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Номер 8.4, страница 58 - гдз по алгебре 10 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
ISBN: 978-601-07-1142-6
Утверждено Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Глава 3. Тригонометрические уравнения и неравенства. Параграф 8. Решение тригонометрических уравнений - номер 8.4, страница 58.
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"*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1048 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113562 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические уравнения и неравенства" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050} "field_page_start" => "48" "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 {#1058} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1048} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:2 [ 0 => App\Models\Branch {#1115 #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" => 1113562 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические уравнения и неравенства" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "48" "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 {#1117 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1118 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113562 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические уравнения и неравенства" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1116} "field_page_start" => "48" "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 {#1117} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1118} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1119 #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" => 1113564 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Решение тригонометрических уравнений" "field_branch_order" => "8" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Term {#1120 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "55" "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 {#1122 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113564 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Решение тригонометрических уравнений" "field_branch_order" => "8" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1123} "field_page_start" => "55" "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 {#1122} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1121} ] #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 {#1104 #items: array:3 [ 0 => App\Models\Element {#1097 #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" => 1290757 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089 #items: array:1 [ 0 => App\Models\Edition {#1096 #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" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1094 …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 {#1092 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1094 …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 {#1092 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1090 …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" => "1113747" "type" => "task" ] "text" => "<p><strong>8.4. a)</strong> $\cos 7x + \cos x = 0;$</p><p><strong>б)</strong> $\sin 7x - \sin x = 0;$</p><p><strong>в)</strong> $\sin^2 x + \sin 2x = 1;$</p><p><strong>г)</strong> $\cos^2 x - \sin 2x = 1.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "2224" "height" => 320 "path" => "/media/algebra_10/Abylkasymova-u/0-08/4-1.webp?ts=1753263047" ] ] ] #original: array:7 [ "id" => 1290757 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1089} "task" => array:2 [ "refs" => "1113747" "type" => "task" ] "text" => "<p><strong>8.4. a)</strong> $\cos 7x + \cos x = 0;$</p><p><strong>б)</strong> $\sin 7x - \sin x = 0;$</p><p><strong>в)</strong> $\sin^2 x + \sin 2x = 1;$</p><p><strong>г)</strong> $\cos^2 x - \sin 2x = 1.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "2224" "height" => 320 "path" => "/media/algebra_10/Abylkasymova-u/0-08/4-1.webp?ts=1753263047" ] ] ] #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 {#1091 #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" => 1291271 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1049 #items: array:1 [ 0 => App\Models\Edition {#1088 #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" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1055 …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 {#1086 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1055 …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 {#1086 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1054 …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" => "1113747" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1637" "height" => 866 "path" => "/media/algebra_10/Abylkasymova-u/1-08/4-1.webp?ts=1753263231" ] 1 => array:5 [ "name" => "4-2.jpg" "alt" => null "width" => "1978" "height" => 3409 "path" => "/media/algebra_10/Abylkasymova-u/1-08/4-2.webp?ts=1753263231" ] ] ] #original: array:6 [ "id" => 1291271 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1049} "task" => array:2 [ "refs" => "1113747" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1637" "height" => 866 "path" => "/media/algebra_10/Abylkasymova-u/1-08/4-1.webp?ts=1753263231" ] 1 => array:5 [ "name" => "4-2.jpg" "alt" => null "width" => "1978" "height" => 3409 "path" => "/media/algebra_10/Abylkasymova-u/1-08/4-2.webp?ts=1753263231" ] ] ] #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 {#1052 #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" => 1549400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1110 #items: array:1 [ 0 => App\Models\Edition {#1051 #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" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041 …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 {#1034 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041 …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 {#1034 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …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" => "1113747" "type" => "task" ] "text" => "<p><strong>а)</strong> Исходное уравнение: $ \cos(7x) + \cos(x) = 0 $.</p><p>Для решения этого уравнения воспользуемся формулой суммы косинусов: $ \cos(\alpha) + \cos(\beta) = 2 \cos\left(\frac{\alpha+\beta}{2}\right) \cos\left(\frac{\alpha-\beta}{2}\right) $.</p><p>Применим эту формулу к нашему уравнению, где $ \alpha = 7x $ и $ \beta = x $:</p><p>$ 2 \cos\left(\frac{7x+x}{2}\right) \cos\left(\frac{7x-x}{2}\right) = 0 $</p><p>$ 2 \cos(4x) \cos(3x) = 0 $</p><p>Произведение равно нулю, когда один из множителей равен нулю. Таким образом, у нас есть два случая:</p><p>1) $ \cos(4x) = 0 $</p><p>Это частный случай тригонометрического уравнения. Решение имеет вид:</p><p>$ 4x = \frac{\pi}{2} + \pi k $, где $ k \in \mathbb{Z} $.</p><p>Разделим обе части на 4:</p><p>$ x = \frac{\pi}{8} + \frac{\pi k}{4} $, где $ k \in \mathbb{Z} $.</p><p>2) $ \cos(3x) = 0 $</p><p>Аналогично, решение имеет вид:</p><p>$ 3x = \frac{\pi}{2} + \pi n $, где $ n \in \mathbb{Z} $.</p><p>Разделим обе части на 3:</p><p>$ x = \frac{\pi}{6} + \frac{\pi n}{3} $, где $ n \in \mathbb{Z} $.</p><p>Объединяя решения из обоих случаев, получаем окончательный ответ.</p><p><strong>Ответ:</strong> $ x = \frac{\pi}{8} + \frac{\pi k}{4}, \; x = \frac{\pi}{6} + \frac{\pi n}{3} $, где $ k, n \in \mathbb{Z} $.</p><p><strong>б)</strong> Исходное уравнение: $ \sin(7x) - \sin(x) = 0 $.</p><p>Для решения этого уравнения воспользуемся формулой разности синусов: $ \sin(\alpha) - \sin(\beta) = 2 \cos\left(\frac{\alpha+\beta}{2}\right) \sin\left(\frac{\alpha-\beta}{2}\right) $.</p><p>Применим эту формулу к нашему уравнению, где $ \alpha = 7x $ и $ \beta = x $:</p><p>$ 2 \cos\left(\frac{7x+x}{2}\right) \sin\left(\frac{7x-x}{2}\right) = 0 $</p><p>$ 2 \cos(4x) \sin(3x) = 0 $</p><p>Произведение равно нулю, когда один из множителей равен нулю. Таким образом, у нас есть два случая:</p><p>1) $ \cos(4x) = 0 $</p><p>Решение этого уравнения:</p><p>$ 4x = \frac{\pi}{2} + \pi k $, где $ k \in \mathbb{Z} $.</p><p>$ x = \frac{\pi}{8} + \frac{\pi k}{4} $, где $ k \in \mathbb{Z} $.</p><p>2) $ \sin(3x) = 0 $</p><p>Это частный случай тригонометрического уравнения. Решение имеет вид:</p><p>$ 3x = \pi n $, где $ n \in \mathbb{Z} $.</p><p>Разделим обе части на 3:</p><p>$ x = \frac{\pi n}{3} $, где $ n \in \mathbb{Z} $.</p><p>Объединяя решения из обоих случаев, получаем окончательный ответ.</p><p><strong>Ответ:</strong> $ x = \frac{\pi}{8} + \frac{\pi k}{4}, \; x = \frac{\pi n}{3} $, где $ k, n \in \mathbb{Z} $.</p><p><strong>в)</strong> Исходное уравнение: $ \sin^2(x) + \sin(2x) = 1 $.</p><p>Используем формулу синуса двойного угла $ \sin(2x) = 2 \sin(x) \cos(x) $ и основное тригонометрическое тождество $ 1 = \sin^2(x) + \cos^2(x) $.</p><p>Подставим эти выражения в исходное уравнение:</p><p>$ \sin^2(x) + 2 \sin(x) \cos(x) = \sin^2(x) + \cos^2(x) $</p><p>Вычтем $ \sin^2(x) $ из обеих частей уравнения:</p><p>$ 2 \sin(x) \cos(x) = \cos^2(x) $</p><p>Перенесем все члены в левую часть:</p><p>$ 2 \sin(x) \cos(x) - \cos^2(x) = 0 $</p><p>Вынесем общий множитель $ \cos(x) $ за скобки:</p><p>$ \cos(x) (2 \sin(x) - \cos(x)) = 0 $</p><p>Произведение равно нулю, когда один из множителей равен нулю. Рассматриваем два случая:</p><p>1) $ \cos(x) = 0 $</p><p>Решение этого уравнения:</p><p>$ x = \frac{\pi}{2} + \pi k $, где $ k \in \mathbb{Z} $.</p><p>2) $ 2 \sin(x) - \cos(x) = 0 $</p><p>$ 2 \sin(x) = \cos(x) $</p><p>Заметим, что $ \cos(x) \neq 0 $, иначе из уравнения следовало бы, что $ \sin(x) = 0 $, что невозможно, так как $ \sin^2(x) + \cos^2(x) = 1 $. Разделим обе части на $ \cos(x) $:</p><p>$ 2 \tan(x) = 1 $</p><p>$ \tan(x) = \frac{1}{2} $</p><p>Решение этого уравнения:</p><p>$ x = \arctan\left(\frac{1}{2}\right) + \pi n $, где $ n \in \mathbb{Z} $.