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Номер 727, страница 208 - 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 727, страница 208.
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"field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 …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 {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1039 …2} 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"field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#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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=> null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Branch {#1094 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. 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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" => 1277082 "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" => "1108468" "type" => "task" ] "text" => "<p><strong>727.</strong> Вычислите значение функции:</p><p><strong>a)</strong> $f(x) = \sin 4x + \cos 4x$, если $\operatorname{ctg} 2x = \frac{1}{2}$, $\frac{\pi}{8} < x < \frac{\pi}{4}$;</p><p><strong>б)</strong> $f(x) = a \sin 2x + b \cos 2x$, если $\operatorname{tg} x = \frac{a}{b}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "727-1.jpg" "alt" => null "width" => "1289" "height" => 373 "path" => "/media/algebra_09/soltan-u19/0-1/727-1.webp?ts=1752834734" ] ] ] #original: array:7 [ "id" => 1277082 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1128} "task" => array:2 [ "refs" => "1108468" "type" => "task" ] "text" => "<p><strong>727.</strong> Вычислите значение функции:</p><p><strong>a)</strong> $f(x) = \sin 4x + \cos 4x$, если $\operatorname{ctg} 2x = \frac{1}{2}$, $\frac{\pi}{8} < x < \frac{\pi}{4}$;</p><p><strong>б)</strong> $f(x) = a \sin 2x + b \cos 2x$, если $\operatorname{tg} x = \frac{a}{b}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "727-1.jpg" "alt" => null "width" => "1289" "height" => 373 "path" => "/media/algebra_09/soltan-u19/0-1/727-1.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" => 1278373 "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" => "1108468" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "727-1.jpg" "alt" => null "width" => "1349" "height" => 1577 "path" => "/media/algebra_09/soltan-u19/1-1/727-1.webp?ts=1752831534" ] 1 => array:5 [ "name" => "727-2.jpg" "alt" => null "width" => "798" "height" => 254 "path" => "/media/algebra_09/soltan-u19/1-1/727-2.webp?ts=1752831534" ] ] ] #original: array:6 [ "id" => 1278373 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1136} "task" => array:2 [ "refs" => "1108468" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "727-1.jpg" "alt" => null "width" => "1349" "height" => 1577 "path" => "/media/algebra_09/soltan-u19/1-1/727-1.webp?ts=1752831534" ] 1 => array:5 [ "name" => "727-2.jpg" "alt" => null "width" => "798" "height" => 254 "path" => "/media/algebra_09/soltan-u19/1-1/727-2.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" => 1554328 "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" => "1108468" "type" => "task" ] "text" => "<p><strong>а)</strong> Нам нужно вычислить значение функции $f(x) = \sin 4x + \cos 4x$, зная, что $\text{ctg } 2x = \frac{1}{2}$ и $\frac{\pi}{8} < x < \frac{\pi}{4}$.</p><p>Аргумент функции $f(x)$ равен $4x$, а в условии дан котангенс угла $2x$. Мы можем использовать формулы двойного угла, чтобы выразить $\sin 4x$ и $\cos 4x$ через тригонометрические функции угла $2x$. В частности, удобно использовать формулы выражения синуса и косинуса через тангенс половинного угла:</p><p>$\sin(2\alpha) = \frac{2 \text{ tg } \alpha}{1 + \text{tg}^2 \alpha}$</p><p>$\cos(2\alpha) = \frac{1 - \text{tg}^2 \alpha}{1 + \text{tg}^2 \alpha}$</p><p>В нашем случае $\alpha = 2x$. Сначала найдем $\text{tg } 2x$:</p><p>$\text{tg } 2x = \frac{1}{\text{ctg } 2x} = \frac{1}{1/2} = 2$.</p><p>Теперь подставим значение $\text{tg } 2x$ в формулы для $\sin 4x$ и $\cos 4x$:</p><p>$\sin 4x = \frac{2 \text{ tg } 2x}{1 + \text{tg}^2 2x} = \frac{2 \cdot 2}{1 + 2^2} = \frac{4}{1 + 4} = \frac{4}{5}$.</p><p>$\cos 4x = \frac{1 - \text{tg}^2 2x}{1 + \text{tg}^2 2x} = \frac{1 - 2^2}{1 + 2^2} = \frac{1 - 4}{1 + 4} = -\frac{3}{5}$.</p><p>Наконец, вычислим значение функции $f(x)$:</p><p>$f(x) = \sin 4x + \cos 4x = \frac{4}{5} + \left(-\frac{3}{5}\right) = \frac{4-3}{5} = \frac{1}{5}$.</p><p>Условие $\frac{\pi}{8} < x < \frac{\pi}{4}$ можно использовать для проверки. Если умножить неравенство на 2, получим $\frac{\pi}{4} < 2x < \frac{\pi}{2}$. Это первая четверть, где тангенс положителен, что соответствует нашему значению $\text{tg } 2x = 2$.</p><p><strong>Ответ:</strong> $\frac{1}{5}$.</p><p><strong>б)</strong> Нам нужно вычислить значение функции $f(x) = a \sin 2x + b \cos 2x$, зная, что $\text{tg } x = \frac{a}{b}$.</p><p>Как и в предыдущем пункте, воспользуемся формулами двойного угла, выражающими синус и косинус через тангенс. Здесь $\alpha=x$.