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Номер 584, 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 584, 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" => 1108317 "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/1-584" "field_display_title" => "584" "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 {#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" => 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 {#1063 #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 {#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: [] #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 {#1063} "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 {#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 => "*" ] } 1 => 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" => 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 {#1095 #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 {#1096 #items: array:1 [ 0 => App\Models\Book {#1097 #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 {#1098 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1109 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1120 …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 {#1121 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1124 …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 {#1098 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1102 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1104 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1108 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1109 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1111 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1113 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1118 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1119 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1120 …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 {#1121 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1122 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1124 …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 {#1125 #items: array:1 [ 0 => App\Models\Branch {#1126 #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 {#1127 …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 {#1128 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1129 …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 {#1127 …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 {#1128 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1129 …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 {#1095} "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 {#1096} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1125} ] #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 {#1130 #items: array:3 [ 0 => App\Models\Element {#1131 #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" => 1276939 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1132 #items: array:1 [ 0 => App\Models\Edition {#1133 #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 {#1134 …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 {#1136 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1138 …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 {#1134 …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 {#1136 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1138 …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" => "1108317" "type" => "task" ] "text" => "<p><strong>584.</strong> Отметьте на координатной окружности множество точек $B_{\alpha}$ и запишите все значения $\alpha$, удовлетворяющие условию:</p><p><strong>а)</strong> $\sin \alpha = -\frac{\sqrt{2}}{2}$;</p><p><strong>в)</strong> $\sin \alpha \le -\frac{1}{2}$;</p><p><strong>б)</strong> $\cos \alpha = -\frac{1}{2}$;</p><p><strong>г)</strong> $\cos \alpha > \frac{\sqrt{2}}{2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "584-1.jpg" "alt" => null "width" => "1577" "height" => 498 "path" => "/media/algebra_09/soltan-u19/0-1/584-1.webp?ts=1752834597" ] ] ] #original: array:7 [ "id" => 1276939 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1132} "task" => array:2 [ "refs" => "1108317" "type" => "task" ] "text" => "<p><strong>584.</strong> Отметьте на координатной окружности множество точек $B_{\alpha}$ и запишите все значения $\alpha$, удовлетворяющие условию:</p><p><strong>а)</strong> $\sin \alpha = -\frac{\sqrt{2}}{2}$;</p><p><strong>в)</strong> $\sin \alpha \le -\frac{1}{2}$;</p><p><strong>б)</strong> $\cos \alpha = -\frac{1}{2}$;</p><p><strong>г)</strong> $\cos \alpha > \frac{\sqrt{2}}{2}$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "584-1.jpg" "alt" => null "width" => "1577" "height" => 498 "path" => "/media/algebra_09/soltan-u19/0-1/584-1.