</p><p>Объединяя решения из обоих случаев, получаем окончательный ответ.</p><p><strong>Ответ:</strong> $ x = \frac{\pi}{2} + \pi k, \; x = \arctan\left(\frac{1}{2}\right) + \pi n $, где $ k, n \in \mathbb{Z} $.</p><p><strong>г)</strong> Исходное уравнение: $ \cos^2(x) - \sin(2x) = 1 $.</p><p>Используем формулу синуса двойного угла $ \sin(2x) = 2 \sin(x) \cos(x) $ и основное тригонометрическое тождество $ 1 = \sin^2(x) + \cos^2(x) $.</p><p>Подставим эти выражения в исходное уравнение:</p><p>$ \cos^2(x) - 2 \sin(x) \cos(x) = \sin^2(x) + \cos^2(x) $</p><p>Вычтем $ \cos^2(x) $ из обеих частей уравнения:</p><p>$ -2 \sin(x) \cos(x) = \sin^2(x) $</p><p>Перенесем все члены в левую часть:</p><p>$ \sin^2(x) + 2 \sin(x) \cos(x) = 0 $</p><p>Вынесем общий множитель $ \sin(x) $ за скобки:</p><p>$ \sin(x) (\sin(x) + 2 \cos(x)) = 0 $</p><p>Произведение равно нулю, когда один из множителей равен нулю. Рассматриваем два случая:</p><p>1) $ \sin(x) = 0 $</p><p>Решение этого уравнения:</p><p>$ x = \pi k $, где $ k \in \mathbb{Z} $.</p><p>2) $ \sin(x) + 2 \cos(x) = 0 $</p><p>$ \sin(x) = -2 \cos(x) $</p><p>Заметим, что $ \cos(x) \neq 0 $, иначе из уравнения следовало бы, что $ \sin(x) = 0 $, что невозможно. Разделим обе части на $ \cos(x) $:</p><p>$ \tan(x) = -2 $</p><p>Решение этого уравнения:</p><p>$ x = \arctan(-2) + \pi n $, или $ x = -\arctan(2) + \pi n $, где $ n \in \mathbb{Z} $.</p><p>Объединяя решения из обоих случаев, получаем окончательный ответ.</p><p><strong>Ответ:</strong> $ x = \pi k, \; x = -\arctan(2) + \pi n $, где $ k, n \in \mathbb{Z} $.</p>" ] #original: array:6 [ "id" => 1549400 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1110} "task" => array:2 [ "refs" => "1113747" "type" => "task" ] "text" => "<p><strong>а)</strong> Исходное уравнение: $ \cos(7x) + \cos(x) = 0 $.</p><p>Для решения этого уравнения воспользуемся формулой суммы косинусов: $ \cos(\alpha) + \cos(\beta) = 2 \cos\left(\frac{\alpha+\beta}{2}\right) \cos\left(\frac{\alpha-\beta}{2}\right) $.</p><p>Применим эту формулу к нашему уравнению, где $ \alpha = 7x $ и $ \beta = x $:</p><p>$ 2 \cos\left(\frac{7x+x}{2}\right) \cos\left(\frac{7x-x}{2}\right) = 0 $</p><p>$ 2 \cos(4x) \cos(3x) = 0 $</p><p>Произведение равно нулю, когда один из множителей равен нулю. Таким образом, у нас есть два случая:</p><p>1) $ \cos(4x) = 0 $</p><p>Это частный случай тригонометрического уравнения. Решение имеет вид:</p><p>$ 4x = \frac{\pi}{2} + \pi k $, где $ k \in \mathbb{Z} $.</p><p>Разделим обе части на 4:</p><p>$ x = \frac{\pi}{8} + \frac{\pi k}{4} $, где $ k \in \mathbb{Z} $.</p><p>2) $ \cos(3x) = 0 $</p><p>Аналогично, решение имеет вид:</p><p>$ 3x = \frac{\pi}{2} + \pi n $, где $ n \in \mathbb{Z} $.</p><p>Разделим обе части на 3:</p><p>$ x = \frac{\pi}{6} + \frac{\pi n}{3} $, где $ n \in \mathbb{Z} $.</p><p>Объединяя решения из обоих случаев, получаем окончательный ответ.</p><p><strong>Ответ:</strong> $ x = \frac{\pi}{8} + \frac{\pi k}{4}, \; x = \frac{\pi}{6} + \frac{\pi n}{3} $, где $ k, n \in \mathbb{Z} $.</p><p><strong>б)</strong> Исходное уравнение: $ \sin(7x) - \sin(x) = 0 $.</p><p>Для решения этого уравнения воспользуемся формулой разности синусов: $ \sin(\alpha) - \sin(\beta) = 2 \cos\left(\frac{\alpha+\beta}{2}\right) \sin\left(\frac{\alpha-\beta}{2}\right) $.</p><p>Применим эту формулу к нашему уравнению, где $ \alpha = 7x $ и $ \beta = x $:</p><p>$ 2 \cos\left(\frac{7x+x}{2}\right) \sin\left(\frac{7x-x}{2}\right) = 0 $</p><p>$ 2 \cos(4x) \sin(3x) = 0 $</p><p>Произведение равно нулю, когда один из множителей равен нулю. Таким образом, у нас есть два случая:</p><p>1) $ \cos(4x) = 0 $</p><p>Решение этого уравнения:</p><p>$ 4x = \frac{\pi}{2} + \pi k $, где $ k \in \mathbb{Z} $.</p><p>$ x = \frac{\pi}{8} + \frac{\pi k}{4} $, где $ k \in \mathbb{Z} $.</p><p>2) $ \sin(3x) = 0 $</p><p>Это частный случай тригонометрического уравнения. Решение имеет вид:</p><p>$ 3x = \pi n $, где $ n \in \mathbb{Z} $.</p><p>Разделим обе части на 3:</p><p>$ x = \frac{\pi n}{3} $, где $ n \in \mathbb{Z} $.</p><p>Объединяя решения из обоих случаев, получаем окончательный ответ.</p><p><strong>Ответ:</strong> $ x = \frac{\pi}{8} + \frac{\pi k}{4}, \; x = \frac{\pi n}{3} $, где $ k, n \in \mathbb{Z} $.</p><p><strong>в)</strong> Исходное уравнение: $ \sin^2(x) + \sin(2x) = 1 $.</p><p>Используем формулу синуса двойного угла $ \sin(2x) = 2 \sin(x) \cos(x) $ и основное тригонометрическое тождество $ 1 = \sin^2(x) + \cos^2(x) $.</p><p>Подставим эти выражения в исходное уравнение:</p><p>$ \sin^2(x) + 2 \sin(x) \cos(x) = \sin^2(x) + \cos^2(x) $</p><p>Вычтем $ \sin^2(x) $ из обеих частей уравнения:</p><p>$ 2 \sin(x) \cos(x) = \cos^2(x) $</p><p>Перенесем все члены в левую часть:</p><p>$ 2 \sin(x) \cos(x) - \cos^2(x) = 0 $</p><p>Вынесем общий множитель $ \cos(x) $ за скобки:</p><p>$ \cos(x) (2 \sin(x) - \cos(x)) = 0 $</p><p>Произведение равно нулю, когда один из множителей равен нулю. Рассматриваем два случая:</p><p>1) $ \cos(x) = 0 $</p><p>Решение этого уравнения:</p><p>$ x = \frac{\pi}{2} + \pi k $, где $ k \in \mathbb{Z} $.