</p><p>$\sin 2x = \frac{2 \text{ tg } x}{1 + \text{tg}^2 x}$</p><p>$\cos 2x = \frac{1 - \text{tg}^2 x}{1 + \text{tg}^2 x}$</p><p>Подставим в эти формулы данное значение $\text{tg } x = \frac{a}{b}$ (предполагая, что $b \neq 0$):</p><p>$\sin 2x = \frac{2 \cdot \frac{a}{b}}{1 + \left(\frac{a}{b}\right)^2} = \frac{\frac{2a}{b}}{1 + \frac{a^2}{b^2}} = \frac{\frac{2a}{b}}{\frac{b^2+a^2}{b^2}} = \frac{2a}{b} \cdot \frac{b^2}{a^2+b^2} = \frac{2ab}{a^2+b^2}$.</p><p>$\cos 2x = \frac{1 - \left(\frac{a}{b}\right)^2}{1 + \left(\frac{a}{b}\right)^2} = \frac{1 - \frac{a^2}{b^2}}{1 + \frac{a^2}{b^2}} = \frac{\frac{b^2-a^2}{b^2}}{\frac{b^2+a^2}{b^2}} = \frac{b^2-a^2}{a^2+b^2}$.</p><p>Теперь подставим полученные выражения для $\sin 2x$ и $\cos 2x$ в исходную функцию:</p><p>$f(x) = a \sin 2x + b \cos 2x = a \left(\frac{2ab}{a^2+b^2}\right) + b \left(\frac{b^2-a^2}{a^2+b^2}\right)$.</p><p>Приведем к общему знаменателю:</p><p>$f(x) = \frac{a \cdot 2ab + b(b^2-a^2)}{a^2+b^2} = \frac{2a^2b + b^3 - a^2b}{a^2+b^2} = \frac{a^2b + b^3}{a^2+b^2}$.</p><p>Вынесем $b$ в числителе за скобки:</p><p>$f(x) = \frac{b(a^2 + b^2)}{a^2+b^2}$.</p><p>Сократим дробь на $(a^2+b^2)$, так как $a^2+b^2 \neq 0$ (если $a$ и $b$ не равны нулю одновременно, в этом случае задача не имеет смысла).</p><p>$f(x) = b$.</p><p>Если $b=0$, то $\text{tg } x$ не определен, что обычно означает $x = \frac{\pi}{2} + k\pi$. Тогда $\sin 2x = 0$, $\cos 2x = -1$. Функция $f(x) = a \cdot 0 + 0 \cdot (-1) = 0$. Наш ответ $f(x)=b$ также дает 0, так что формула верна и в этом случае.</p><p><strong>Ответ:</strong> $b$.</p>" ] #original: array:6 [ "id" => 1554328 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1144} "task" => array:2 [ "refs" => "1108468" "type" => "task" ] "text" => "<p><strong>а)</strong> Нам нужно вычислить значение функции $f(x) = \sin 4x + \cos 4x$, зная, что $\text{ctg } 2x = \frac{1}{2}$ и $\frac{\pi}{8} < x < \frac{\pi}{4}$.</p><p>Аргумент функции $f(x)$ равен $4x$, а в условии дан котангенс угла $2x$. Мы можем использовать формулы двойного угла, чтобы выразить $\sin 4x$ и $\cos 4x$ через тригонометрические функции угла $2x$. В частности, удобно использовать формулы выражения синуса и косинуса через тангенс половинного угла:</p><p>$\sin(2\alpha) = \frac{2 \text{ tg } \alpha}{1 + \text{tg}^2 \alpha}$</p><p>$\cos(2\alpha) = \frac{1 - \text{tg}^2 \alpha}{1 + \text{tg}^2 \alpha}$</p><p>В нашем случае $\alpha = 2x$. Сначала найдем $\text{tg } 2x$:</p><p>$\text{tg } 2x = \frac{1}{\text{ctg } 2x} = \frac{1}{1/2} = 2$.</p><p>Теперь подставим значение $\text{tg } 2x$ в формулы для $\sin 4x$ и $\cos 4x$:</p><p>$\sin 4x = \frac{2 \text{ tg } 2x}{1 + \text{tg}^2 2x} = \frac{2 \cdot 2}{1 + 2^2} = \frac{4}{1 + 4} = \frac{4}{5}$.</p><p>$\cos 4x = \frac{1 - \text{tg}^2 2x}{1 + \text{tg}^2 2x} = \frac{1 - 2^2}{1 + 2^2} = \frac{1 - 4}{1 + 4} = -\frac{3}{5}$.</p><p>Наконец, вычислим значение функции $f(x)$:</p><p>$f(x) = \sin 4x + \cos 4x = \frac{4}{5} + \left(-\frac{3}{5}\right) = \frac{4-3}{5} = \frac{1}{5}$.</p><p>Условие $\frac{\pi}{8} < x < \frac{\pi}{4}$ можно использовать для проверки. Если умножить неравенство на 2, получим $\frac{\pi}{4} < 2x < \frac{\pi}{2}$. Это первая четверть, где тангенс положителен, что соответствует нашему значению $\text{tg } 2x = 2$.</p><p><strong>Ответ:</strong> $\frac{1}{5}$.</p><p><strong>б)</strong> Нам нужно вычислить значение функции $f(x) = a \sin 2x + b \cos 2x$, зная, что $\text{tg } x = \frac{a}{b}$.</p><p>Как и в предыдущем пункте, воспользуемся формулами двойного угла, выражающими синус и косинус через тангенс. Здесь $\alpha=x$.</p><p>$\sin 2x = \frac{2 \text{ tg } x}{1 + \text{tg}^2 x}$</p><p>$\cos 2x = \frac{1 - \text{tg}^2 x}{1 + \text{tg}^2 x}$</p><p>Подставим в эти формулы данное значение $\text{tg } x = \frac{a}{b}$ (предполагая, что $b \neq 0$):</p><p>$\sin 2x = \frac{2 \cdot \frac{a}{b}}{1 + \left(\frac{a}{b}\right)^2} = \frac{\frac{2a}{b}}{1 + \frac{a^2}{b^2}} = \frac{\frac{2a}{b}}{\frac{b^2+a^2}{b^2}} = \frac{2a}{b} \cdot \frac{b^2}{a^2+b^2} = \frac{2ab}{a^2+b^2}$.</p><p>$\cos 2x = \frac{1 - \left(\frac{a}{b}\right)^2}{1 + \left(\frac{a}{b}\right)^2} = \frac{1 - \frac{a^2}{b^2}}{1 + \frac{a^2}{b^2}} = \frac{\frac{b^2-a^2}{b^2}}{\frac{b^2+a^2}{b^2}} = \frac{b^2-a^2}{a^2+b^2}$.</p><p>Теперь подставим полученные выражения для $\sin 2x$ и $\cos 2x$ в исходную функцию:</p><p>$f(x) = a \sin 2x + b \cos 2x = a \left(\frac{2ab}{a^2+b^2}\right) + b \left(\frac{b^2-a^2}{a^2+b^2}\right)$.</p><p>Приведем к общему знаменателю:</p><p>$f(x) = \frac{a \cdot 2ab + b(b^2-a^2)}{a^2+b^2} = \frac{2a^2b + b^3 - a^2b}{a^2+b^2} = \frac{a^2b + b^3}{a^2+b^2}$.</p><p>Вынесем $b$ в числителе за скобки:</p><p>$f(x) = \frac{b(a^2 + b^2)}{a^2+b^2}$.</p><p>Сократим дробь на $(a^2+b^2)$, так как $a^2+b^2 \neq 0$ (если $a$ и $b$ не равны нулю одновременно, в этом случае задача не имеет смысла).</p><p>$f(x) = b$.