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 {#1139 #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" => 1278230 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1140 #items: array:1 [ 0 => App\Models\Edition {#1141 #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 {#1142 …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 {#1144 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1145 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1146 …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 {#1142 …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 {#1144 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1145 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1146 …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" => "1108317" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "584-1.jpg" "alt" => null "width" => "1457" "height" => 2149 "path" => "/media/algebra_09/soltan-u19/1-1/584-1.webp?ts=1752831368" ] 1 => array:5 [ "name" => "584-2.jpg" "alt" => null "width" => "1328" "height" => 629 "path" => "/media/algebra_09/soltan-u19/1-1/584-2.webp?ts=1752831368" ] ] ] #original: array:6 [ "id" => 1278230 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1140} "task" => array:2 [ "refs" => "1108317" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "584-1.jpg" "alt" => null "width" => "1457" "height" => 2149 "path" => "/media/algebra_09/soltan-u19/1-1/584-1.webp?ts=1752831368" ] 1 => array:5 [ "name" => "584-2.jpg" "alt" => null "width" => "1328" "height" => 629 "path" => "/media/algebra_09/soltan-u19/1-1/584-2.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 {#1147 #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" => 1554185 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1148 #items: array:1 [ 0 => App\Models\Edition {#1149 #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 {#1150 …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 {#1152 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1154 …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 {#1150 …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 {#1152 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1153 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1154 …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" => "1108317" "type" => "task" ] "text" => "<p><strong>а)</strong> $ \sin \alpha = -\frac{\sqrt{2}}{2} $</p><p>На координатной (единичной) окружности значение синуса угла $\alpha$ соответствует ординате (координате y) точки $B_\alpha$. Таким образом, нам нужно найти точки на окружности, у которых ордината равна $-\frac{\sqrt{2}}{2}$.</p><p>Для этого проведем горизонтальную прямую $y = -\frac{\sqrt{2}}{2}$. Эта прямая пересечет окружность в двух точках, расположенных в III и IV координатных четвертях.</p><p>Точке в IV четверти соответствует угол $\alpha_1 = -\frac{\pi}{4}$.</p><p>Точке в III четверти соответствует угол $\alpha_2 = \pi - (-\frac{\pi}{4}) = \frac{5\pi}{4}$ или $\alpha_2 = -\frac{3\pi}{4}$.</p><p>Так как функция синуса периодична с периодом $2\pi$, все значения $\alpha$, удовлетворяющие условию, можно записать в виде двух серий решений:</p><p>$\alpha = -\frac{\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$</p><p>$\alpha = \frac{5\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$ (что эквивалентно $\alpha = -\frac{3\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$)</p><p>Эти две серии можно объединить в одну формулу: $\alpha = (-1)^{k+1}\frac{\pi}{4} + \pi k, \quad k \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $\alpha = -\frac{\pi}{4} + 2\pi k, \quad \alpha = \frac{5\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$.</p><p><strong>б)</strong> $ \cos \alpha = -\frac{1}{2} $</p><p>Значение косинуса угла $\alpha$ соответствует абсциссе (координате x) точки $B_\alpha$ на координатной окружности. Нам нужно найти точки на окружности с абсциссой, равной $-\frac{1}{2}$.</p><p>Проведем вертикальную прямую $x = -\frac{1}{2}$. Она пересечет окружность в двух точках, расположенных во II и III координатных четвертях.</p><p>Точке во II четверти соответствует угол $\alpha_1 = \frac{2\pi}{3}$.</p><p>Точке в III четверти соответствует угол $\alpha_2 = \frac{4\pi}{3}$ или $\alpha_2 = -\frac{2\pi}{3}$.</p><p>Учитывая периодичность функции косинуса, общее решение можно записать в виде:</p><p>$\alpha = \pm \arccos(-\frac{1}{2}) + 2\pi k, \quad k \in \mathbb{Z}$</p><p>$\alpha = \pm \frac{2\pi}{3} + 2\pi k, \quad k \in \mathbb{Z}$</p><p><strong>Ответ:</strong> $\alpha = \pm \frac{2\pi}{3} + 2\pi k, \quad k \in \mathbb{Z}$.</p><p><strong>в)</strong> $ \sin \alpha \le -\frac{1}{2} $</p><p>Нам нужно найти на координатной окружности все точки $B_\alpha$, ордината которых меньше или равна $-\frac{1}{2}$. Эти точки образуют дугу, расположенную ниже прямой $y = -\frac{1}{2}$, включая концы дуги.</p><p>Сначала найдем граничные точки, решив уравнение $\sin \alpha = -\frac{1}{2}$.</p><p>Решениями являются углы $\alpha_1 = -\frac{\pi}{6}$ (в IV четверти) и $\alpha_2 = \pi - (-\frac{\pi}{6}) = \frac{7\pi}{6}$ (в III четверти). Угол $\frac{7\pi}{6}$ также можно записать как $-\frac{5\pi}{6}$.</p><p>Неравенству $\sin \alpha \le -\frac{1}{2}$ удовлетворяют все углы $\alpha$, расположенные на дуге от точки $-\frac{5\pi}{6}$ до точки $-\frac{\pi}{6}$ при движении против часовой стрелки.</p><p>С учетом периодичности, получаем решение в виде двойного неравенства:</p><p>$-\frac{5\pi}{6} + 2\pi k \le \alpha \le -\frac{\pi}{6} + 2\pi k, \quad k \in \mathbb{Z}$</p><p><strong>Ответ:</strong> $\alpha \in [-\frac{5\pi}{6} + 2\pi k; -\frac{\pi}{6} + 2\pi k], \quad k \in \mathbb{Z}$.