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Номер 765, страница 217 - 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. 29. Преобразование произведения тригонометрических функций в сумму и разность - номер 765, страница 217.
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Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1033 #items: array:1 [ 0 => App\Models\Book {#1030 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1035 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:3 [ …3] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1048 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1050 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ …4] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1056 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032} "field_class" => Illuminate\Database\Eloquent\Collection {#1035} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037} "field_author" => Illuminate\Database\Eloquent\Collection {#1039} "field_author_foreign" => 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=> null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. 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Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1066} "field_class" => Illuminate\Database\Eloquent\Collection {#1068} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070} "field_author" => Illuminate\Database\Eloquent\Collection {#1072} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077} "field_country" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1081 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_publication_number" => null "field_publication_type" => 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Illuminate\Database\Eloquent\Collection {#1090 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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: [] 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Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1095 …2} "field_page_start" => "147" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1096 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1097 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "29. Преобразование произведения тригонометрических функций в сумму и разность" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "216" "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 {#1098 #items: array:3 [ 0 => App\Models\Element {#1099 #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" => 1277120 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Edition {#1101 #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 {#1102 …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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1106 …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 {#1102 …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 {#1104 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1105 …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" => "1108508" "type" => "task" ] "text" => "<p><strong>765.</strong> Представьте в виде суммы произведение:</p><p><strong>a)</strong> $\cos(\frac{\pi}{6} + \alpha) \cdot \cos(\frac{\pi}{6} - \alpha)$;</p><p><strong>б)</strong> $\cos(\frac{\pi}{8} + 2\alpha) \cdot \cos(\frac{\pi}{8} - 2\alpha)$;</p><p><strong>в)</strong> $\sin(\frac{\pi}{3} - 3\alpha) \cdot \cos(\frac{\pi}{3} + 3\alpha)$;</p><p><strong>г)</strong> $2\cos(\frac{\pi}{4} - \alpha) \cdot \sin(\frac{\pi}{4} + \alpha)$;</p><p><strong>д)</strong> $2\sin(2\alpha - \frac{\pi}{3}) \cdot \sin(\frac{\pi}{3} + 2\alpha)$;</p><p><strong>е)</strong> $2\sin(3\alpha - \frac{\pi}{4}) \cdot \sin(2\alpha + \frac{\pi}{4})$.</p><p><strong>766.</strong> Вычислите:</p>" "img" => array:1 [ 0 => array:5 [ "name" => "765-1.jpg" "alt" => null "width" => "1531" "height" => 486 "path" => "/media/algebra_09/soltan-u19/0-1/765-1.webp?ts=1752834761" ] ] ] #original: array:7 [ "id" => 1277120 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1108508" "type" => "task" ] "text" => "<p><strong>765.</strong> Представьте в виде суммы произведение:</p><p><strong>a)</strong> $\cos(\frac{\pi}{6} + \alpha) \cdot \cos(\frac{\pi}{6} - \alpha)$;</p><p><strong>б)</strong> $\cos(\frac{\pi}{8} + 2\alpha) \cdot \cos(\frac{\pi}{8} - 2\alpha)$;</p><p><strong>в)</strong> $\sin(\frac{\pi}{3} - 3\alpha) \cdot \cos(\frac{\pi}{3} + 3\alpha)$;</p><p><strong>г)</strong> $2\cos(\frac{\pi}{4} - \alpha) \cdot \sin(\frac{\pi}{4} + \alpha)$;</p><p><strong>д)</strong> $2\sin(2\alpha - \frac{\pi}{3}) \cdot \sin(\frac{\pi}{3} + 2\alpha)$;</p><p><strong>е)</strong> $2\sin(3\alpha - \frac{\pi}{4}) \cdot \sin(2\alpha + \frac{\pi}{4})$.