frac{1}{4} $
$ \sin\alpha = \frac{1}{2} $
Теперь найдем все значения $ \alpha $ из интервала $ [0; 2\pi] $, которые удовлетворяют этому уравнению. Такими значениями являются $ \alpha_1 = \frac{\pi}{6} $ и $ \alpha_2 = \pi - \frac{\pi}{6} = \frac{5\pi}{6} $.
Ответ: $ \frac{\pi}{6}; \frac{5\pi}{6} $.
б) Дано уравнение $ \sin\alpha \cdot \cos\alpha = \frac{1}{2} $. Используем ту же формулу синуса двойного угла $ \sin(2\alpha) = 2\sin\alpha\cos\alpha $, из которой $ \sin\alpha\cos\alpha = \frac{1}{2}\sin(2\alpha) $. Подставим в уравнение:
$ \frac{1}{2}\sin(2\alpha) = \frac{1}{2} $
$ \sin(2\alpha) = 1 $
Пусть $ t = 2\alpha $. Так как по условию $ \alpha \in [0; 2\pi] $, то $ t \in [0; 4\pi] $. Решим уравнение $ \sin t = 1 $ на интервале $ [0; 4\pi] $. Решениями являются $ t_1 = \frac{\pi}{2} $ и $ t_2 = \frac{\pi}{2} + 2\pi = \frac{5\pi}{2} $. Теперь вернемся к $ \alpha $:
$ 2\alpha_1 = \frac{\pi}{2} \Rightarrow \alpha_1 = \frac{\pi}{4} $
$ 2\alpha_2 = \frac{5\pi}{2} \Rightarrow \alpha_2 = \frac{5\pi}{4} $
Оба найденных значения принадлежат заданному интервалу $ [0; 2\pi] $.
Ответ: $ \frac{\pi}{4}; \frac{5\pi}{4} $.
в) Дано уравнение $ \sin^2\frac{\alpha}{4} - \cos^2\frac{\alpha}{4} = \frac{1}{2} $. Воспользуемся формулой косинуса двойного угла $ \cos(2x) = \cos^2 x - \sin^2 x $. Левая часть уравнения представляет собой $ -(\cos^2\frac{\alpha}{4} - \sin^2\frac{\alpha}{4}) $, что равно $ -\cos(2 \cdot \frac{\alpha}{4}) = -\cos\frac{\alpha}{2} $. Таким образом, уравнение принимает вид:
$ -\cos\frac{\alpha}{2} = \frac{1}{2} $
$ \cos\frac{\alpha}{2} = -\frac{1}{2} $
Пусть $ t = \frac{\alpha}{2} $. Поскольку $ \alpha \in [0; 2\pi] $, то $ t \in [0; \pi] $. Нам нужно решить уравнение $ \cos t = -\frac{1}{2} $ на отрезке $ [0; \pi] $. Единственным решением на этом отрезке является $ t = \frac{2\pi}{3} $. Выполним обратную замену:
$ \frac{\alpha}{2} = \frac{2\pi}{3} \Rightarrow \alpha = \frac{4\pi}{3} $.
Данное значение находится в пределах интервала $ [0; 2\pi] $.
Ответ: $ \frac{4\pi}{3} $.
г) Дано уравнение $ \cos^2(2\alpha) - \sin^2(2\alpha) = 1 $. Применим формулу косинуса двойного угла $ \cos(2x) = \cos^2 x - \sin^2 x $. При $ x = 2\alpha $ левая часть уравнения становится равной $ \cos(2 \cdot 2\alpha) = \cos(4\alpha) $. Уравнение сводится к следующему:
$ \cos(4\alpha) = 1 $
Пусть $ t = 4\alpha $. Учитывая, что $ \alpha \in [0; 2\pi] $, соответствующий интервал для $ t $ будет $ [0; 8\pi] $. Решим уравнение $ \cos t = 1 $ на интервале $ [0; 8\pi] $. Решениями являются значения $ t = 2\pi k $, где $ k $ — целое число. Выберем значения $ t $, попадающие в интервал $ [0; 8\pi] $:
$ t_0 = 0 $ (при k=0)
$ t_1 = 2\pi $ (при k=1)
$ t_2 = 4\pi $ (при k=2)
$ t_3 = 6\pi $ (при k=3)
$ t_4 = 8\pi $ (при k=4)
Теперь найдем соответствующие значения $ \alpha $:
$ 4\alpha_0 = 0 \Rightarrow \alpha_0 = 0 $
$ 4\alpha_1 = 2\pi \Rightarrow \alpha_1 = \frac{\pi}{2} $
$ 4\alpha_2 = 4\pi \Rightarrow \alpha_2 = \pi $
$ 4\alpha_3 = 6\pi \Rightarrow \alpha_3 = \frac{3\pi}{2} $
$ 4\alpha_4 = 8\pi \Rightarrow \alpha_4 = 2\pi $
Все пять значений входят в заданный интервал $ [0; 2\pi] $.
Ответ: $ 0; \frac{\pi}{2}; \pi; \frac{3\pi}{2}; 2\pi $.
Другие задания:
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