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Номер 583, страница 169 - 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. 22. Синус, косинус, тангенс и котангенс произвольного угла - номер 583, страница 169.
App\Models\Task {#1024 // 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" => 1108316 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "169" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-583" "field_display_title" => "583" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #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 {#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 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 …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 {#1056 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4298 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1032 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1035 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1037 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1039 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1043 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1050 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1055 …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 {#1056 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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. Тригонометрия" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333} "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} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1060} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1061 #items: array:2 [ 0 => App\Models\Branch {#1023} 1 => App\Models\Branch {#1062 #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" => 1107664 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "22. Синус, косинус, тангенс и котангенс произвольного угла" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "158" "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 #items: array:1 [ 0 => App\Models\Book {#1065 #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 {#1066 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 …2} "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" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …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 {#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 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1072 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1076 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1077 …2} "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" => Illuminate\Database\Eloquent\Collection {#1087 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1088 …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 {#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] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1093 #items: array:1 [ 0 => App\Models\Branch {#1094 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1107662 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "IV. 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} ] #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 {#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" => 1107664 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "22. Синус, косинус, тангенс и котангенс произвольного угла" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1063} "field_page_start" => "158" "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" => 1276938 "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" => "1108316" "type" => "task" ] "text" => "<p><strong>583.</strong> Исследуйте, какие значения $\beta$ удовлетворяют равенству</p><p>$\sin \beta = -\frac{1}{2}$, если:</p><p><strong>а)</strong> $\pi \le \beta \le \frac{3\pi}{2}$;</p><p><strong>б)</strong> $\frac{3\pi}{2} \le \beta \le 2\pi$;</p><p><strong>в)</strong> $-\pi \le \beta \le 0$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "583-1.jpg" "alt" => null "width" => "1573" "height" => 258 "path" => "/media/algebra_09/soltan-u19/0-1/583-1.webp?ts=1752834597" ] ] ] #original: array:7 [ "id" => 1276938 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1100} "task" => array:2 [ "refs" => "1108316" "type" => "task" ] "text" => "<p><strong>583.