</p><p><strong>766.</strong> Вычислите:</p>" "img" => array:1 [ 0 => array:5 [ "name" => "765-1.jpg" "alt" => null "width" => "1531" "height" => 486 "path" => "/media/algebra_09/soltan-u19/0-1/765-1.webp?ts=1752834761" ] ] ] #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 {#1107 #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" => 1278411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1109 #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 {#1110 …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 {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …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 {#1110 …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 {#1112 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1114 …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" => "1108508" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "765-1.jpg" "alt" => null "width" => "1270" "height" => 1884 "path" => "/media/algebra_09/soltan-u19/1-1/765-1.webp?ts=1752831566" ] 1 => array:5 [ "name" => "765-2.jpg" "alt" => null "width" => "1388" "height" => 409 "path" => "/media/algebra_09/soltan-u19/1-1/765-2.webp?ts=1752831566" ] ] ] #original: array:6 [ "id" => 1278411 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1108508" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "765-1.jpg" "alt" => null "width" => "1270" "height" => 1884 "path" => "/media/algebra_09/soltan-u19/1-1/765-1.webp?ts=1752831566" ] 1 => array:5 [ "name" => "765-2.jpg" "alt" => null "width" => "1388" "height" => 409 "path" => "/media/algebra_09/soltan-u19/1-1/765-2.webp?ts=1752831566" ] ] ] #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 {#1115 #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" => 1554366 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116 #items: array:1 [ 0 => App\Models\Edition {#1117 #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 {#1118 …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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …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 {#1118 …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 {#1120 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1122 …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" => "1108508" "type" => "task" ] "text" => "<p><strong>а)</strong> Для преобразования произведения $cos(\frac{\pi}{6} + \alpha) \cdot cos(\frac{\pi}{6} - \alpha)$ в сумму используем формулу произведения косинусов: $cos(x)cos(y) = \frac{1}{2}(cos(x - y) + cos(x + y))$.<br>В данном случае $x = \frac{\pi}{6} + \alpha$ и $y = \frac{\pi}{6} - \alpha$.<br>Найдем сумму и разность аргументов:<br>$x + y = (\frac{\pi}{6} + \alpha) + (\frac{\pi}{6} - \alpha) = \frac{2\pi}{6} = \frac{\pi}{3}$<br>$x - y = (\frac{\pi}{6} + \alpha) - (\frac{\pi}{6} - \alpha) = \frac{\pi}{6} + \alpha - \frac{\pi}{6} + \alpha = 2\alpha$<br>Подставим найденные значения в формулу:<br>$cos(\frac{\pi}{6} + \alpha) \cdot cos(\frac{\pi}{6} - \alpha) = \frac{1}{2}(cos(2\alpha) + cos(\frac{\pi}{3}))$<br>Так как $cos(\frac{\pi}{3}) = \frac{1}{2}$, получаем:<br>$\frac{1}{2}(cos(2\alpha) + \frac{1}{2}) = \frac{1}{2}cos(2\alpha) + \frac{1}{4}$<br><strong>Ответ:</strong> $\frac{1}{2}cos(2\alpha) + \frac{1}{4}$.</p><p><strong>б)</strong> Аналогично предыдущему пункту, используем формулу $cos(x)cos(y) = \frac{1}{2}(cos(x - y) + cos(x + y))$.<br>Здесь $x = \frac{\pi}{8} + 2\alpha$ и $y = \frac{\pi}{8} - 2\alpha$.<br>Сумма и разность аргументов:<br>$x + y = (\frac{\pi}{8} + 2\alpha) + (\frac{\pi}{8} - 2\alpha) = \frac{2\pi}{8} = \frac{\pi}{4}$<br>$x - y = (\frac{\pi}{8} + 2\alpha) - (\frac{\pi}{8} - 2\alpha) = 4\alpha$<br>Подставляем в формулу:<br>$cos(\frac{\pi}{8} + 2\alpha) \cdot cos(\frac{\pi}{8} - 2\alpha) = \frac{1}{2}(cos(4\alpha) + cos(\frac{\pi}{4}))$<br>Так как $cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, получаем:<br>$\frac{1}{2}(cos(4\alpha) + \frac{\sqrt{2}}{2}) = \frac{1}{2}cos(4\alpha) + \frac{\sqrt{2}}{4}$<br><strong>Ответ:</strong> $\frac{1}{2}cos(4\alpha) + \frac{\sqrt{2}}{4}$.</p><p><strong>в)</strong> Для преобразования произведения $sin(\frac{\pi}{3} - 3\alpha) \cdot cos(\frac{\pi}{3} + 3\alpha)$ используем формулу произведения синуса на косинус: $sin(x)cos(y) = \frac{1}{2}(sin(x + y) + sin(x - y))$.<br>Здесь $x = \frac{\pi}{3} - 3\alpha$ и $y = \frac{\pi}{3} + 3\alpha$.