</p><p>2) $ 2 \sin(x) - \cos(x) = 0 $</p><p>$ 2 \sin(x) = \cos(x) $</p><p>Заметим, что $ \cos(x) \neq 0 $, иначе из уравнения следовало бы, что $ \sin(x) = 0 $, что невозможно, так как $ \sin^2(x) + \cos^2(x) = 1 $. Разделим обе части на $ \cos(x) $:</p><p>$ 2 \tan(x) = 1 $</p><p>$ \tan(x) = \frac{1}{2} $</p><p>Решение этого уравнения:</p><p>$ x = \arctan\left(\frac{1}{2}\right) + \pi n $, где $ n \in \mathbb{Z} $.</p><p>Объединяя решения из обоих случаев, получаем окончательный ответ.</p><p><strong>Ответ:</strong> $ x = \frac{\pi}{2} + \pi k, \; x = \arctan\left(\frac{1}{2}\right) + \pi n $, где $ k, n \in \mathbb{Z} $.</p><p><strong>г)</strong> Исходное уравнение: $ \cos^2(x) - \sin(2x) = 1 $.</p><p>Используем формулу синуса двойного угла $ \sin(2x) = 2 \sin(x) \cos(x) $ и основное тригонометрическое тождество $ 1 = \sin^2(x) + \cos^2(x) $.</p><p>Подставим эти выражения в исходное уравнение:</p><p>$ \cos^2(x) - 2 \sin(x) \cos(x) = \sin^2(x) + \cos^2(x) $</p><p>Вычтем $ \cos^2(x) $ из обеих частей уравнения:</p><p>$ -2 \sin(x) \cos(x) = \sin^2(x) $</p><p>Перенесем все члены в левую часть:</p><p>$ \sin^2(x) + 2 \sin(x) \cos(x) = 0 $</p><p>Вынесем общий множитель $ \sin(x) $ за скобки:</p><p>$ \sin(x) (\sin(x) + 2 \cos(x)) = 0 $</p><p>Произведение равно нулю, когда один из множителей равен нулю. Рассматриваем два случая:</p><p>1) $ \sin(x) = 0 $</p><p>Решение этого уравнения:</p><p>$ x = \pi k $, где $ k \in \mathbb{Z} $.</p><p>2) $ \sin(x) + 2 \cos(x) = 0 $</p><p>$ \sin(x) = -2 \cos(x) $</p><p>Заметим, что $ \cos(x) \neq 0 $, иначе из уравнения следовало бы, что $ \sin(x) = 0 $, что невозможно. Разделим обе части на $ \cos(x) $:</p><p>$ \tan(x) = -2 $</p><p>Решение этого уравнения:</p><p>$ x = \arctan(-2) + \pi n $, или $ x = -\arctan(2) + \pi n $, где $ n \in \mathbb{Z} $.</p><p>Объединяя решения из обоих случаев, получаем окончательный ответ.</p><p><strong>Ответ:</strong> $ x = \pi k, \; x = -\arctan(2) + \pi n $, где $ k, n \in \mathbb{Z} $.</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" => "1113748" "type" => "task" ] "previous" => array:2 [ "refs" => "1113746" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Book {#1057} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:1 [ 0 => App\Models\BookPage {#1042 #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" => 1098395 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/page-58" "field_display_title" => "58" "field_folder" => "1" "field_image_name" => "58" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1040 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:1 [ 0 => App\Models\Book {#1057} ] #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 {#1114 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1130 #items: array:1 [ 0 => App\Models\Element {#1129 #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" => 1261820 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1131 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261820 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1131 …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" => "1098396" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098394" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1199 #items: array:13 [ 0 => App\Models\Task {#1219 #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" => 1113743 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/vopr-8" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1220 …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 {#1221 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1222 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1223 …2} "content" => Illuminate\Database\Eloquent\Collection {#1226 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1232 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113743 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/vopr-8" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1220 …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 {#1221 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1222 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1223 …2} "content" => Illuminate\Database\Eloquent\Collection {#1226 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1232 …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 {#1225 #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" => 1113744 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-1" "field_display_title" => "8.