</strong> Исследуйте, какие значения $\beta$ удовлетворяют равенству</p><p>$\sin \beta = -\frac{1}{2}$, если:</p><p><strong>а)</strong> $\pi \le \beta \le \frac{3\pi}{2}$;</p><p><strong>б)</strong> $\frac{3\pi}{2} \le \beta \le 2\pi$;</p><p><strong>в)</strong> $-\pi \le \beta \le 0$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "583-1.jpg" "alt" => null "width" => "1573" "height" => 258 "path" => "/media/algebra_09/soltan-u19/0-1/583-1.webp?ts=1752834597" ] ] ] #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" => 1278229 "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" => "1108316" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "583-1.jpg" "alt" => null "width" => "816" "height" => 732 "path" => "/media/algebra_09/soltan-u19/1-1/583-1.webp?ts=1752831368" ] ] ] #original: array:6 [ "id" => 1278229 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1108316" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "583-1.jpg" "alt" => null "width" => "816" "height" => 732 "path" => "/media/algebra_09/soltan-u19/1-1/583-1.webp?ts=1752831368" ] ] ] #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" => 1554184 "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" => "1108316" "type" => "task" ] "text" => "<p>Для решения задачи сначала найдем общее решение уравнения $\sin\beta = -\frac{1}{2}$.</p><p>Общее решение уравнения $\sin x = a$ имеет вид $x = (-1)^n \arcsin(a) + \pi n$, где $n \in \mathbb{Z}$.</p><p>В нашем случае $a = -\frac{1}{2}$, а $\arcsin(-\frac{1}{2}) = -\frac{\pi}{6}$.</p><p>Следовательно, общее решение для $\beta$ можно записать как $\beta = (-1)^n(-\frac{\pi}{6}) + \pi n = (-1)^{n+1}\frac{\pi}{6} + \pi n$, где $n \in \mathbb{Z}$.</p><p>Более наглядно для отбора корней на промежутках представить общее решение в виде двух серий, соответствующих углам в III и IV четвертях единичной окружности, где синус отрицателен:</p><p><strong>1.</strong> $\beta = \pi + \frac{\pi}{6} + 2\pi k = \frac{7\pi}{6} + 2\pi k$, где $k \in \mathbb{Z}$.</p><p><strong>2.</strong> $\beta = -\frac{\pi}{6} + 2\pi k$, где $k \in \mathbb{Z}$.</p><p>Теперь исследуем, какие значения $\beta$ из этих серий попадают в заданные промежутки.</p><p><strong>а)</strong> Исследуем промежуток $\pi \le \beta \le \frac{3\pi}{2}$. Этот промежуток соответствует III четверти единичной окружности.</p><p>Рассмотрим первую серию корней $\beta = \frac{7\pi}{6} + 2\pi k$.<br>Подставим целые значения $k$ для поиска подходящих корней. При $k=0$, получаем $\beta = \frac{7\pi}{6}$. Проверим принадлежность этого значения промежутку: $\pi \le \frac{7\pi}{6} \le \frac{3\pi}{2}$. Приведя к общему знаменателю 6, получим $\frac{6\pi}{6} \le \frac{7\pi}{6} \le \frac{9\pi}{6}$. Неравенство верно, следовательно, $\beta = \frac{7\pi}{6}$ является решением.<br>При $k=1$, $\beta = \frac{7\pi}{6} + 2\pi = \frac{19\pi}{6}$, что больше $\frac{3\pi}{2}$.<br>При $k=-1$, $\beta = \frac{7\pi}{6} - 2\pi = -\frac{5\pi}{6}$, что меньше $\pi$.</p><p>Рассмотрим вторую серию корней $\beta = -\frac{\pi}{6} + 2\pi k$.<br>При $k=1$, $\beta = -\frac{\pi}{6} + 2\pi = \frac{11\pi}{6}$. Это значение не входит в промежуток, так как $\frac{11\pi}{6} > \frac{9\pi}{6} = \frac{3\pi}{2}$.<br>Другие значения $k$ также не дадут решений в данном промежутке.</p><p>Таким образом, на промежутке $\pi \le \beta \le \frac{3\pi}{2}$ есть только одно решение.</p><p><strong>Ответ:</strong> $\beta = \frac{7\pi}{6}$.</p><p><strong>б)</strong> Исследуем промежуток $\frac{3\pi}{2} \le \beta \le 2\pi$. Этот промежуток соответствует IV четверти единичной окружности.</p><p>Рассмотрим первую серию корней $\beta = \frac{7\pi}{6} + 2\pi k$.<br>Как мы видели в пункте а), при $k=0$, $\beta = \frac{7\pi}{6}$. Это значение меньше $\frac{3\pi}{2}$, поэтому оно не входит в заданный промежуток.</p><p>Рассмотрим вторую серию корней $\beta = -\frac{\pi}{6} + 2\pi k$.<br>При $k=1$, получаем $\beta = -\frac{\pi}{6} + 2\pi = \frac{11\pi}{6}$. Проверим принадлежность этого значения промежутку: $\frac{3\pi}{2} \le \frac{11\pi}{6} \le 2\pi$. Приведя к общему знаменателю 6, получим $\frac{9\pi}{6} \le \frac{11\pi}{6} \le \frac{12\pi}{6}$. Неравенство верно, следовательно, $\beta = \frac{11\pi}{6}$ является решением.<br>При $k=0$, $\beta = -\frac{\pi}{6}$, что меньше $\frac{3\pi}{2}$.<br>При $k=2$, $\beta = -\frac{\pi}{6} + 4\pi = \frac{23\pi}{6}$, что больше $2\pi$.</p><p>Таким образом, на промежутке $\frac{3\pi}{2} \le \beta \le 2\pi$ есть только одно решение.</p><p><strong>Ответ:</strong> $\beta = \frac{11\pi}{6}$.</p><p><strong>в)</strong> Исследуем промежуток $-\pi \le \beta \le 0$. Этот промежуток включает IV и III четверти при отрицательном направлении обхода единичной окружности.</p><p>Рассмотрим первую серию корней $\beta = \frac{7\pi}{6} + 2\pi k$.