<br>Сумма и разность аргументов:<br>$x + y = (\frac{\pi}{3} - 3\alpha) + (\frac{\pi}{3} + 3\alpha) = \frac{2\pi}{3}$<br>$x - y = (\frac{\pi}{3} - 3\alpha) - (\frac{\pi}{3} + 3\alpha) = -6\alpha$<br>Подставляем в формулу:<br>$sin(\frac{\pi}{3} - 3\alpha) \cdot cos(\frac{\pi}{3} + 3\alpha) = \frac{1}{2}(sin(\frac{2\pi}{3}) + sin(-6\alpha))$<br>Так как $sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2}$ и $sin(-6\alpha) = -sin(6\alpha)$, получаем:<br>$\frac{1}{2}(\frac{\sqrt{3}}{2} - sin(6\alpha)) = \frac{\sqrt{3}}{4} - \frac{1}{2}sin(6\alpha)$<br><strong>Ответ:</strong> $\frac{\sqrt{3}}{4} - \frac{1}{2}sin(6\alpha)$.</p><p><strong>г)</strong> Для преобразования выражения $2cos(\frac{\pi}{4} - \alpha) \cdot sin(\frac{\pi}{4} + \alpha)$ используем формулу $2cos(x)sin(y) = sin(x + y) - sin(x - y)$.<br>Здесь $x = \frac{\pi}{4} - \alpha$ и $y = \frac{\pi}{4} + \alpha$.<br>Сумма и разность аргументов:<br>$x + y = (\frac{\pi}{4} - \alpha) + (\frac{\pi}{4} + \alpha) = \frac{\pi}{2}$<br>$x - y = (\frac{\pi}{4} - \alpha) - (\frac{\pi}{4} + \alpha) = -2\alpha$<br>Подставляем в формулу:<br>$2cos(\frac{\pi}{4} - \alpha) \cdot sin(\frac{\pi}{4} + \alpha) = sin(\frac{\pi}{2}) - sin(-2\alpha)$<br>Так как $sin(\frac{\pi}{2}) = 1$ и $sin(-2\alpha) = -sin(2\alpha)$, получаем:<br>$1 - (-sin(2\alpha)) = 1 + sin(2\alpha)$<br><strong>Ответ:</strong> $1 + sin(2\alpha)$.</p><p><strong>д)</strong> Для преобразования выражения $2sin(2\alpha - \frac{\pi}{3}) \cdot sin(\frac{\pi}{3} + 2\alpha)$ используем формулу $2sin(x)sin(y) = cos(x - y) - cos(x + y)$.<br>Представим выражение как $2sin(2\alpha + \frac{\pi}{3}) \cdot sin(2\alpha - \frac{\pi}{3})$.<br>Здесь $x = 2\alpha + \frac{\pi}{3}$ и $y = 2\alpha - \frac{\pi}{3}$.<br>Разность и сумма аргументов:<br>$x - y = (2\alpha + \frac{\pi}{3}) - (2\alpha - \frac{\pi}{3}) = \frac{2\pi}{3}$<br>$x + y = (2\alpha + \frac{\pi}{3}) + (2\alpha - \frac{\pi}{3}) = 4\alpha$<br>Подставляем в формулу:<br>$2sin(2\alpha - \frac{\pi}{3}) \cdot sin(\frac{\pi}{3} + 2\alpha) = cos(\frac{2\pi}{3}) - cos(4\alpha)$<br>Так как $cos(\frac{2\pi}{3}) = -cos(\frac{\pi}{3}) = -\frac{1}{2}$, получаем:<br>$-\frac{1}{2} - cos(4\alpha)$<br><strong>Ответ:</strong> $-\frac{1}{2} - cos(4\alpha)$.</p><p><strong>е)</strong> Для преобразования выражения $2sin(3\alpha - \frac{\pi}{4}) \cdot sin(2\alpha + \frac{\pi}{4})$ используем формулу $2sin(x)sin(y) = cos(x - y) - cos(x + y)$.<br>Здесь $x = 3\alpha - \frac{\pi}{4}$ и $y = 2\alpha + \frac{\pi}{4}$.<br>Разность и сумма аргументов:<br>$x - y = (3\alpha - \frac{\pi}{4}) - (2\alpha + \frac{\pi}{4}) = \alpha - \frac{2\pi}{4} = \alpha - \frac{\pi}{2}$<br>$x + y = (3\alpha - \frac{\pi}{4}) + (2\alpha + \frac{\pi}{4}) = 5\alpha$<br>Подставляем в формулу:<br>$2sin(3\alpha - \frac{\pi}{4}) \cdot sin(2\alpha + \frac{\pi}{4}) = cos(\alpha - \frac{\pi}{2}) - cos(5\alpha)$<br>Используя формулу приведения $cos(\alpha - \frac{\pi}{2}) = cos(\frac{\pi}{2} - \alpha) = sin(\alpha)$, получаем:<br>$sin(\alpha) - cos(5\alpha)$<br><strong>Ответ:</strong> $sin(\alpha) - cos(5\alpha)$.</p>" ] #original: array:6 [ "id" => 1554366 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1108508" "type" => "task" ] "text" => "<p><strong>а)</strong> Для преобразования произведения $cos(\frac{\pi}{6} + \alpha) \cdot cos(\frac{\pi}{6} - \alpha)$ в сумму используем формулу произведения косинусов: $cos(x)cos(y) = \frac{1}{2}(cos(x - y) + cos(x + y))$.<br>В данном случае $x = \frac{\pi}{6} + \alpha$ и $y = \frac{\pi}{6} - \alpha$.<br>Найдем сумму и разность аргументов:<br>$x + y = (\frac{\pi}{6} + \alpha) + (\frac{\pi}{6} - \alpha) = \frac{2\pi}{6} = \frac{\pi}{3}$<br>$x - y = (\frac{\pi}{6} + \alpha) - (\frac{\pi}{6} - \alpha) = \frac{\pi}{6} + \alpha - \frac{\pi}{6} + \alpha = 2\alpha$<br>Подставим найденные значения в формулу:<br>$cos(\frac{\pi}{6} + \alpha) \cdot cos(\frac{\pi}{6} - \alpha) = \frac{1}{2}(cos(2\alpha) + cos(\frac{\pi}{3}))$<br>Так как $cos(\frac{\pi}{3}) = \frac{1}{2}$, получаем:<br>$\frac{1}{2}(cos(2\alpha) + \frac{1}{2}) = \frac{1}{2}cos(2\alpha) + \frac{1}{4}$<br><strong>Ответ:</strong> $\frac{1}{2}cos(2\alpha) + \frac{1}{4}$.</p><p><strong>б)</strong> Аналогично предыдущему пункту, используем формулу $cos(x)cos(y) = \frac{1}{2}(cos(x - y) + cos(x + y))$.<br>Здесь $x = \frac{\pi}{8} + 2\alpha$ и $y = \frac{\pi}{8} - 2\alpha$.<br>Сумма и разность аргументов:<br>$x + y = (\frac{\pi}{8} + 2\alpha) + (\frac{\pi}{8} - 2\alpha) = \frac{2\pi}{8} = \frac{\pi}{4}$<br>$x - y = (\frac{\pi}{8} + 2\alpha) - (\frac{\pi}{8} - 2\alpha) = 4\alpha$<br>Подставляем в формулу:<br>$cos(\frac{\pi}{8} + 2\alpha) \cdot cos(\frac{\pi}{8} - 2\alpha) = \frac{1}{2}(cos(4\alpha) + cos(\frac{\pi}{4}))$<br>Так как $cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, получаем:<br>$\frac{1}{2}(cos(4\alpha) + \frac{\sqrt{2}}{2}) = \frac{1}{2}cos(4\alpha) + \frac{\sqrt{2}}{4}$<br><strong>Ответ:</strong> $\frac{1}{2}cos(4\alpha) + \frac{\sqrt{2}}{4}$.