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Номер 725, страница 207 - гдз по алгебре 9 класс учебник Солтан, Солтан
Авторы: Солтан Г. Н., Солтан А. Е., Жумадилова А. Ж.
Тип: Учебник
Издательство: Кокшетау
Год издания: 2019 - 2026
ISBN: 978-601-317-424-2
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 9 классе
IV. Тригонометрия. 27. Формулы тригонометрических функций двойного и половинного углов - номер 725, страница 207.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #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" => 1108466 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-725" "field_display_title" => "725" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #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" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "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_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 {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #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 {#1038 #items: array:1 [ 0 => App\Models\Book {#1040 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false 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"field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …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 {#1041 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1047 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1063 …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" => "" 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"field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068 #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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"field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1097 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1099 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1100 …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 {#1074 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1078 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1080 …2} 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"field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1101 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1072} "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 {#1070} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1101} ] #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 {#1102 #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" => 1107669 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "27. Формулы тригонометрических функций двойного и половинного углов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "203" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1104 #items: array:1 [ 0 => App\Models\Book {#1105 #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 {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …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 {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …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 {#1106 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1110 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1112 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1126 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1127 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1128 …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 {#1129 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1130 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1132 …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 {#1133 #items: array:1 [ 0 => App\Models\Branch {#1134 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1135 …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 {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1165 …2} ] #original: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1135 …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 {#1136 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1165 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1107669 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "27. Формулы тригонометрических функций двойного и половинного углов" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1103} "field_page_start" => "203" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1104} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1133} ] #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 {#1184 #items: array:3 [ 0 => App\Models\Element {#1194 #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" => 1277080 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203 #items: array:1 [ 0 => App\Models\Edition {#1195 #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 {#1197 …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 {#1198 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1200 …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 {#1197 …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 {#1198 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1199 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1200 …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" => "1108466" "type" => "task" ] "text" => "<p><strong>725.</strong> Вычислите:</p><p><strong>а)</strong> $tg \alpha$, если $tg 2\alpha = -\frac{5}{12}$;</p><p><strong>б)</strong> $tg(\alpha + \frac{\pi}{4})$, если $sin 2\alpha = 0,96$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "725-1.jpg" "alt" => null "width" => "1557" "height" => 219 "path" => "/media/algebra_09/soltan-u19/0-1/725-1.webp?ts=1752834710" ] ] ] #original: array:7 [ "id" => 1277080 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1203} "task" => array:2 [ "refs" => "1108466" "type" => "task" ] "text" => "<p><strong>725.