<br>При $k=-1$, получаем $\beta = \frac{7\pi}{6} - 2\pi = \frac{7\pi - 12\pi}{6} = -\frac{5\pi}{6}$. Проверим принадлежность этого значения промежутку: $-\pi \le -\frac{5\pi}{6} \le 0$. Неравенство верно, так как $-\frac{6\pi}{6} \le -\frac{5\pi}{6} \le 0$. Следовательно, $\beta = -\frac{5\pi}{6}$ является решением.</p><p>Рассмотрим вторую серию корней $\beta = -\frac{\pi}{6} + 2\pi k$.<br>При $k=0$, получаем $\beta = -\frac{\pi}{6}$. Проверим принадлежность этого значения промежутку: $-\pi \le -\frac{\pi}{6} \le 0$. Неравенство верно. Следовательно, $\beta = -\frac{\pi}{6}$ является решением.<br>При $k=-1$, $\beta = -\frac{\pi}{6} - 2\pi = -\frac{13\pi}{6}$, что меньше $-\pi$.</p><p>Таким образом, на промежутке $-\pi \le \beta \le 0$ есть два решения.</p><p><strong>Ответ:</strong> $\beta = -\frac{5\pi}{6}$; $\beta = -\frac{\pi}{6}$.</p>" ] #original: array:6 [ "id" => 1554184 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1116} "task" => array:2 [ "refs" => "1108316" "type" => "task" ] "text" => "<p>Для решения задачи сначала найдем общее решение уравнения $\sin\beta = -\frac{1}{2}$.</p><p>Общее решение уравнения $\sin x = a$ имеет вид $x = (-1)^n \arcsin(a) + \pi n$, где $n \in \mathbb{Z}$.</p><p>В нашем случае $a = -\frac{1}{2}$, а $\arcsin(-\frac{1}{2}) = -\frac{\pi}{6}$.</p><p>Следовательно, общее решение для $\beta$ можно записать как $\beta = (-1)^n(-\frac{\pi}{6}) + \pi n = (-1)^{n+1}\frac{\pi}{6} + \pi n$, где $n \in \mathbb{Z}$.</p><p>Более наглядно для отбора корней на промежутках представить общее решение в виде двух серий, соответствующих углам в III и IV четвертях единичной окружности, где синус отрицателен:</p><p><strong>1.</strong> $\beta = \pi + \frac{\pi}{6} + 2\pi k = \frac{7\pi}{6} + 2\pi k$, где $k \in \mathbb{Z}$.</p><p><strong>2.</strong> $\beta = -\frac{\pi}{6} + 2\pi k$, где $k \in \mathbb{Z}$.</p><p>Теперь исследуем, какие значения $\beta$ из этих серий попадают в заданные промежутки.</p><p><strong>а)</strong> Исследуем промежуток $\pi \le \beta \le \frac{3\pi}{2}$. Этот промежуток соответствует III четверти единичной окружности.</p><p>Рассмотрим первую серию корней $\beta = \frac{7\pi}{6} + 2\pi k$.<br>Подставим целые значения $k$ для поиска подходящих корней. При $k=0$, получаем $\beta = \frac{7\pi}{6}$. Проверим принадлежность этого значения промежутку: $\pi \le \frac{7\pi}{6} \le \frac{3\pi}{2}$. Приведя к общему знаменателю 6, получим $\frac{6\pi}{6} \le \frac{7\pi}{6} \le \frac{9\pi}{6}$. Неравенство верно, следовательно, $\beta = \frac{7\pi}{6}$ является решением.<br>При $k=1$, $\beta = \frac{7\pi}{6} + 2\pi = \frac{19\pi}{6}$, что больше $\frac{3\pi}{2}$.<br>При $k=-1$, $\beta = \frac{7\pi}{6} - 2\pi = -\frac{5\pi}{6}$, что меньше $\pi$.</p><p>Рассмотрим вторую серию корней $\beta = -\frac{\pi}{6} + 2\pi k$.<br>При $k=1$, $\beta = -\frac{\pi}{6} + 2\pi = \frac{11\pi}{6}$. Это значение не входит в промежуток, так как $\frac{11\pi}{6} > \frac{9\pi}{6} = \frac{3\pi}{2}$.<br>Другие значения $k$ также не дадут решений в данном промежутке.</p><p>Таким образом, на промежутке $\pi \le \beta \le \frac{3\pi}{2}$ есть только одно решение.</p><p><strong>Ответ:</strong> $\beta = \frac{7\pi}{6}$.</p><p><strong>б)</strong> Исследуем промежуток $\frac{3\pi}{2} \le \beta \le 2\pi$. Этот промежуток соответствует IV четверти единичной окружности.</p><p>Рассмотрим первую серию корней $\beta = \frac{7\pi}{6} + 2\pi k$.<br>Как мы видели в пункте а), при $k=0$, $\beta = \frac{7\pi}{6}$. Это значение меньше $\frac{3\pi}{2}$, поэтому оно не входит в заданный промежуток.</p><p>Рассмотрим вторую серию корней $\beta = -\frac{\pi}{6} + 2\pi k$.<br>При $k=1$, получаем $\beta = -\frac{\pi}{6} + 2\pi = \frac{11\pi}{6}$. Проверим принадлежность этого значения промежутку: $\frac{3\pi}{2} \le \frac{11\pi}{6} \le 2\pi$. Приведя к общему знаменателю 6, получим $\frac{9\pi}{6} \le \frac{11\pi}{6} \le \frac{12\pi}{6}$. Неравенство верно, следовательно, $\beta = \frac{11\pi}{6}$ является решением.<br>При $k=0$, $\beta = -\frac{\pi}{6}$, что меньше $\frac{3\pi}{2}$.<br>При $k=2$, $\beta = -\frac{\pi}{6} + 4\pi = \frac{23\pi}{6}$, что больше $2\pi$.</p><p>Таким образом, на промежутке $\frac{3\pi}{2} \le \beta \le 2\pi$ есть только одно решение.</p><p><strong>Ответ:</strong> $\beta = \frac{11\pi}{6}$.</p><p><strong>в)</strong> Исследуем промежуток $-\pi \le \beta \le 0$. Этот промежуток включает IV и III четверти при отрицательном направлении обхода единичной окружности.</p><p>Рассмотрим первую серию корней $\beta = \frac{7\pi}{6} + 2\pi k$.