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1224 …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 {#1231 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1233 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1229 …2} "content" => Illuminate\Database\Eloquent\Collection {#1228 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1246 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113744 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-1" "field_display_title" => "8.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1224 …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 {#1231 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1233 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1229 …2} "content" => Illuminate\Database\Eloquent\Collection {#1228 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1246 …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 {#1230 #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" => 1113745 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-2" "field_display_title" => "8.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1227 …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 {#1245 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1247 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1243 …2} "content" => Illuminate\Database\Eloquent\Collection {#1242 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1260 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113745 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-2" "field_display_title" => "8.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1227 …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 {#1245 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1247 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1243 …2} "content" => Illuminate\Database\Eloquent\Collection {#1242 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1260 …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 {#1244 #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" => 1113746 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-3" "field_display_title" => "8.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1241 …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 {#1259 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1261 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1257 …2} "content" => Illuminate\Database\Eloquent\Collection {#1256 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1274 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113746 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-3" "field_display_title" => "8.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1241 …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 {#1259 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1261 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1257 …2} "content" => Illuminate\Database\Eloquent\Collection {#1256 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1274 …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 {#1258 #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" => 1113747 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-4" "field_display_title" => "8.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1255 …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 {#1273 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1275 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1271 …2} "content" => Illuminate\Database\Eloquent\Collection {#1269 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1272 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113747 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-4" "field_display_title" => "8.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1255 …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 {#1273 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1275 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1271 …2} "content" => Illuminate\Database\Eloquent\Collection {#1269 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1272 …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 {#1270 #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" => 1113748 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-5" "field_display_title" => "8.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1283 …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 {#1284 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1285 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1286 …2} "content" => Illuminate\Database\Eloquent\Collection {#1289 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1295 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113748 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-5" "field_display_title" => "8.