</strong> Вычислите:</p><p><strong>а)</strong> $tg \alpha$, если $tg 2\alpha = -\frac{5}{12}$;</p><p><strong>б)</strong> $tg(\alpha + \frac{\pi}{4})$, если $sin 2\alpha = 0,96$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "725-1.jpg" "alt" => null "width" => "1557" "height" => 219 "path" => "/media/algebra_09/soltan-u19/0-1/725-1.webp?ts=1752834710" ] ] ] #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 {#1201 #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" => 1278371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1211 #items: array:1 [ 0 => App\Models\Edition {#1202 #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 {#1205 …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 {#1206 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1208 …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 {#1205 …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 {#1206 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1208 …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" => "1108466" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "725-1.jpg" "alt" => null "width" => "1382" "height" => 2204 "path" => "/media/algebra_09/soltan-u19/1-1/725-1.webp?ts=1752831506" ] 1 => array:5 [ "name" => "725-2.jpg" "alt" => null "width" => "1058" "height" => 438 "path" => "/media/algebra_09/soltan-u19/1-1/725-2.webp?ts=1752831506" ] ] ] #original: array:6 [ "id" => 1278371 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1211} "task" => array:2 [ "refs" => "1108466" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "725-1.jpg" "alt" => null "width" => "1382" "height" => 2204 "path" => "/media/algebra_09/soltan-u19/1-1/725-1.webp?ts=1752831506" ] 1 => array:5 [ "name" => "725-2.jpg" "alt" => null "width" => "1058" "height" => 438 "path" => "/media/algebra_09/soltan-u19/1-1/725-2.webp?ts=1752831506" ] ] ] #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 {#1209 #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" => 1554326 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ 0 => App\Models\Edition {#1210 #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 {#1213 …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 {#1214 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1216 …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 {#1213 …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 {#1214 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1215 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1216 …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" => "1108466" "type" => "task" ] "text" => "<p><strong>а)</strong> Для вычисления $tg\,\alpha$ воспользуемся формулой тангенса двойного угла:<br>$tg\,2\alpha = \frac{2tg\,\alpha}{1 - tg^2\alpha}$<br>Подставим в формулу известное значение $tg\,2\alpha = -\frac{5}{12}$:<br>$-\frac{5}{12} = \frac{2tg\,\alpha}{1 - tg^2\alpha}$<br>Для удобства введем замену: пусть $x = tg\,\alpha$. Уравнение примет вид:<br>$-\frac{5}{12} = \frac{2x}{1 - x^2}$<br>Решим это уравнение. Используем свойство пропорции:<br>$-5(1 - x^2) = 12(2x)$<br>$-5 + 5x^2 = 24x$<br>Перенесем все члены в левую часть, чтобы получить стандартное квадратное уравнение:<br>$5x^2 - 24x - 5 = 0$<br>Найдем корни этого уравнения с помощью дискриминанта $D = b^2 - 4ac$:<br>$D = (-24)^2 - 4 \cdot 5 \cdot (-5) = 576 + 100 = 676 = 26^2$<br>Так как дискриминант положительный, уравнение имеет два различных корня:<br>$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{24 + 26}{2 \cdot 5} = \frac{50}{10} = 5$<br>$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{24 - 26}{2 \cdot 5} = \frac{-2}{10} = -\frac{1}{5}$<br>Поскольку $x = tg\,\alpha$, мы нашли возможные значения для $tg\,\alpha$.<br>Ответ: $5$ или $-\frac{1}{5}$.</p><p><strong>б)</strong> Для вычисления $tg(\alpha + \frac{\pi}{4})$ можно использовать готовую формулу, связывающую его с $sin\,2\alpha$, или вывести ее. Проведем вывод.<br>Сначала воспользуемся формулой тангенса суммы углов:<br>$tg(\alpha + \frac{\pi}{4}) = \frac{tg\,\alpha + tg\frac{\pi}{4}}{1 - tg\,\alpha \cdot tg\frac{\pi}{4}}$<br>Так как $tg\frac{\pi}{4} = 1$, формула упрощается:<br>$tg(\alpha + \frac{\pi}{4}) = \frac{1+tg\,\alpha}{1-tg\,\alpha}$<br>Выразим тангенс через синус и косинус: $tg\,\alpha = \frac{sin\alpha}{cos\alpha}$.<br>$tg(\alpha + \frac{\pi}{4}) = \frac{1+\frac{sin\alpha}{cos\alpha}}{1-\frac{sin\alpha}{cos\alpha}} = \frac{\frac{cos\alpha+sin\alpha}{cos\alpha}}{\frac{cos\alpha-sin\alpha}{cos\alpha}} = \frac{cos\alpha + sin\alpha}{cos\alpha - sin\alpha}$<br>Возведем обе части равенства в квадрат:<br>$tg^2(\alpha + \frac{\pi}{4}) = \left(\frac{cos\alpha + sin\alpha}{cos\alpha - sin\alpha}\right)^2 = \frac{(cos\alpha + sin\alpha)^2}{(cos\alpha - sin\alpha)^2}$<br>Раскроем скобки, используя формулы квадрата суммы и разности:<br>$tg^2(\alpha + \frac{\pi}{4}) = \frac{cos^2\alpha + 2sin\alpha cos\alpha + sin^2\alpha}{cos^2\alpha - 2sin\alpha cos\alpha + sin^2\alpha}$<br>Применим основное тригонометрическое тождество $sin^2\alpha + cos^2\alpha = 1$ и формулу синуса двойного угла $2sin\alpha cos\alpha = sin\,2\alpha$:<br>$tg^2(\alpha + \frac{\pi}{4}) = \frac{1 + sin\,2\alpha}{1 - sin\,2\alpha}$<br>Теперь подставим в полученную формулу данное значение $sin\,2\alpha = 0,96$:<br>$tg^2(\alpha + \frac{\pi}{4}) = \frac{1 + 0,96}{1 - 0,96} = \frac{1,96}{0,04} = \frac{196}{4} = 49$<br>Из этого уравнения находим возможные значения для $tg(\alpha + \frac{\pi}{4})$:<br>$tg(\alpha + \frac{\pi}{4}) = \sqrt{49} = 7$ или $tg(\alpha + \frac{\pi}{4}) = -\sqrt{49} = -7$.<br>Ответ: $7$ или $-7$.