<br>При $k=-1$, получаем $\beta = \frac{7\pi}{6} - 2\pi = \frac{7\pi - 12\pi}{6} = -\frac{5\pi}{6}$. Проверим принадлежность этого значения промежутку: $-\pi \le -\frac{5\pi}{6} \le 0$. Неравенство верно, так как $-\frac{6\pi}{6} \le -\frac{5\pi}{6} \le 0$. Следовательно, $\beta = -\frac{5\pi}{6}$ является решением.</p><p>Рассмотрим вторую серию корней $\beta = -\frac{\pi}{6} + 2\pi k$.<br>При $k=0$, получаем $\beta = -\frac{\pi}{6}$. Проверим принадлежность этого значения промежутку: $-\pi \le -\frac{\pi}{6} \le 0$. Неравенство верно. Следовательно, $\beta = -\frac{\pi}{6}$ является решением.<br>При $k=-1$, $\beta = -\frac{\pi}{6} - 2\pi = -\frac{13\pi}{6}$, что меньше $-\pi$.</p><p>Таким образом, на промежутке $-\pi \le \beta \le 0$ есть два решения.</p><p><strong>Ответ:</strong> $\beta = -\frac{5\pi}{6}$; $\beta = -\frac{\pi}{6}$.</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" => "1108317" "type" => "task" ] "previous" => array:2 [ "refs" => "1108315" "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 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"field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/page-169" "field_display_title" => "169" "field_folder" => "folder1" "field_image_name" => "169" "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" => 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=> "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 {#1207 #items: array:1 [ 0 => App\Models\Element {#1206 #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" => 1261062 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1216 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261062 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1216 …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" => "1097912" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1097910" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1849 #items: array:7 [ 0 => App\Models\Task {#1863 #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" => 1108311 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "169" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-578" "field_display_title" => "578" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1867 …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 {#1866 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1901 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1968 …2} "content" => Illuminate\Database\Eloquent\Collection {#1978 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1985 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108311 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "169" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-578" "field_display_title" => "578" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1867 …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 {#1866 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1901 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1968 …2} "content" => Illuminate\Database\Eloquent\Collection {#1978 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1985 …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 {#1970 #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" => 1108312 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "169" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-579" "field_display_title" => "579" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1971 …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 {#1969 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1976 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1975 …2} "content" => Illuminate\Database\Eloquent\Collection {#1983 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2011 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108312 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "169" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-579" "field_display_title" => "579" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1971 …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 {#1969 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1976 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1975 …2} "content" => Illuminate\Database\Eloquent\Collection {#1983 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2011 …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 {#1974 #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" => 1108313 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "169" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-580" "field_display_title" => "580" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1973 …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 {#1972 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1982 