</p><p><strong>г)</strong> $ \cos \alpha > \frac{\sqrt{2}}{2} $</p><p>Нам нужно найти на координатной окружности все точки $B_\alpha$, абсцисса которых строго больше $\frac{\sqrt{2}}{2}$. Эти точки образуют дугу, расположенную правее прямой $x = \frac{\sqrt{2}}{2}$, не включая концы дуги.</p><p>Найдем граничные точки из уравнения $\cos \alpha = \frac{\sqrt{2}}{2}$.</p><p>Решениями являются углы $\alpha_1 = \frac{\pi}{4}$ (в I четверти) и $\alpha_2 = -\frac{\pi}{4}$ (в IV четверти).</p><p>Неравенству $\cos \alpha > \frac{\sqrt{2}}{2}$ удовлетворяют все углы $\alpha$, расположенные на дуге между точками $-\frac{\pi}{4}$ и $\frac{\pi}{4}$.</p><p>С учетом периодичности, решение записывается в виде строгого двойного неравенства:</p><p>$-\frac{\pi}{4} + 2\pi k < \alpha < \frac{\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$</p><p><strong>Ответ:</strong> $\alpha \in (-\frac{\pi}{4} + 2\pi k; \frac{\pi}{4} + 2\pi k), \quad k \in \mathbb{Z}$.</p>" ] #original: array:6 [ "id" => 1554185 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1148} "task" => array:2 [ "refs" => "1108317" "type" => "task" ] "text" => "<p><strong>а)</strong> $ \sin \alpha = -\frac{\sqrt{2}}{2} $</p><p>На координатной (единичной) окружности значение синуса угла $\alpha$ соответствует ординате (координате y) точки $B_\alpha$. Таким образом, нам нужно найти точки на окружности, у которых ордината равна $-\frac{\sqrt{2}}{2}$.</p><p>Для этого проведем горизонтальную прямую $y = -\frac{\sqrt{2}}{2}$. Эта прямая пересечет окружность в двух точках, расположенных в III и IV координатных четвертях.</p><p>Точке в IV четверти соответствует угол $\alpha_1 = -\frac{\pi}{4}$.</p><p>Точке в III четверти соответствует угол $\alpha_2 = \pi - (-\frac{\pi}{4}) = \frac{5\pi}{4}$ или $\alpha_2 = -\frac{3\pi}{4}$.</p><p>Так как функция синуса периодична с периодом $2\pi$, все значения $\alpha$, удовлетворяющие условию, можно записать в виде двух серий решений:</p><p>$\alpha = -\frac{\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$</p><p>$\alpha = \frac{5\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$ (что эквивалентно $\alpha = -\frac{3\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$)</p><p>Эти две серии можно объединить в одну формулу: $\alpha = (-1)^{k+1}\frac{\pi}{4} + \pi k, \quad k \in \mathbb{Z}$.</p><p><strong>Ответ:</strong> $\alpha = -\frac{\pi}{4} + 2\pi k, \quad \alpha = \frac{5\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$.</p><p><strong>б)</strong> $ \cos \alpha = -\frac{1}{2} $</p><p>Значение косинуса угла $\alpha$ соответствует абсциссе (координате x) точки $B_\alpha$ на координатной окружности. Нам нужно найти точки на окружности с абсциссой, равной $-\frac{1}{2}$.</p><p>Проведем вертикальную прямую $x = -\frac{1}{2}$. Она пересечет окружность в двух точках, расположенных во II и III координатных четвертях.</p><p>Точке во II четверти соответствует угол $\alpha_1 = \frac{2\pi}{3}$.</p><p>Точке в III четверти соответствует угол $\alpha_2 = \frac{4\pi}{3}$ или $\alpha_2 = -\frac{2\pi}{3}$.</p><p>Учитывая периодичность функции косинуса, общее решение можно записать в виде:</p><p>$\alpha = \pm \arccos(-\frac{1}{2}) + 2\pi k, \quad k \in \mathbb{Z}$</p><p>$\alpha = \pm \frac{2\pi}{3} + 2\pi k, \quad k \in \mathbb{Z}$</p><p><strong>Ответ:</strong> $\alpha = \pm \frac{2\pi}{3} + 2\pi k, \quad k \in \mathbb{Z}$.</p><p><strong>в)</strong> $ \sin \alpha \le -\frac{1}{2} $</p><p>Нам нужно найти на координатной окружности все точки $B_\alpha$, ордината которых меньше или равна $-\frac{1}{2}$. Эти точки образуют дугу, расположенную ниже прямой $y = -\frac{1}{2}$, включая концы дуги.</p><p>Сначала найдем граничные точки, решив уравнение $\sin \alpha = -\frac{1}{2}$.</p><p>Решениями являются углы $\alpha_1 = -\frac{\pi}{6}$ (в IV четверти) и $\alpha_2 = \pi - (-\frac{\pi}{6}) = \frac{7\pi}{6}$ (в III четверти). Угол $\frac{7\pi}{6}$ также можно записать как $-\frac{5\pi}{6}$.</p><p>Неравенству $\sin \alpha \le -\frac{1}{2}$ удовлетворяют все углы $\alpha$, расположенные на дуге от точки $-\frac{5\pi}{6}$ до точки $-\frac{\pi}{6}$ при движении против часовой стрелки.</p><p>С учетом периодичности, получаем решение в виде двойного неравенства:</p><p>$-\frac{5\pi}{6} + 2\pi k \le \alpha \le -\frac{\pi}{6} + 2\pi k, \quad k \in \mathbb{Z}$</p><p><strong>Ответ:</strong> $\alpha \in [-\frac{5\pi}{6} + 2\pi k; -\frac{\pi}{6} + 2\pi k], \quad k \in \mathbb{Z}$.</p><p><strong>г)</strong> $ \cos \alpha > \frac{\sqrt{2}}{2} $</p><p>Нам нужно найти на координатной окружности все точки $B_\alpha$, абсцисса которых строго больше $\frac{\sqrt{2}}{2}$. Эти точки образуют дугу, расположенную правее прямой $x = \frac{\sqrt{2}}{2}$, не включая концы дуги.</p><p>Найдем граничные точки из уравнения $\cos \alpha = \frac{\sqrt{2}}{2}$.</p><p>Решениями являются углы $\alpha_1 = \frac{\pi}{4}$ (в I четверти) и $\alpha_2 = -\frac{\pi}{4}$ (в IV четверти).</p><p>Неравенству $\cos \alpha > \frac{\sqrt{2}}{2}$ удовлетворяют все углы $\alpha$, расположенные на дуге между точками $-\frac{\pi}{4}$ и $\frac{\pi}{4}$.