gdz.top
Номер 726, 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 726, 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 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"field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1066} "field_class" => Illuminate\Database\Eloquent\Collection {#1068} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070} "field_author" => Illuminate\Database\Eloquent\Collection {#1072} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077} "field_country" => Illuminate\Database\Eloquent\Collection {#1079} "field_city" => Illuminate\Database\Eloquent\Collection {#1081} "field_series" => Illuminate\Database\Eloquent\Collection {#1083} "field_umk" => 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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" => 1277081 "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" => "1108467" "type" => "task" ] "text" => "<p><strong>726.</strong> Учащимся было дано задание: найти при $\alpha = 120^\circ$ значение выражения $\sqrt{(1 - \cos \alpha)^2 + \sin^2 \alpha}$.</p><p><strong>Николай выполнил задание так:</strong></p><p>$\sqrt{(1 - \cos \alpha)^2 + \sin^2 \alpha} = \sqrt{(1 - \cos 60^\circ)^2 + \sin^2 60^\circ} = \sqrt{\frac{1}{4} + \frac{3}{4}} = 1$.</p><p><strong>Василий решал так:</strong></p><p>$\sqrt{(1 - \cos \alpha)^2 + \sin^2 \alpha} = \sqrt{1 - 2 \cos \alpha + \cos^2 \alpha + \sin^2 \alpha} = \sqrt{2(1 - \cos \alpha)} = \sqrt{4 \sin^2 \frac{\alpha}{2}} = 2 \left| \sin \frac{\alpha}{2} \right| = 2 \sin 60^\circ = \sqrt{3}$.</p><p>Чье решение было правильным?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "726-1.jpg" "alt" => null "width" => "1567" "height" => 598 "path" => "/media/algebra_09/soltan-u19/0-1/726-1.webp?ts=1752834734" ] ] ] #original: array:7 [ "id" => 1277081 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128} "task" => array:2 [ "refs" => "1108467" "type" => "task" ] "text" => "<p><strong>726.</strong> Учащимся было дано задание: найти при $\alpha = 120^\circ$ значение выражения $\sqrt{(1 - \cos \alpha)^2 + \sin^2 \alpha}$.</p><p><strong>Николай выполнил задание так:</strong></p><p>$\sqrt{(1 - \cos \alpha)^2 + \sin^2 \alpha} = \sqrt{(1 - \cos 60^\circ)^2 + \sin^2 60^\circ} = \sqrt{\frac{1}{4} + \frac{3}{4}} = 1$.</p><p><strong>Василий решал так:</strong></p><p>$\sqrt{(1 - \cos \alpha)^2 + \sin^2 \alpha} = \sqrt{1 - 2 \cos \alpha + \cos^2 \alpha + \sin^2 \alpha} = \sqrt{2(1 - \cos \alpha)} = \sqrt{4 \sin^2 \frac{\alpha}{2}} = 2 \left| \sin \frac{\alpha}{2} \right| = 2 \sin 60^\circ = \sqrt{3}$.</p><p>Чье решение было правильным?</p>" "img" => array:1 [ 0 => array:5 [ "name" => "726-1.jpg" "alt" => null "width" => "1567" "height" => 598 "path" => "/media/algebra_09/soltan-u19/0-1/726-1.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" => 1278372 "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" => "1108467" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "726-1.jpg" "alt" => null "width" => "1373" "height" => 328 "path" => "/media/algebra_09/soltan-u19/1-1/726-1.webp?ts=1752831534" ] ] ] #original: array:6 [ "id" => 1278372 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136} "task" => array:2 [ "refs" => "1108467" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "726-1.jpg" "alt" => null "width" => "1373" "height" => 328 "path" => "/media/algebra_09/soltan-u19/1-1/726-1.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" => 1554327 "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" => "1108467" "type" => "task" ] "text" => "<p>Чтобы определить, чьё решение является правильным, необходимо проанализировать шаги каждого ученика и сравнить их с верным решением. Задание состоит в том, чтобы найти значение выражения $\sqrt{(1 - \cos \alpha)^2 + \sin^2 \alpha}$ при $\alpha = 120^\circ$.</p><p>Сначала рассмотрим решение <strong>Николая</strong>. Он начал с подстановки $\sqrt{(1 - \cos 60^\circ)^2 + \sin^2 60^\circ}$. Это является ошибкой, так как по условию задачи $\alpha = 120^\circ$, а не $60^\circ$. Хотя его дальнейшие вычисления для угла $60^\circ$ верны ($\sqrt{(1 - \frac{1}{2})^2 + (\frac{\sqrt{3}}{2})^2} = \sqrt{(\frac{1}{2})^2 + \frac{3}{4}} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1$), результат не является решением исходной задачи из-за неверной подстановки.</p><p>Теперь проанализируем решение <strong>Василия</strong>. Он выбрал путь упрощения выражения перед подстановкой значения угла.<br>1. Он раскрывает скобки под корнем: $\sqrt{1 - 2\cos \alpha + \cos^2 \alpha + \sin^2 \alpha}$.<br>2. Затем использует основное тригонометрическое тождество $\sin^2 \alpha + \cos^2 \alpha = 1$, что приводит выражение к виду $\sqrt{1 - 2\cos \alpha + 1} = \sqrt{2 - 2\cos \alpha}$.<br>3. Далее он выносит общий множитель и применяет формулу половинного угла $1 - \cos \alpha = 2\sin^2 \frac{\alpha}{2}$: $\sqrt{2(1 - \cos \alpha)} = \sqrt{2 \cdot 2\sin^2 \frac{\alpha}{2}} = \sqrt{4\sin^2 \frac{\alpha}{2}}$.