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1986 …2} "content" => Illuminate\Database\Eloquent\Collection {#2026 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2033 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108313 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "169" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-580" "field_display_title" => "580" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1973 …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 {#1972 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1982 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1986 …2} "content" => Illuminate\Database\Eloquent\Collection {#2026 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2033 …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 {#2005 #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" => 1108314 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "169" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-581" "field_display_title" => "581" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2009 …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 {#2006 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2024 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2023 …2} "content" => Illuminate\Database\Eloquent\Collection {#2031 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2047 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108314 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "169" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-581" "field_display_title" => "581" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#2009 …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 {#2006 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2024 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2023 …2} "content" => Illuminate\Database\Eloquent\Collection {#2031 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2047 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#2010 #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" => 1108315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "169" "field_page_end" => null "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik/1-582" "field_display_title" => "582" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1981 …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 {#1980 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#2030 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#2034 …2} "content" => Illuminate\Database\Eloquent\Collection {#2045 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#2061 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1108315 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "169" "field_page_end" => null "field_url" 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№583 (с. 169)
Условие. №583 (с. 169)
скриншот условия
583. Исследуйте, какие значения $\beta$ удовлетворяют равенству
$\sin \beta = -\frac{1}{2}$, если:
а) $\pi \le \beta \le \frac{3\pi}{2}$;
б) $\frac{3\pi}{2} \le \beta \le 2\pi$;
в) $-\pi \le \beta \le 0$.
Решение. №583 (с. 169)
Решение 2 (rus). №583 (с. 169)
Для решения задачи сначала найдем общее решение уравнения $\sin\beta = -\frac{1}{2}$.
Общее решение уравнения $\sin x = a$ имеет вид $x = (-1)^n \arcsin(a) + \pi n$, где $n \in \mathbb{Z}$.
В нашем случае $a = -\frac{1}{2}$, а $\arcsin(-\frac{1}{2}) = -\frac{\pi}{6}$.
Следовательно, общее решение для $\beta$ можно записать как $\beta = (-1)^n(-\frac{\pi}{6}) + \pi n = (-1)^{n+1}\frac{\pi}{6} + \pi n$, где $n \in \mathbb{Z}$.
Более наглядно для отбора корней на промежутках представить общее решение в виде двух серий, соответствующих углам в III и IV четвертях единичной окружности, где синус отрицателен:
1. $\beta = \pi + \frac{\pi}{6} + 2\pi k = \frac{7\pi}{6} + 2\pi k$, где $k \in \mathbb{Z}$.
2. $\beta = -\frac{\pi}{6} + 2\pi k$, где $k \in \mathbb{Z}$.
Теперь исследуем, какие значения $\beta$ из этих серий попадают в заданные промежутки.
а) Исследуем промежуток $\pi \le \beta \le \frac{3\pi}{2}$. Этот промежуток соответствует III четверти единичной окружности.
Рассмотрим первую серию корней $\beta = \frac{7\pi}{6} + 2\pi k$.
Подставим целые значения $k$ для поиска подходящих корней. При $k=0$, получаем $\beta = \frac{7\pi}{6}$. Проверим принадлежность этого значения промежутку: $\pi \le \frac{7\pi}{6} \le \frac{3\pi}{2}$. Приведя к общему знаменателю 6, получим $\frac{6\pi}{6} \le \frac{7\pi}{6} \le \frac{9\pi}{6}$. Неравенство верно, следовательно, $\beta = \frac{7\pi}{6}$ является решением.