</p><p>С учетом периодичности, решение записывается в виде строгого двойного неравенства:</p><p>$-\frac{\pi}{4} + 2\pi k < \alpha < \frac{\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$</p><p><strong>Ответ:</strong> $\alpha \in (-\frac{\pi}{4} + 2\pi k; \frac{\pi}{4} + 2\pi k), \quad k \in \mathbb{Z}$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1108318" "type" => "task" ] "previous" => array:2 [ "refs" => "1108316" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1155 #items: array:1 [ 0 => App\Models\Book {#1156 #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 {#1157 #items: array:1 [ 0 => App\Models\Term {#1158 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1159 #items: array:1 [ 0 => App\Models\Term {#1160 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #original: array:6 [ "id" => 5458 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "9" "field_cases" => array:6 [ …6] "field_translit" => "devjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1161 #items: array:1 [ 0 => App\Models\Term {#1162 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #original: array:6 [ "id" => 7002 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Кокшетау" "field_cases" => array:6 [ …6] "field_translit" => "Kokshetau" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1163 #items: array:3 [ 0 => App\Models\Term {#1164 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #original: array:12 [ "id" => 7003 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Генадий" "field_patronymic" => "Николаевич" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1165 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7004 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Солтан" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алла" "field_patronymic" => "Евгеньевна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1166 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7005 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумадилова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Аманбала" "field_patronymic" => "Жумадиловна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1167 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1168 #items: array:1 [ 0 => App\Models\Term {#1169 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1170 #items: array:1 [ 0 => App\Models\Term {#1171 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1172 #items: array:1 [ 0 => App\Models\Term {#1173 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1174 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1175 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1176 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1177 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1178 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1179 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 9 класса общеобразовательных школ" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1291" "field_priority" => "4" "field_default_folder" => "/algebra_09/soltan-u19/" "field_isbn" => "978-601-317-424-2" "field_cover" => array:1 [ 0 => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_09/soltan-u19/covers/cover1.webp?ts=1752140452" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1180 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1181 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1182 #items: [] #escapeWhenCastingToString: false } "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1183 #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 {#1157} "field_class" => Illuminate\Database\Eloquent\Collection {#1159} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1161} "field_author" => Illuminate\Database\Eloquent\Collection {#1163} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1167} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1168} "field_country" => Illuminate\Database\Eloquent\Collection {#1170} "field_city" => Illuminate\Database\Eloquent\Collection {#1172} "field_series" => Illuminate\Database\Eloquent\Collection {#1174} "field_umk" => Illuminate\Database\Eloquent\Collection {#1175} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1176} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1177} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1178} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1179} "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 {#1180} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1181} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1182} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1183} "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 {#1198 #items: array:1 [ 0 => App\Models\BookPage {#1197 #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" => 1097911 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "169" "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 {#1199 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1230 #items: array:1 [ 0 => App\Models\Book {#1200 #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 {#1202 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1203 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1205 