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1283 …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 {#1284 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1285 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1286 …2} "content" => Illuminate\Database\Eloquent\Collection {#1289 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1295 …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 {#1288 #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" => 1113749 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-6" "field_display_title" => "8.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1287 …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 {#1294 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1296 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1292 …2} "content" => Illuminate\Database\Eloquent\Collection {#1291 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1309 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113749 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-6" "field_display_title" => "8.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1287 …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 {#1294 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1296 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1292 …2} "content" => Illuminate\Database\Eloquent\Collection {#1291 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1309 …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 => "*" ] } 7 => App\Models\Task {#1293 #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" => 1113750 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-7" "field_display_title" => "8.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1290 …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 {#1308 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1310 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1306 …2} "content" => Illuminate\Database\Eloquent\Collection {#1305 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1323 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113750 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-7" "field_display_title" => "8.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1290 …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 {#1308 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1310 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1306 …2} "content" => Illuminate\Database\Eloquent\Collection {#1305 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1323 …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 => "*" ] } 8 => App\Models\Task {#1307 #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" => 1113751 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-8" "field_display_title" => "8.8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1304 …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 {#1322 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1324 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1320 …2} "content" => Illuminate\Database\Eloquent\Collection {#1319 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1337 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113751 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-8" "field_display_title" => "8.8" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1304 …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 {#1322 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1324 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1320 …2} "content" => Illuminate\Database\Eloquent\Collection {#1319 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1337 …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 => "*" ] } 9 => App\Models\Task {#1321 #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" => 1113752 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-9" "field_display_title" => "8.9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1318 …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 {#1336 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1338 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1334 …2} "content" => Illuminate\Database\Eloquent\Collection {#1333 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1351 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113752 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "58" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/08-9" "field_display_title" => "8.9" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1318 …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 {#1336 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1338 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1334 …2} "content" => Illuminate\Database\Eloquent\Collection {#1333 …2} "next" => array:2 [ …2] …3 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: 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№8.4 (с. 58)
Условие. №8.4 (с. 58)
скриншот условия
8.4. a) $\cos 7x + \cos x = 0;$
б) $\sin 7x - \sin x = 0;$
в) $\sin^2 x + \sin 2x = 1;$
г) $\cos^2 x - \sin 2x = 1.$
Решение. №8.4 (с. 58)
Решение 2. №8.4 (с. 58)
а) Исходное уравнение: $ \cos(7x) + \cos(x) = 0 $.