</p><p>Если $b=0$, то $\text{tg } x$ не определен, что обычно означает $x = \frac{\pi}{2} + k\pi$. Тогда $\sin 2x = 0$, $\cos 2x = -1$. Функция $f(x) = a \cdot 0 + 0 \cdot (-1) = 0$. Наш ответ $f(x)=b$ также дает 0, так что формула верна и в этом случае.</p><p><strong>Ответ:</strong> $b$.</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" => "1108469" "type" => "task" ] "previous" => array:2 [ "refs" => "1108467" "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 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"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 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"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 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{#1187 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1186 #items: array:1 [ 0 => App\Models\Book {#1188 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1202 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1204 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1209 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1211 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1191 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1195 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1200 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1202 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1204 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1206 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1208 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1209 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1210 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1211 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1213 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1216 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ 0 => App\Models\Element {#1218 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1261101 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261101 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1097951" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097949" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1226 #items: array:7 [ 0 => App\Models\Task {#1227 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108468 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "208" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-727" "field_display_title" => "727" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1230 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1231 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1264 …2} "content" => Illuminate\Database\Eloquent\Collection {#1301 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1326 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108468 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "208" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-727" "field_display_title" => "727" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1228 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1230 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1231 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1264 …2} "content" => Illuminate\Database\Eloquent\Collection {#1301 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1326 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1327 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108469 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "208" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-728" "field_display_title" => "728" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1328 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1330 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1331 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1364 …2} "content" => Illuminate\Database\Eloquent\Collection {#1401 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1426 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108469 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "208" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-728" "field_display_title" => "728" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1328 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1330 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1331 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1364 …2} "content" => Illuminate\Database\Eloquent\Collection {#1401 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1426 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1427 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108470 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "208" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-729" "field_display_title" => "729" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1428 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1430 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1431 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1464 …2} "content" => Illuminate\Database\Eloquent\Collection {#1501 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1526 