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1211 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1216 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1218 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1220 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1221 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1222 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1223 …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 {#1224 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1225 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1226 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1227 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "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 {#1202 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1203 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1205 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1207 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1211 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1212 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1214 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1216 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1218 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1219 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1220 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1221 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1222 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1223 …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 {#1224 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1225 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1226 …2} "field_url" => "/9-klass/algebra/kokshetau-soltan-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1227 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "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 {#1228 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1239 #items: array:1 [ 0 => App\Models\Element {#1238 #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 {#1248 …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 {#1248 …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 {#1657 #items: array:7 [ 0 => App\Models\Task {#1671 #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 {#1675 …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 {#1674 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1709 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1776 …2} "content" => Illuminate\Database\Eloquent\Collection {#1779 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1785 …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 {#1675 …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 {#1674 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1709 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1776 …2} "content" => Illuminate\Database\Eloquent\Collection {#1779 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1785 …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 {#1777 #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 {#1778 …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 {#1784 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1786 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1782 …2} "content" => Illuminate\Database\Eloquent\Collection {#1812 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1819 …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 {#1778 …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 {#1784 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1786 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1782 …2} "content" => Illuminate\Database\Eloquent\Collection {#1812 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1819 …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 {#1780 #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 …16 ] #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] } 3 => App\Models\Task {#1808 #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 {#1818 #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: 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№584 (с. 169)
Условие. №584 (с. 169)
скриншот условия
584. Отметьте на координатной окружности множество точек $B_{\alpha}$ и запишите все значения $\alpha$, удовлетворяющие условию:
а) $\sin \alpha = -\frac{\sqrt{2}}{2}$;
в) $\sin \alpha \le -\frac{1}{2}$;
б) $\cos \alpha = -\frac{1}{2}$;
г) $\cos \alpha > \frac{\sqrt{2}}{2}$.
Решение. №584 (с. 169)
Решение 2 (rus). №584 (с. 169)
а) $ \sin \alpha = -\frac{\sqrt{2}}{2} $
На координатной (единичной) окружности значение синуса угла $\alpha$ соответствует ординате (координате y) точки $B_\alpha$. Таким образом, нам нужно найти точки на окружности, у которых ордината равна $-\frac{\sqrt{2}}{2}$.
Для этого проведем горизонтальную прямую $y = -\frac{\sqrt{2}}{2}$. Эта прямая пересечет окружность в двух точках, расположенных в III и IV координатных четвертях.
Точке в IV четверти соответствует угол $\alpha_1 = -\frac{\pi}{4}$.
Точке в III четверти соответствует угол $\alpha_2 = \pi - (-\frac{\pi}{4}) = \frac{5\pi}{4}$ или $\alpha_2 = -\frac{3\pi}{4}$.