<br>4. Наконец, он правильно извлекает корень: $\sqrt{4\sin^2 \frac{\alpha}{2}} = |2\sin \frac{\alpha}{2}| = 2|\sin \frac{\alpha}{2}|$.<br>Все преобразования выполнены верно.<br>После этого Василий подставляет $\alpha = 120^\circ$:<br>$2|\sin \frac{120^\circ}{2}| = 2|\sin 60^\circ|$.<br>Так как $\sin 60^\circ > 0$, модуль можно убрать: $2\sin 60^\circ = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3}$.<br>Все шаги Василия логичны и математически корректны.</p><p>Для окончательной проверки можно подставить значение $\alpha = 120^\circ$ напрямую в исходное выражение:<br>$\cos 120^\circ = -\frac{1}{2}$<br>$\sin 120^\circ = \frac{\sqrt{3}}{2}$<br>$\sqrt{(1 - (-\frac{1}{2}))^2 + (\frac{\sqrt{3}}{2})^2} = \sqrt{(1 + \frac{1}{2})^2 + \frac{3}{4}} = \sqrt{(\frac{3}{2})^2 + \frac{3}{4}} = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{\frac{12}{4}} = \sqrt{3}$.<br>Этот результат совпадает с ответом Василия.</p><p><strong>Ответ:</strong> Правильным было решение Василия.</p>" ] #original: array:6 [ "id" => 1554327 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144} "task" => array:2 [ "refs" => "1108467" "type" => "task" ] "text" => "<p>Чтобы определить, чьё решение является правильным, необходимо проанализировать шаги каждого ученика и сравнить их с верным решением. Задание состоит в том, чтобы найти значение выражения $\sqrt{(1 - \cos \alpha)^2 + \sin^2 \alpha}$ при $\alpha = 120^\circ$.</p><p>Сначала рассмотрим решение <strong>Николая</strong>. Он начал с подстановки $\sqrt{(1 - \cos 60^\circ)^2 + \sin^2 60^\circ}$. Это является ошибкой, так как по условию задачи $\alpha = 120^\circ$, а не $60^\circ$. Хотя его дальнейшие вычисления для угла $60^\circ$ верны ($\sqrt{(1 - \frac{1}{2})^2 + (\frac{\sqrt{3}}{2})^2} = \sqrt{(\frac{1}{2})^2 + \frac{3}{4}} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1$), результат не является решением исходной задачи из-за неверной подстановки.</p><p>Теперь проанализируем решение <strong>Василия</strong>. Он выбрал путь упрощения выражения перед подстановкой значения угла.<br>1. Он раскрывает скобки под корнем: $\sqrt{1 - 2\cos \alpha + \cos^2 \alpha + \sin^2 \alpha}$.<br>2. Затем использует основное тригонометрическое тождество $\sin^2 \alpha + \cos^2 \alpha = 1$, что приводит выражение к виду $\sqrt{1 - 2\cos \alpha + 1} = \sqrt{2 - 2\cos \alpha}$.<br>3. Далее он выносит общий множитель и применяет формулу половинного угла $1 - \cos \alpha = 2\sin^2 \frac{\alpha}{2}$: $\sqrt{2(1 - \cos \alpha)} = \sqrt{2 \cdot 2\sin^2 \frac{\alpha}{2}} = \sqrt{4\sin^2 \frac{\alpha}{2}}$.<br>4. Наконец, он правильно извлекает корень: $\sqrt{4\sin^2 \frac{\alpha}{2}} = |2\sin \frac{\alpha}{2}| = 2|\sin \frac{\alpha}{2}|$.<br>Все преобразования выполнены верно.<br>После этого Василий подставляет $\alpha = 120^\circ$:<br>$2|\sin \frac{120^\circ}{2}| = 2|\sin 60^\circ|$.<br>Так как $\sin 60^\circ > 0$, модуль можно убрать: $2\sin 60^\circ = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3}$.<br>Все шаги Василия логичны и математически корректны.</p><p>Для окончательной проверки можно подставить значение $\alpha = 120^\circ$ напрямую в исходное выражение:<br>$\cos 120^\circ = -\frac{1}{2}$<br>$\sin 120^\circ = \frac{\sqrt{3}}{2}$<br>$\sqrt{(1 - (-\frac{1}{2}))^2 + (\frac{\sqrt{3}}{2})^2} = \sqrt{(1 + \frac{1}{2})^2 + \frac{3}{4}} = \sqrt{(\frac{3}{2})^2 + \frac{3}{4}} = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{\frac{12}{4}} = \sqrt{3}$.<br>Этот результат совпадает с ответом Василия.</p><p><strong>Ответ:</strong> Правильным было решение Василия.</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" => "1108468" "type" => "task" ] "previous" => array:2 [ "refs" => "1108466" "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 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#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 [ …6] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#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 [ …6] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#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 +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1163 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1164 #items: array:1 [ 0 => App\Models\Term {#1165 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1166 #items: array:1 [ 0 => App\Models\Term {#1167 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#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 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#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" => 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№726 (с. 207)
Условие. №726 (с. 207)
скриншот условия
726. Учащимся было дано задание: найти при $\alpha = 120^\circ$ значение выражения $\sqrt{(1 - \cos \alpha)^2 + \sin^2 \alpha}$.