</p>" ] #original: array:6 [ "id" => 1554326 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1219} "task" => array:2 [ "refs" => "1108466" "type" => "task" ] "text" => "<p><strong>а)</strong> Для вычисления $tg\,\alpha$ воспользуемся формулой тангенса двойного угла:<br>$tg\,2\alpha = \frac{2tg\,\alpha}{1 - tg^2\alpha}$<br>Подставим в формулу известное значение $tg\,2\alpha = -\frac{5}{12}$:<br>$-\frac{5}{12} = \frac{2tg\,\alpha}{1 - tg^2\alpha}$<br>Для удобства введем замену: пусть $x = tg\,\alpha$. Уравнение примет вид:<br>$-\frac{5}{12} = \frac{2x}{1 - x^2}$<br>Решим это уравнение. Используем свойство пропорции:<br>$-5(1 - x^2) = 12(2x)$<br>$-5 + 5x^2 = 24x$<br>Перенесем все члены в левую часть, чтобы получить стандартное квадратное уравнение:<br>$5x^2 - 24x - 5 = 0$<br>Найдем корни этого уравнения с помощью дискриминанта $D = b^2 - 4ac$:<br>$D = (-24)^2 - 4 \cdot 5 \cdot (-5) = 576 + 100 = 676 = 26^2$<br>Так как дискриминант положительный, уравнение имеет два различных корня:<br>$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{24 + 26}{2 \cdot 5} = \frac{50}{10} = 5$<br>$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{24 - 26}{2 \cdot 5} = \frac{-2}{10} = -\frac{1}{5}$<br>Поскольку $x = tg\,\alpha$, мы нашли возможные значения для $tg\,\alpha$.<br>Ответ: $5$ или $-\frac{1}{5}$.</p><p><strong>б)</strong> Для вычисления $tg(\alpha + \frac{\pi}{4})$ можно использовать готовую формулу, связывающую его с $sin\,2\alpha$, или вывести ее. Проведем вывод.<br>Сначала воспользуемся формулой тангенса суммы углов:<br>$tg(\alpha + \frac{\pi}{4}) = \frac{tg\,\alpha + tg\frac{\pi}{4}}{1 - tg\,\alpha \cdot tg\frac{\pi}{4}}$<br>Так как $tg\frac{\pi}{4} = 1$, формула упрощается:<br>$tg(\alpha + \frac{\pi}{4}) = \frac{1+tg\,\alpha}{1-tg\,\alpha}$<br>Выразим тангенс через синус и косинус: $tg\,\alpha = \frac{sin\alpha}{cos\alpha}$.<br>$tg(\alpha + \frac{\pi}{4}) = \frac{1+\frac{sin\alpha}{cos\alpha}}{1-\frac{sin\alpha}{cos\alpha}} = \frac{\frac{cos\alpha+sin\alpha}{cos\alpha}}{\frac{cos\alpha-sin\alpha}{cos\alpha}} = \frac{cos\alpha + sin\alpha}{cos\alpha - sin\alpha}$<br>Возведем обе части равенства в квадрат:<br>$tg^2(\alpha + \frac{\pi}{4}) = \left(\frac{cos\alpha + sin\alpha}{cos\alpha - sin\alpha}\right)^2 = \frac{(cos\alpha + sin\alpha)^2}{(cos\alpha - sin\alpha)^2}$<br>Раскроем скобки, используя формулы квадрата суммы и разности:<br>$tg^2(\alpha + \frac{\pi}{4}) = \frac{cos^2\alpha + 2sin\alpha cos\alpha + sin^2\alpha}{cos^2\alpha - 2sin\alpha cos\alpha + sin^2\alpha}$<br>Применим основное тригонометрическое тождество $sin^2\alpha + cos^2\alpha = 1$ и формулу синуса двойного угла $2sin\alpha cos\alpha = sin\,2\alpha$:<br>$tg^2(\alpha + \frac{\pi}{4}) = \frac{1 + sin\,2\alpha}{1 - sin\,2\alpha}$<br>Теперь подставим в полученную формулу данное значение $sin\,2\alpha = 0,96$:<br>$tg^2(\alpha + \frac{\pi}{4}) = \frac{1 + 0,96}{1 - 0,96} = \frac{1,96}{0,04} = \frac{196}{4} = 49$<br>Из этого уравнения находим возможные значения для $tg(\alpha + \frac{\pi}{4})$:<br>$tg(\alpha + \frac{\pi}{4}) = \sqrt{49} = 7$ или $tg(\alpha + \frac{\pi}{4}) = -\sqrt{49} = -7$.<br>Ответ: $7$ или $-7$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: 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"width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1187 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1217 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1218 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1220 #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 {#1176} "field_class" => Illuminate\Database\Eloquent\Collection {#1177} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1181} "field_author" => Illuminate\Database\Eloquent\Collection {#1178} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1167} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1174} "field_country" => Illuminate\Database\Eloquent\Collection {#1172} "field_city" => Illuminate\Database\Eloquent\Collection {#1171} "field_series" => Illuminate\Database\Eloquent\Collection {#1183} "field_umk" => Illuminate\Database\Eloquent\Collection {#1190} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1192} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1188} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1186} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1189} "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 {#1187} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1217} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1218} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1220} "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 {#1863 #items: array:1 [ 0 => App\Models\BookPage {#1032 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1097949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-207" "field_display_title" => "207" "field_folder" => "folder1" "field_image_name" => "207" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1023 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1026 #items: array:1 [ 0 => App\Models\Book {#1221 #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 {#1222 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1225 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1227 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1229 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1233 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1234 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1236 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1238 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1240 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1241 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1242 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1243 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1244 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1245 …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 {#1246 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1247 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1248 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1249 …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 {#1222 