Для решения этого уравнения воспользуемся формулой суммы косинусов: $ \cos(\alpha) + \cos(\beta) = 2 \cos\left(\frac{\alpha+\beta}{2}\right) \cos\left(\frac{\alpha-\beta}{2}\right) $.
Применим эту формулу к нашему уравнению, где $ \alpha = 7x $ и $ \beta = x $:
$ 2 \cos\left(\frac{7x+x}{2}\right) \cos\left(\frac{7x-x}{2}\right) = 0 $
$ 2 \cos(4x) \cos(3x) = 0 $
Произведение равно нулю, когда один из множителей равен нулю. Таким образом, у нас есть два случая:
1) $ \cos(4x) = 0 $
Это частный случай тригонометрического уравнения. Решение имеет вид:
$ 4x = \frac{\pi}{2} + \pi k $, где $ k \in \mathbb{Z} $.
Разделим обе части на 4:
$ x = \frac{\pi}{8} + \frac{\pi k}{4} $, где $ k \in \mathbb{Z} $.
2) $ \cos(3x) = 0 $
Аналогично, решение имеет вид:
$ 3x = \frac{\pi}{2} + \pi n $, где $ n \in \mathbb{Z} $.
Разделим обе части на 3:
$ x = \frac{\pi}{6} + \frac{\pi n}{3} $, где $ n \in \mathbb{Z} $.
Объединяя решения из обоих случаев, получаем окончательный ответ.
Ответ: $ x = \frac{\pi}{8} + \frac{\pi k}{4}, \; x = \frac{\pi}{6} + \frac{\pi n}{3} $, где $ k, n \in \mathbb{Z} $.
б) Исходное уравнение: $ \sin(7x) - \sin(x) = 0 $.
Для решения этого уравнения воспользуемся формулой разности синусов: $ \sin(\alpha) - \sin(\beta) = 2 \cos\left(\frac{\alpha+\beta}{2}\right) \sin\left(\frac{\alpha-\beta}{2}\right) $.
Применим эту формулу к нашему уравнению, где $ \alpha = 7x $ и $ \beta = x $:
$ 2 \cos\left(\frac{7x+x}{2}\right) \sin\left(\frac{7x-x}{2}\right) = 0 $
$ 2 \cos(4x) \sin(3x) = 0 $
Произведение равно нулю, когда один из множителей равен нулю. Таким образом, у нас есть два случая:
1) $ \cos(4x) = 0 $
Решение этого уравнения:
$ 4x = \frac{\pi}{2} + \pi k $, где $ k \in \mathbb{Z} $.
$ x = \frac{\pi}{8} + \frac{\pi k}{4} $, где $ k \in \mathbb{Z} $.
2) $ \sin(3x) = 0 $
Это частный случай тригонометрического уравнения. Решение имеет вид:
$ 3x = \pi n $, где $ n \in \mathbb{Z} $.
Разделим обе части на 3:
$ x = \frac{\pi n}{3} $, где $ n \in \mathbb{Z} $.