Николай выполнил задание так:
$\sqrt{(1 - \cos \alpha)^2 + \sin^2 \alpha} = \sqrt{(1 - \cos 60^\circ)^2 + \sin^2 60^\circ} = \sqrt{\frac{1}{4} + \frac{3}{4}} = 1$.
Василий решал так:
$\sqrt{(1 - \cos \alpha)^2 + \sin^2 \alpha} = \sqrt{1 - 2 \cos \alpha + \cos^2 \alpha + \sin^2 \alpha} = \sqrt{2(1 - \cos \alpha)} = \sqrt{4 \sin^2 \frac{\alpha}{2}} = 2 \left| \sin \frac{\alpha}{2} \right| = 2 \sin 60^\circ = \sqrt{3}$.
Чье решение было правильным?
Решение. №726 (с. 207)
Решение 2 (rus). №726 (с. 207)
Чтобы определить, чьё решение является правильным, необходимо проанализировать шаги каждого ученика и сравнить их с верным решением. Задание состоит в том, чтобы найти значение выражения $\sqrt{(1 - \cos \alpha)^2 + \sin^2 \alpha}$ при $\alpha = 120^\circ$.
Сначала рассмотрим решение Николая. Он начал с подстановки $\sqrt{(1 - \cos 60^\circ)^2 + \sin^2 60^\circ}$. Это является ошибкой, так как по условию задачи $\alpha = 120^\circ$, а не $60^\circ$. Хотя его дальнейшие вычисления для угла $60^\circ$ верны ($\sqrt{(1 - \frac{1}{2})^2 + (\frac{\sqrt{3}}{2})^2} = \sqrt{(\frac{1}{2})^2 + \frac{3}{4}} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1$), результат не является решением исходной задачи из-за неверной подстановки.
Теперь проанализируем решение Василия. Он выбрал путь упрощения выражения перед подстановкой значения угла.
1. Он раскрывает скобки под корнем: $\sqrt{1 - 2\cos \alpha + \cos^2 \alpha + \sin^2 \alpha}$.
2. Затем использует основное тригонометрическое тождество $\sin^2 \alpha + \cos^2 \alpha = 1$, что приводит выражение к виду $\sqrt{1 - 2\cos \alpha + 1} = \sqrt{2 - 2\cos \alpha}$.
3. Далее он выносит общий множитель и применяет формулу половинного угла $1 - \cos \alpha = 2\sin^2 \frac{\alpha}{2}$: $\sqrt{2(1 - \cos \alpha)} = \sqrt{2 \cdot 2\sin^2 \frac{\alpha}{2}} = \sqrt{4\sin^2 \frac{\alpha}{2}}$.
4. Наконец, он правильно извлекает корень: $\sqrt{4\sin^2 \frac{\alpha}{2}} = |2\sin \frac{\alpha}{2}| = 2|\sin \frac{\alpha}{2}|$.
Все преобразования выполнены верно.
После этого Василий подставляет $\alpha = 120^\circ$:
$2|\sin \frac{120^\circ}{2}| = 2|\sin 60^\circ|$.
Так как $\sin 60^\circ > 0$, модуль можно убрать: $2\sin 60^\circ = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3}$.
Все шаги Василия логичны и математически корректны.
Для окончательной проверки можно подставить значение $\alpha = 120^\circ$ напрямую в исходное выражение:
$\cos 120^\circ = -\frac{1}{2}$
$\sin 120^\circ = \frac{\sqrt{3}}{2}$
$\sqrt{(1 - (-\frac{1}{2}))^2 + (\frac{\sqrt{3}}{2})^2} = \sqrt{(1 + \frac{1}{2})^2 + \frac{3}{4}} = \sqrt{(\frac{3}{2})^2 + \frac{3}{4}} = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{\frac{12}{4}} = \sqrt{3}$.
Этот результат совпадает с ответом Василия.
Ответ: Правильным было решение Василия.
Другие задания:
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