При $k=1$, $\beta = \frac{7\pi}{6} + 2\pi = \frac{19\pi}{6}$, что больше $\frac{3\pi}{2}$.
При $k=-1$, $\beta = \frac{7\pi}{6} - 2\pi = -\frac{5\pi}{6}$, что меньше $\pi$.
Рассмотрим вторую серию корней $\beta = -\frac{\pi}{6} + 2\pi k$.
При $k=1$, $\beta = -\frac{\pi}{6} + 2\pi = \frac{11\pi}{6}$. Это значение не входит в промежуток, так как $\frac{11\pi}{6} > \frac{9\pi}{6} = \frac{3\pi}{2}$.
Другие значения $k$ также не дадут решений в данном промежутке.
Таким образом, на промежутке $\pi \le \beta \le \frac{3\pi}{2}$ есть только одно решение.
Ответ: $\beta = \frac{7\pi}{6}$.
б) Исследуем промежуток $\frac{3\pi}{2} \le \beta \le 2\pi$. Этот промежуток соответствует IV четверти единичной окружности.
Рассмотрим первую серию корней $\beta = \frac{7\pi}{6} + 2\pi k$.
Как мы видели в пункте а), при $k=0$, $\beta = \frac{7\pi}{6}$. Это значение меньше $\frac{3\pi}{2}$, поэтому оно не входит в заданный промежуток.
Рассмотрим вторую серию корней $\beta = -\frac{\pi}{6} + 2\pi k$.
При $k=1$, получаем $\beta = -\frac{\pi}{6} + 2\pi = \frac{11\pi}{6}$. Проверим принадлежность этого значения промежутку: $\frac{3\pi}{2} \le \frac{11\pi}{6} \le 2\pi$. Приведя к общему знаменателю 6, получим $\frac{9\pi}{6} \le \frac{11\pi}{6} \le \frac{12\pi}{6}$. Неравенство верно, следовательно, $\beta = \frac{11\pi}{6}$ является решением.
При $k=0$, $\beta = -\frac{\pi}{6}$, что меньше $\frac{3\pi}{2}$.
При $k=2$, $\beta = -\frac{\pi}{6} + 4\pi = \frac{23\pi}{6}$, что больше $2\pi$.
Таким образом, на промежутке $\frac{3\pi}{2} \le \beta \le 2\pi$ есть только одно решение.
Ответ: $\beta = \frac{11\pi}{6}$.
в) Исследуем промежуток $-\pi \le \beta \le 0$. Этот промежуток включает IV и III четверти при отрицательном направлении обхода единичной окружности.
Рассмотрим первую серию корней $\beta = \frac{7\pi}{6} + 2\pi k$.
При $k=-1$, получаем $\beta = \frac{7\pi}{6} - 2\pi = \frac{7\pi - 12\pi}{6} = -\frac{5\pi}{6}$. Проверим принадлежность этого значения промежутку: $-\pi \le -\frac{5\pi}{6} \le 0$. Неравенство верно, так как $-\frac{6\pi}{6} \le -\frac{5\pi}{6} \le 0$. Следовательно, $\beta = -\frac{5\pi}{6}$ является решением.
Рассмотрим вторую серию корней $\beta = -\frac{\pi}{6} + 2\pi k$.
При $k=0$, получаем $\beta = -\frac{\pi}{6}$. Проверим принадлежность этого значения промежутку: $-\pi \le -\frac{\pi}{6} \le 0$. Неравенство верно. Следовательно, $\beta = -\frac{\pi}{6}$ является решением.
При $k=-1$, $\beta = -\frac{\pi}{6} - 2\pi = -\frac{13\pi}{6}$, что меньше $-\pi$.
Таким образом, на промежутке $-\pi \le \beta \le 0$ есть два решения.
Ответ: $\beta = -\frac{5\pi}{6}$; $\beta = -\frac{\pi}{6}$.
Другие задания:
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