</p><p><strong>в)</strong> Для преобразования произведения $sin(\frac{\pi}{3} - 3\alpha) \cdot cos(\frac{\pi}{3} + 3\alpha)$ используем формулу произведения синуса на косинус: $sin(x)cos(y) = \frac{1}{2}(sin(x + y) + sin(x - y))$.<br>Здесь $x = \frac{\pi}{3} - 3\alpha$ и $y = \frac{\pi}{3} + 3\alpha$.<br>Сумма и разность аргументов:<br>$x + y = (\frac{\pi}{3} - 3\alpha) + (\frac{\pi}{3} + 3\alpha) = \frac{2\pi}{3}$<br>$x - y = (\frac{\pi}{3} - 3\alpha) - (\frac{\pi}{3} + 3\alpha) = -6\alpha$<br>Подставляем в формулу:<br>$sin(\frac{\pi}{3} - 3\alpha) \cdot cos(\frac{\pi}{3} + 3\alpha) = \frac{1}{2}(sin(\frac{2\pi}{3}) + sin(-6\alpha))$<br>Так как $sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2}$ и $sin(-6\alpha) = -sin(6\alpha)$, получаем:<br>$\frac{1}{2}(\frac{\sqrt{3}}{2} - sin(6\alpha)) = \frac{\sqrt{3}}{4} - \frac{1}{2}sin(6\alpha)$<br><strong>Ответ:</strong> $\frac{\sqrt{3}}{4} - \frac{1}{2}sin(6\alpha)$.</p><p><strong>г)</strong> Для преобразования выражения $2cos(\frac{\pi}{4} - \alpha) \cdot sin(\frac{\pi}{4} + \alpha)$ используем формулу $2cos(x)sin(y) = sin(x + y) - sin(x - y)$.<br>Здесь $x = \frac{\pi}{4} - \alpha$ и $y = \frac{\pi}{4} + \alpha$.<br>Сумма и разность аргументов:<br>$x + y = (\frac{\pi}{4} - \alpha) + (\frac{\pi}{4} + \alpha) = \frac{\pi}{2}$<br>$x - y = (\frac{\pi}{4} - \alpha) - (\frac{\pi}{4} + \alpha) = -2\alpha$<br>Подставляем в формулу:<br>$2cos(\frac{\pi}{4} - \alpha) \cdot sin(\frac{\pi}{4} + \alpha) = sin(\frac{\pi}{2}) - sin(-2\alpha)$<br>Так как $sin(\frac{\pi}{2}) = 1$ и $sin(-2\alpha) = -sin(2\alpha)$, получаем:<br>$1 - (-sin(2\alpha)) = 1 + sin(2\alpha)$<br><strong>Ответ:</strong> $1 + sin(2\alpha)$.</p><p><strong>д)</strong> Для преобразования выражения $2sin(2\alpha - \frac{\pi}{3}) \cdot sin(\frac{\pi}{3} + 2\alpha)$ используем формулу $2sin(x)sin(y) = cos(x - y) - cos(x + y)$.<br>Представим выражение как $2sin(2\alpha + \frac{\pi}{3}) \cdot sin(2\alpha - \frac{\pi}{3})$.<br>Здесь $x = 2\alpha + \frac{\pi}{3}$ и $y = 2\alpha - \frac{\pi}{3}$.<br>Разность и сумма аргументов:<br>$x - y = (2\alpha + \frac{\pi}{3}) - (2\alpha - \frac{\pi}{3}) = \frac{2\pi}{3}$<br>$x + y = (2\alpha + \frac{\pi}{3}) + (2\alpha - \frac{\pi}{3}) = 4\alpha$<br>Подставляем в формулу:<br>$2sin(2\alpha - \frac{\pi}{3}) \cdot sin(\frac{\pi}{3} + 2\alpha) = cos(\frac{2\pi}{3}) - cos(4\alpha)$<br>Так как $cos(\frac{2\pi}{3}) = -cos(\frac{\pi}{3}) = -\frac{1}{2}$, получаем:<br>$-\frac{1}{2} - cos(4\alpha)$<br><strong>Ответ:</strong> $-\frac{1}{2} - cos(4\alpha)$.</p><p><strong>е)</strong> Для преобразования выражения $2sin(3\alpha - \frac{\pi}{4}) \cdot sin(2\alpha + \frac{\pi}{4})$ используем формулу $2sin(x)sin(y) = cos(x - y) - cos(x + y)$.<br>Здесь $x = 3\alpha - \frac{\pi}{4}$ и $y = 2\alpha + \frac{\pi}{4}$.<br>Разность и сумма аргументов:<br>$x - y = (3\alpha - \frac{\pi}{4}) - (2\alpha + \frac{\pi}{4}) = \alpha - \frac{2\pi}{4} = \alpha - \frac{\pi}{2}$<br>$x + y = (3\alpha - \frac{\pi}{4}) + (2\alpha + \frac{\pi}{4}) = 5\alpha$<br>Подставляем в формулу:<br>$2sin(3\alpha - \frac{\pi}{4}) \cdot sin(2\alpha + \frac{\pi}{4}) = cos(\alpha - \frac{\pi}{2}) - cos(5\alpha)$<br>Используя формулу приведения $cos(\alpha - \frac{\pi}{2}) = cos(\frac{\pi}{2} - \alpha) = sin(\alpha)$, получаем:<br>$sin(\alpha) - cos(5\alpha)$<br><strong>Ответ:</strong> $sin(\alpha) - cos(5\alpha)$.</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" => "1108509" "type" => "task" ] "previous" => array:2 [ "refs" => "1108507" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1123 #items: array:1 [ 0 => App\Models\Book {#1124 #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 {#1125 #items: array:1 [ 0 => App\Models\Term {#1126 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1127 #items: array:1 [ 0 => App\Models\Term {#1128 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1129 #items: array:1 [ 0 => App\Models\Term {#1130 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:3 [ 0 => App\Models\Term {#1132 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1133 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1134 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1135 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1136 #items: array:1 [ 0 => App\Models\Term {#1137 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1138 #items: array:1 [ 0 => App\Models\Term {#1139 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1140 #items: array:1 [ 0 => App\Models\Term {#1141 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1142 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1143 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1144 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1146 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1147 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1148 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1125} "field_class" => Illuminate\Database\Eloquent\Collection {#1127} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1129} "field_author" => Illuminate\Database\Eloquent\Collection {#1131} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1135} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1136} "field_country" => Illuminate\Database\Eloquent\Collection {#1138} "field_city" => Illuminate\Database\Eloquent\Collection {#1140} "field_series" => Illuminate\Database\Eloquent\Collection {#1142} "field_umk" => Illuminate\Database\Eloquent\Collection {#1143} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1144} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1145} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1146} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1147} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1148} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1149} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1150} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1151} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 380 "subject" => 542 "class_subject" => 386 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1166 #items: array:1 [ 0 => App\Models\BookPage {#1165 #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" => 1097959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "217" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-217" "field_display_title" => "217" "field_folder" => "folder1" "field_image_name" => "217" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1167 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1198 #items: array:1 [ 0 => App\Models\Book {#1168 #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 {#1170 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1191 …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 {#1192 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1195 …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 {#1170 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1171 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1173 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1175 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1179 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1180 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1182 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1184 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1186 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1187 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1188 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1189 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1190 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1191 …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 {#1192 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1193 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1194 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1195 …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 {#1196 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1208 #items: array:1 [ 0 => App\Models\Element {#1197 #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" => 1261110 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1200 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261110 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1200 …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" => "1097960" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097958" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#2006 #items: array:6 [ 0 => App\Models\Task {#1206 #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" => 1108506 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "217" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/vopr-217" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1209 …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 {#1207 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1210 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1243 …2} "content" => Illuminate\Database\Eloquent\Collection {#1312 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1337 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108506 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "217" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/vopr-217" "field_display_title" => "Вопросы" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1209 …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 {#1207 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1210 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1243 …2} "content" => Illuminate\Database\Eloquent\Collection {#1312 …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 => "*" ] } 1 => App\Models\Task {#1366 #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" => 1108507 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "217" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-764" "field_display_title" => "764" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1367 …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 {#1369 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1370 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1403 …2} "content" => Illuminate\Database\Eloquent\Collection {#1440 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1465 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108507 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "217" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-764" "field_display_title" => "764" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1367 …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 {#1369 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1370 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1403 …2} "content" => Illuminate\Database\Eloquent\Collection {#1440 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1465 …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 {#1494 #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" => 1108508 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "217" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-765" "field_display_title" => "765" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1495 …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 {#1497 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1498 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1531 …2} "content" => Illuminate\Database\Eloquent\Collection {#1568 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1593 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108508 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "217" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-765" "field_display_title" => "765" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1495 …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 {#1497 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1498 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1531 …2} "content" => Illuminate\Database\Eloquent\Collection {#1568 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1593 …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 {#1622 #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" => 1108509 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "217" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-766" "field_display_title" => "766" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1623 …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 {#1625 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1626 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1659 …2} "content" => Illuminate\Database\Eloquent\Collection {#1696 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1721 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108509 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "217" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-766" "field_display_title" => "766" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1623 …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" => 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№765 (с. 217)
Условие. №765 (с. 217)
скриншот условия
765. Представьте в виде суммы произведение:
a) $\cos(\frac{\pi}{6} + \alpha) \cdot \cos(\frac{\pi}{6} - \alpha)$;
б) $\cos(\frac{\pi}{8} + 2\alpha) \cdot \cos(\frac{\pi}{8} - 2\alpha)$;
в) $\sin(\frac{\pi}{3} - 3\alpha) \cdot \cos(\frac{\pi}{3} + 3\alpha)$;
г) $2\cos(\frac{\pi}{4} - \alpha) \cdot \sin(\frac{\pi}{4} + \alpha)$;
д) $2\sin(2\alpha - \frac{\pi}{3}) \cdot \sin(\frac{\pi}{3} + 2\alpha)$;
е) $2\sin(3\alpha - \frac{\pi}{4}) \cdot \sin(2\alpha + \frac{\pi}{4})$.