Объединяя решения из обоих случаев, получаем окончательный ответ.
Ответ: $ x = \frac{\pi}{8} + \frac{\pi k}{4}, \; x = \frac{\pi n}{3} $, где $ k, n \in \mathbb{Z} $.
в) Исходное уравнение: $ \sin^2(x) + \sin(2x) = 1 $.
Используем формулу синуса двойного угла $ \sin(2x) = 2 \sin(x) \cos(x) $ и основное тригонометрическое тождество $ 1 = \sin^2(x) + \cos^2(x) $.
Подставим эти выражения в исходное уравнение:
$ \sin^2(x) + 2 \sin(x) \cos(x) = \sin^2(x) + \cos^2(x) $
Вычтем $ \sin^2(x) $ из обеих частей уравнения:
$ 2 \sin(x) \cos(x) = \cos^2(x) $
Перенесем все члены в левую часть:
$ 2 \sin(x) \cos(x) - \cos^2(x) = 0 $
Вынесем общий множитель $ \cos(x) $ за скобки:
$ \cos(x) (2 \sin(x) - \cos(x)) = 0 $
Произведение равно нулю, когда один из множителей равен нулю. Рассматриваем два случая:
1) $ \cos(x) = 0 $
Решение этого уравнения:
$ x = \frac{\pi}{2} + \pi k $, где $ k \in \mathbb{Z} $.
2) $ 2 \sin(x) - \cos(x) = 0 $
$ 2 \sin(x) = \cos(x) $
Заметим, что $ \cos(x) \neq 0 $, иначе из уравнения следовало бы, что $ \sin(x) = 0 $, что невозможно, так как $ \sin^2(x) + \cos^2(x) = 1 $. Разделим обе части на $ \cos(x) $:
$ 2 \tan(x) = 1 $
$ \tan(x) = \frac{1}{2} $
Решение этого уравнения:
$ x = \arctan\left(\frac{1}{2}\right) + \pi n $, где $ n \in \mathbb{Z} $.
Объединяя решения из обоих случаев, получаем окончательный ответ.
Ответ: $ x = \frac{\pi}{2} + \pi k, \; x = \arctan\left(\frac{1}{2}\right) + \pi n $, где $ k, n \in \mathbb{Z} $.
г) Исходное уравнение: $ \cos^2(x) - \sin(2x) = 1 $.
Используем формулу синуса двойного угла $ \sin(2x) = 2 \sin(x) \cos(x) $ и основное тригонометрическое тождество $ 1 = \sin^2(x) + \cos^2(x) $.
Подставим эти выражения в исходное уравнение:
$ \cos^2(x) - 2 \sin(x) \cos(x) = \sin^2(x) + \cos^2(x) $
Вычтем $ \cos^2(x) $ из обеих частей уравнения:
$ -2 \sin(x) \cos(x) = \sin^2(x) $
Перенесем все члены в левую часть:
$ \sin^2(x) + 2 \sin(x) \cos(x) = 0 $
Вынесем общий множитель $ \sin(x) $ за скобки:
$ \sin(x) (\sin(x) + 2 \cos(x)) = 0 $
Произведение равно нулю, когда один из множителей равен нулю. Рассматриваем два случая:
1) $ \sin(x) = 0 $
Решение этого уравнения:
$ x = \pi k $, где $ k \in \mathbb{Z} $.
2) $ \sin(x) + 2 \cos(x) = 0 $
$ \sin(x) = -2 \cos(x) $
Заметим, что $ \cos(x) \neq 0 $, иначе из уравнения следовало бы, что $ \sin(x) = 0 $, что невозможно. Разделим обе части на $ \cos(x) $:
$ \tan(x) = -2 $
Решение этого уравнения:
$ x = \arctan(-2) + \pi n $, или $ x = -\arctan(2) + \pi n $, где $ n \in \mathbb{Z} $.
Объединяя решения из обоих случаев, получаем окончательный ответ.
Ответ: $ x = \pi k, \; x = -\arctan(2) + \pi n $, где $ k, n \in \mathbb{Z} $.
Другие задания:
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