Так как функция синуса периодична с периодом $2\pi$, все значения $\alpha$, удовлетворяющие условию, можно записать в виде двух серий решений:
$\alpha = -\frac{\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$
$\alpha = \frac{5\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$ (что эквивалентно $\alpha = -\frac{3\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$)
Эти две серии можно объединить в одну формулу: $\alpha = (-1)^{k+1}\frac{\pi}{4} + \pi k, \quad k \in \mathbb{Z}$.
Ответ: $\alpha = -\frac{\pi}{4} + 2\pi k, \quad \alpha = \frac{5\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$.
б) $ \cos \alpha = -\frac{1}{2} $
Значение косинуса угла $\alpha$ соответствует абсциссе (координате x) точки $B_\alpha$ на координатной окружности. Нам нужно найти точки на окружности с абсциссой, равной $-\frac{1}{2}$.
Проведем вертикальную прямую $x = -\frac{1}{2}$. Она пересечет окружность в двух точках, расположенных во II и III координатных четвертях.
Точке во II четверти соответствует угол $\alpha_1 = \frac{2\pi}{3}$.
Точке в III четверти соответствует угол $\alpha_2 = \frac{4\pi}{3}$ или $\alpha_2 = -\frac{2\pi}{3}$.
Учитывая периодичность функции косинуса, общее решение можно записать в виде:
$\alpha = \pm \arccos(-\frac{1}{2}) + 2\pi k, \quad k \in \mathbb{Z}$
$\alpha = \pm \frac{2\pi}{3} + 2\pi k, \quad k \in \mathbb{Z}$
Ответ: $\alpha = \pm \frac{2\pi}{3} + 2\pi k, \quad k \in \mathbb{Z}$.
в) $ \sin \alpha \le -\frac{1}{2} $
Нам нужно найти на координатной окружности все точки $B_\alpha$, ордината которых меньше или равна $-\frac{1}{2}$. Эти точки образуют дугу, расположенную ниже прямой $y = -\frac{1}{2}$, включая концы дуги.
Сначала найдем граничные точки, решив уравнение $\sin \alpha = -\frac{1}{2}$.
Решениями являются углы $\alpha_1 = -\frac{\pi}{6}$ (в IV четверти) и $\alpha_2 = \pi - (-\frac{\pi}{6}) = \frac{7\pi}{6}$ (в III четверти). Угол $\frac{7\pi}{6}$ также можно записать как $-\frac{5\pi}{6}$.
Неравенству $\sin \alpha \le -\frac{1}{2}$ удовлетворяют все углы $\alpha$, расположенные на дуге от точки $-\frac{5\pi}{6}$ до точки $-\frac{\pi}{6}$ при движении против часовой стрелки.
С учетом периодичности, получаем решение в виде двойного неравенства:
$-\frac{5\pi}{6} + 2\pi k \le \alpha \le -\frac{\pi}{6} + 2\pi k, \quad k \in \mathbb{Z}$
Ответ: $\alpha \in [-\frac{5\pi}{6} + 2\pi k; -\frac{\pi}{6} + 2\pi k], \quad k \in \mathbb{Z}$.
г) $ \cos \alpha > \frac{\sqrt{2}}{2} $
Нам нужно найти на координатной окружности все точки $B_\alpha$, абсцисса которых строго больше $\frac{\sqrt{2}}{2}$. Эти точки образуют дугу, расположенную правее прямой $x = \frac{\sqrt{2}}{2}$, не включая концы дуги.
Найдем граничные точки из уравнения $\cos \alpha = \frac{\sqrt{2}}{2}$.
Решениями являются углы $\alpha_1 = \frac{\pi}{4}$ (в I четверти) и $\alpha_2 = -\frac{\pi}{4}$ (в IV четверти).
Неравенству $\cos \alpha > \frac{\sqrt{2}}{2}$ удовлетворяют все углы $\alpha$, расположенные на дуге между точками $-\frac{\pi}{4}$ и $\frac{\pi}{4}$.
С учетом периодичности, решение записывается в виде строгого двойного неравенства:
$-\frac{\pi}{4} + 2\pi k < \alpha < \frac{\pi}{4} + 2\pi k, \quad k \in \mathbb{Z}$
Ответ: $\alpha \in (-\frac{\pi}{4} + 2\pi k; \frac{\pi}{4} + 2\pi k), \quad k \in \mathbb{Z}$.
Другие задания:
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