766. Вычислите:
Решение. №765 (с. 217)
Решение 2 (rus). №765 (с. 217)
а) Для преобразования произведения $cos(\frac{\pi}{6} + \alpha) \cdot cos(\frac{\pi}{6} - \alpha)$ в сумму используем формулу произведения косинусов: $cos(x)cos(y) = \frac{1}{2}(cos(x - y) + cos(x + y))$.
В данном случае $x = \frac{\pi}{6} + \alpha$ и $y = \frac{\pi}{6} - \alpha$.
Найдем сумму и разность аргументов:
$x + y = (\frac{\pi}{6} + \alpha) + (\frac{\pi}{6} - \alpha) = \frac{2\pi}{6} = \frac{\pi}{3}$
$x - y = (\frac{\pi}{6} + \alpha) - (\frac{\pi}{6} - \alpha) = \frac{\pi}{6} + \alpha - \frac{\pi}{6} + \alpha = 2\alpha$
Подставим найденные значения в формулу:
$cos(\frac{\pi}{6} + \alpha) \cdot cos(\frac{\pi}{6} - \alpha) = \frac{1}{2}(cos(2\alpha) + cos(\frac{\pi}{3}))$
Так как $cos(\frac{\pi}{3}) = \frac{1}{2}$, получаем:
$\frac{1}{2}(cos(2\alpha) + \frac{1}{2}) = \frac{1}{2}cos(2\alpha) + \frac{1}{4}$
Ответ: $\frac{1}{2}cos(2\alpha) + \frac{1}{4}$.
б) Аналогично предыдущему пункту, используем формулу $cos(x)cos(y) = \frac{1}{2}(cos(x - y) + cos(x + y))$.
Здесь $x = \frac{\pi}{8} + 2\alpha$ и $y = \frac{\pi}{8} - 2\alpha$.
Сумма и разность аргументов:
$x + y = (\frac{\pi}{8} + 2\alpha) + (\frac{\pi}{8} - 2\alpha) = \frac{2\pi}{8} = \frac{\pi}{4}$
$x - y = (\frac{\pi}{8} + 2\alpha) - (\frac{\pi}{8} - 2\alpha) = 4\alpha$
Подставляем в формулу:
$cos(\frac{\pi}{8} + 2\alpha) \cdot cos(\frac{\pi}{8} - 2\alpha) = \frac{1}{2}(cos(4\alpha) + cos(\frac{\pi}{4}))$
Так как $cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, получаем:
$\frac{1}{2}(cos(4\alpha) + \frac{\sqrt{2}}{2}) = \frac{1}{2}cos(4\alpha) + \frac{\sqrt{2}}{4}$
Ответ: $\frac{1}{2}cos(4\alpha) + \frac{\sqrt{2}}{4}$.
в) Для преобразования произведения $sin(\frac{\pi}{3} - 3\alpha) \cdot cos(\frac{\pi}{3} + 3\alpha)$ используем формулу произведения синуса на косинус: $sin(x)cos(y) = \frac{1}{2}(sin(x + y) + sin(x - y))$.
Здесь $x = \frac{\pi}{3} - 3\alpha$ и $y = \frac{\pi}{3} + 3\alpha$.