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108470 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "208" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-729" "field_display_title" => "729" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1428 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1430 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1431 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1464 …2} "content" => Illuminate\Database\Eloquent\Collection {#1501 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1526 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1527 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1108471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "208" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-730" "field_display_title" => "730" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1528 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1530 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1531 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1564 …2} "content" => Illuminate\Database\Eloquent\Collection {#1601 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1626 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "208" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-730" "field_display_title" => "730" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1528 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1530 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1531 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1564 …2} "content" => Illuminate\Database\Eloquent\Collection {#1601 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1626 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1627 #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" => 1108472 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "208" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-731" "field_display_title" => "731" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1628 …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 {#1630 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1631 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1664 …2} "content" => Illuminate\Database\Eloquent\Collection {#1701 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1726 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108472 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "208" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-731" "field_display_title" => "731" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1628 …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 {#1630 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1631 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1664 …2} "content" => Illuminate\Database\Eloquent\Collection {#1701 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1726 …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 => "*" ] } 5 => App\Models\Task {#1727 #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" => 1108473 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "208" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-732" "field_display_title" => "732" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1728 …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 {#1730 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1731 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1764 …2} "content" => Illuminate\Database\Eloquent\Collection {#1801 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1826 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108473 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "208" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-732" "field_display_title" => "732" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1728 …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 {#1730 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1731 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1764 …2} "content" => Illuminate\Database\Eloquent\Collection {#1801 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1826 …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 => "*" ] } 6 => App\Models\Task {#1827 #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" => 1108474 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "208" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-733" "field_display_title" => "733" "field_outside_task" => 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№727 (с. 208)
Условие. №727 (с. 208)
скриншот условия
727. Вычислите значение функции:
a) $f(x) = \sin 4x + \cos 4x$, если $\operatorname{ctg} 2x = \frac{1}{2}$, $\frac{\pi}{8} < x < \frac{\pi}{4}$;
б) $f(x) = a \sin 2x + b \cos 2x$, если $\operatorname{tg} x = \frac{a}{b}$.
Решение. №727 (с. 208)
Решение 2 (rus). №727 (с. 208)
а) Нам нужно вычислить значение функции $f(x) = \sin 4x + \cos 4x$, зная, что $\text{ctg } 2x = \frac{1}{2}$ и $\frac{\pi}{8} < x < \frac{\pi}{4}$.