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1225 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1227 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1229 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1233 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1234 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1236 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1238 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1240 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1241 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1242 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1243 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1244 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1245 …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 {#1246 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1247 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1248 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1249 …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 {#1250 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1251 #items: array:1 [ 0 => App\Models\Element {#1252 #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" => 1261100 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1253 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261100 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1253 …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" => "1097950" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097948" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1260 #items: array:6 [ 0 => App\Models\Task {#1261 #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" => 1108462 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-721" "field_display_title" => "721" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1262 …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 {#1264 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1265 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1298 …2} "content" => Illuminate\Database\Eloquent\Collection {#1335 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1360 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108462 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-721" "field_display_title" => "721" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1262 …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 {#1264 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1265 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1298 …2} "content" => Illuminate\Database\Eloquent\Collection {#1335 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1360 …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 {#1361 #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" => 1108463 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-722" "field_display_title" => "722" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1362 …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 {#1364 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1365 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1398 …2} "content" => Illuminate\Database\Eloquent\Collection {#1435 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1460 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108463 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-722" "field_display_title" => "722" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1362 …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 {#1364 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1365 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1398 …2} "content" => Illuminate\Database\Eloquent\Collection {#1435 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1460 …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 {#1461 #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" => 1108464 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-723" "field_display_title" => "723" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1462 …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 {#1464 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1465 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1498 …2} "content" => Illuminate\Database\Eloquent\Collection {#1535 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1560 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108464 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-723" "field_display_title" => "723" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1462 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null …9 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 3 => App\Models\Task {#1561 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 4 => App\Models\Task {#1661 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 5 => App\Models\Task {#1761 #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 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1097949 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "207" "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-207" "field_display_title" => "207" "field_folder" => "folder1" "field_image_name" => "207" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1023} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1026} "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 {#1250} "content" => Illuminate\Database\Eloquent\Collection {#1251} "next" => array:2 [ "refs" => "1097950" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097948" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1260} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] 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№725 (с. 207)
Условие. №725 (с. 207)
скриншот условия
725. Вычислите:
а) $tg \alpha$, если $tg 2\alpha = -\frac{5}{12}$;
б) $tg(\alpha + \frac{\pi}{4})$, если $sin 2\alpha = 0,96$.