Сумма и разность аргументов:
$x + y = (\frac{\pi}{3} - 3\alpha) + (\frac{\pi}{3} + 3\alpha) = \frac{2\pi}{3}$
$x - y = (\frac{\pi}{3} - 3\alpha) - (\frac{\pi}{3} + 3\alpha) = -6\alpha$
Подставляем в формулу:
$sin(\frac{\pi}{3} - 3\alpha) \cdot cos(\frac{\pi}{3} + 3\alpha) = \frac{1}{2}(sin(\frac{2\pi}{3}) + sin(-6\alpha))$
Так как $sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2}$ и $sin(-6\alpha) = -sin(6\alpha)$, получаем:
$\frac{1}{2}(\frac{\sqrt{3}}{2} - sin(6\alpha)) = \frac{\sqrt{3}}{4} - \frac{1}{2}sin(6\alpha)$
Ответ: $\frac{\sqrt{3}}{4} - \frac{1}{2}sin(6\alpha)$.
г) Для преобразования выражения $2cos(\frac{\pi}{4} - \alpha) \cdot sin(\frac{\pi}{4} + \alpha)$ используем формулу $2cos(x)sin(y) = sin(x + y) - sin(x - y)$.
Здесь $x = \frac{\pi}{4} - \alpha$ и $y = \frac{\pi}{4} + \alpha$.
Сумма и разность аргументов:
$x + y = (\frac{\pi}{4} - \alpha) + (\frac{\pi}{4} + \alpha) = \frac{\pi}{2}$
$x - y = (\frac{\pi}{4} - \alpha) - (\frac{\pi}{4} + \alpha) = -2\alpha$
Подставляем в формулу:
$2cos(\frac{\pi}{4} - \alpha) \cdot sin(\frac{\pi}{4} + \alpha) = sin(\frac{\pi}{2}) - sin(-2\alpha)$
Так как $sin(\frac{\pi}{2}) = 1$ и $sin(-2\alpha) = -sin(2\alpha)$, получаем:
$1 - (-sin(2\alpha)) = 1 + sin(2\alpha)$
Ответ: $1 + sin(2\alpha)$.
д) Для преобразования выражения $2sin(2\alpha - \frac{\pi}{3}) \cdot sin(\frac{\pi}{3} + 2\alpha)$ используем формулу $2sin(x)sin(y) = cos(x - y) - cos(x + y)$.
Представим выражение как $2sin(2\alpha + \frac{\pi}{3}) \cdot sin(2\alpha - \frac{\pi}{3})$.
Здесь $x = 2\alpha + \frac{\pi}{3}$ и $y = 2\alpha - \frac{\pi}{3}$.
Разность и сумма аргументов:
$x - y = (2\alpha + \frac{\pi}{3}) - (2\alpha - \frac{\pi}{3}) = \frac{2\pi}{3}$
$x + y = (2\alpha + \frac{\pi}{3}) + (2\alpha - \frac{\pi}{3}) = 4\alpha$
Подставляем в формулу:
$2sin(2\alpha - \frac{\pi}{3}) \cdot sin(\frac{\pi}{3} + 2\alpha) = cos(\frac{2\pi}{3}) - cos(4\alpha)$
Так как $cos(\frac{2\pi}{3}) = -cos(\frac{\pi}{3}) = -\frac{1}{2}$, получаем:
$-\frac{1}{2} - cos(4\alpha)$
Ответ: $-\frac{1}{2} - cos(4\alpha)$.
е) Для преобразования выражения $2sin(3\alpha - \frac{\pi}{4}) \cdot sin(2\alpha + \frac{\pi}{4})$ используем формулу $2sin(x)sin(y) = cos(x - y) - cos(x + y)$.
Здесь $x = 3\alpha - \frac{\pi}{4}$ и $y = 2\alpha + \frac{\pi}{4}$.
Разность и сумма аргументов:
$x - y = (3\alpha - \frac{\pi}{4}) - (2\alpha + \frac{\pi}{4}) = \alpha - \frac{2\pi}{4} = \alpha - \frac{\pi}{2}$
$x + y = (3\alpha - \frac{\pi}{4}) + (2\alpha + \frac{\pi}{4}) = 5\alpha$
Подставляем в формулу:
$2sin(3\alpha - \frac{\pi}{4}) \cdot sin(2\alpha + \frac{\pi}{4}) = cos(\alpha - \frac{\pi}{2}) - cos(5\alpha)$
Используя формулу приведения $cos(\alpha - \frac{\pi}{2}) = cos(\frac{\pi}{2} - \alpha) = sin(\alpha)$, получаем:
$sin(\alpha) - cos(5\alpha)$
Ответ: $sin(\alpha) - cos(5\alpha)$.
Другие задания:
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