Аргумент функции $f(x)$ равен $4x$, а в условии дан котангенс угла $2x$. Мы можем использовать формулы двойного угла, чтобы выразить $\sin 4x$ и $\cos 4x$ через тригонометрические функции угла $2x$. В частности, удобно использовать формулы выражения синуса и косинуса через тангенс половинного угла:
$\sin(2\alpha) = \frac{2 \text{ tg } \alpha}{1 + \text{tg}^2 \alpha}$
$\cos(2\alpha) = \frac{1 - \text{tg}^2 \alpha}{1 + \text{tg}^2 \alpha}$
В нашем случае $\alpha = 2x$. Сначала найдем $\text{tg } 2x$:
$\text{tg } 2x = \frac{1}{\text{ctg } 2x} = \frac{1}{1/2} = 2$.
Теперь подставим значение $\text{tg } 2x$ в формулы для $\sin 4x$ и $\cos 4x$:
$\sin 4x = \frac{2 \text{ tg } 2x}{1 + \text{tg}^2 2x} = \frac{2 \cdot 2}{1 + 2^2} = \frac{4}{1 + 4} = \frac{4}{5}$.
$\cos 4x = \frac{1 - \text{tg}^2 2x}{1 + \text{tg}^2 2x} = \frac{1 - 2^2}{1 + 2^2} = \frac{1 - 4}{1 + 4} = -\frac{3}{5}$.
Наконец, вычислим значение функции $f(x)$:
$f(x) = \sin 4x + \cos 4x = \frac{4}{5} + \left(-\frac{3}{5}\right) = \frac{4-3}{5} = \frac{1}{5}$.
Условие $\frac{\pi}{8} < x < \frac{\pi}{4}$ можно использовать для проверки. Если умножить неравенство на 2, получим $\frac{\pi}{4} < 2x < \frac{\pi}{2}$. Это первая четверть, где тангенс положителен, что соответствует нашему значению $\text{tg } 2x = 2$.
Ответ: $\frac{1}{5}$.
б) Нам нужно вычислить значение функции $f(x) = a \sin 2x + b \cos 2x$, зная, что $\text{tg } x = \frac{a}{b}$.
Как и в предыдущем пункте, воспользуемся формулами двойного угла, выражающими синус и косинус через тангенс. Здесь $\alpha=x$.
$\sin 2x = \frac{2 \text{ tg } x}{1 + \text{tg}^2 x}$
$\cos 2x = \frac{1 - \text{tg}^2 x}{1 + \text{tg}^2 x}$
Подставим в эти формулы данное значение $\text{tg } x = \frac{a}{b}$ (предполагая, что $b \neq 0$):
$\sin 2x = \frac{2 \cdot \frac{a}{b}}{1 + \left(\frac{a}{b}\right)^2} = \frac{\frac{2a}{b}}{1 + \frac{a^2}{b^2}} = \frac{\frac{2a}{b}}{\frac{b^2+a^2}{b^2}} = \frac{2a}{b} \cdot \frac{b^2}{a^2+b^2} = \frac{2ab}{a^2+b^2}$.
$\cos 2x = \frac{1 - \left(\frac{a}{b}\right)^2}{1 + \left(\frac{a}{b}\right)^2} = \frac{1 - \frac{a^2}{b^2}}{1 + \frac{a^2}{b^2}} = \frac{\frac{b^2-a^2}{b^2}}{\frac{b^2+a^2}{b^2}} = \frac{b^2-a^2}{a^2+b^2}$.
Теперь подставим полученные выражения для $\sin 2x$ и $\cos 2x$ в исходную функцию:
$f(x) = a \sin 2x + b \cos 2x = a \left(\frac{2ab}{a^2+b^2}\right) + b \left(\frac{b^2-a^2}{a^2+b^2}\right)$.
Приведем к общему знаменателю:
$f(x) = \frac{a \cdot 2ab + b(b^2-a^2)}{a^2+b^2} = \frac{2a^2b + b^3 - a^2b}{a^2+b^2} = \frac{a^2b + b^3}{a^2+b^2}$.
Вынесем $b$ в числителе за скобки:
$f(x) = \frac{b(a^2 + b^2)}{a^2+b^2}$.
Сократим дробь на $(a^2+b^2)$, так как $a^2+b^2 \neq 0$ (если $a$ и $b$ не равны нулю одновременно, в этом случае задача не имеет смысла).
$f(x) = b$.
Если $b=0$, то $\text{tg } x$ не определен, что обычно означает $x = \frac{\pi}{2} + k\pi$. Тогда $\sin 2x = 0$, $\cos 2x = -1$. Функция $f(x) = a \cdot 0 + 0 \cdot (-1) = 0$. Наш ответ $f(x)=b$ также дает 0, так что формула верна и в этом случае.
Ответ: $b$.
Другие задания:
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