Решение. №725 (с. 207)
Решение 2 (rus). №725 (с. 207)
а) Для вычисления $tg\,\alpha$ воспользуемся формулой тангенса двойного угла:
$tg\,2\alpha = \frac{2tg\,\alpha}{1 - tg^2\alpha}$
Подставим в формулу известное значение $tg\,2\alpha = -\frac{5}{12}$:
$-\frac{5}{12} = \frac{2tg\,\alpha}{1 - tg^2\alpha}$
Для удобства введем замену: пусть $x = tg\,\alpha$. Уравнение примет вид:
$-\frac{5}{12} = \frac{2x}{1 - x^2}$
Решим это уравнение. Используем свойство пропорции:
$-5(1 - x^2) = 12(2x)$
$-5 + 5x^2 = 24x$
Перенесем все члены в левую часть, чтобы получить стандартное квадратное уравнение:
$5x^2 - 24x - 5 = 0$
Найдем корни этого уравнения с помощью дискриминанта $D = b^2 - 4ac$:
$D = (-24)^2 - 4 \cdot 5 \cdot (-5) = 576 + 100 = 676 = 26^2$
Так как дискриминант положительный, уравнение имеет два различных корня:
$x_1 = \frac{-b + \sqrt{D}}{2a} = \frac{24 + 26}{2 \cdot 5} = \frac{50}{10} = 5$
$x_2 = \frac{-b - \sqrt{D}}{2a} = \frac{24 - 26}{2 \cdot 5} = \frac{-2}{10} = -\frac{1}{5}$
Поскольку $x = tg\,\alpha$, мы нашли возможные значения для $tg\,\alpha$.
Ответ: $5$ или $-\frac{1}{5}$.
б) Для вычисления $tg(\alpha + \frac{\pi}{4})$ можно использовать готовую формулу, связывающую его с $sin\,2\alpha$, или вывести ее. Проведем вывод.
Сначала воспользуемся формулой тангенса суммы углов:
$tg(\alpha + \frac{\pi}{4}) = \frac{tg\,\alpha + tg\frac{\pi}{4}}{1 - tg\,\alpha \cdot tg\frac{\pi}{4}}$
Так как $tg\frac{\pi}{4} = 1$, формула упрощается:
$tg(\alpha + \frac{\pi}{4}) = \frac{1+tg\,\alpha}{1-tg\,\alpha}$
Выразим тангенс через синус и косинус: $tg\,\alpha = \frac{sin\alpha}{cos\alpha}$.
$tg(\alpha + \frac{\pi}{4}) = \frac{1+\frac{sin\alpha}{cos\alpha}}{1-\frac{sin\alpha}{cos\alpha}} = \frac{\frac{cos\alpha+sin\alpha}{cos\alpha}}{\frac{cos\alpha-sin\alpha}{cos\alpha}} = \frac{cos\alpha + sin\alpha}{cos\alpha - sin\alpha}$
Возведем обе части равенства в квадрат:
$tg^2(\alpha + \frac{\pi}{4}) = \left(\frac{cos\alpha + sin\alpha}{cos\alpha - sin\alpha}\right)^2 = \frac{(cos\alpha + sin\alpha)^2}{(cos\alpha - sin\alpha)^2}$
Раскроем скобки, используя формулы квадрата суммы и разности:
$tg^2(\alpha + \frac{\pi}{4}) = \frac{cos^2\alpha + 2sin\alpha cos\alpha + sin^2\alpha}{cos^2\alpha - 2sin\alpha cos\alpha + sin^2\alpha}$
Применим основное тригонометрическое тождество $sin^2\alpha + cos^2\alpha = 1$ и формулу синуса двойного угла $2sin\alpha cos\alpha = sin\,2\alpha$:
$tg^2(\alpha + \frac{\pi}{4}) = \frac{1 + sin\,2\alpha}{1 - sin\,2\alpha}$
Теперь подставим в полученную формулу данное значение $sin\,2\alpha = 0,96$:
$tg^2(\alpha + \frac{\pi}{4}) = \frac{1 + 0,96}{1 - 0,96} = \frac{1,96}{0,04} = \frac{196}{4} = 49$
Из этого уравнения находим возможные значения для $tg(\alpha + \frac{\pi}{4})$:
$tg(\alpha + \frac{\pi}{4}) = \sqrt{49} = 7$ или $tg(\alpha + \frac{\pi}{4}) = -\sqrt{49} = -7$.
Ответ: $7$ или $-7$.
Другие задания:
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