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Номер 7.8, страница 54 - гдз по алгебре 10 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
ISBN: 978-601-07-1142-6
Утверждено Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Глава 3. Тригонометрические уравнения и неравенства. Параграф 7. Простейшие тригонометрические уравнения - номер 7.8, страница 54.
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"updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1071 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113562 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические уравнения и неравенства" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => 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_page_start" => "48" 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Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1054 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1056 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1058 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_under_the_edition" => 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Illuminate\Database\Eloquent\Collection {#1066 …2} "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4293 "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 {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1051 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_country" => 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null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113562 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические уравнения и неравенства" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "48" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null 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[] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "48" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1144 #items: array:1 [ 0 => App\Models\Book {#1115 #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" => 4293 "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 {#1117 #items: array:1 [ 0 => App\Models\Term {#1114 #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 {#1118 #items: array:1 [ 0 => App\Models\Term {#1119 #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" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #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 {#1120 #items: array:1 [ 0 => App\Models\Term {#1121 #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" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ …6] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ …6] "field_translit" => "mektep" ] #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 {#1122 #items: array:2 [ 0 => App\Models\Term {#1123 #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" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "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 {#1124 #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" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "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 {#1125 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:1 [ 0 => App\Models\Term {#1127 #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 {#1128 #items: array:1 [ 0 => App\Models\Term {#1129 #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 {#1130 #items: array:1 [ 0 => App\Models\Term {#1131 #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 {#1132 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1133 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1134 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1135 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1136 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1137 #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 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1138 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1140 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1141 #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" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4293 "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 {#1117} "field_class" => Illuminate\Database\Eloquent\Collection {#1118} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1120} "field_author" => Illuminate\Database\Eloquent\Collection {#1122} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1125} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1126} "field_country" => Illuminate\Database\Eloquent\Collection {#1128} "field_city" => Illuminate\Database\Eloquent\Collection {#1130} "field_series" => Illuminate\Database\Eloquent\Collection {#1132} "field_umk" => Illuminate\Database\Eloquent\Collection {#1133} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1134} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1135} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1136} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1137} "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 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1138} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1139} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1140} "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1141} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #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 {#1142 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113562 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические уравнения и неравенства" "field_branch_order" => "3" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1116} "field_page_start" => "48" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1144} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1142} ] #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 {#1143 #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" => 1113563 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Простейшие тригонометрические уравнения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148 #items: array:1 [ 0 => App\Models\Term {#1145 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "48" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1147 #items: array:1 [ 0 => App\Models\Book {#1115} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113563 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Простейшие тригонометрические уравнения" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1148} "field_page_start" => "48" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1147} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1146} ] #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 {#1087 #items: array:3 [ 0 => App\Models\Element {#1077 #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" => 1290749 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107 #items: array:1 [ 0 => App\Models\Edition {#1078 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1076 …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 {#1072 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1076 …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 {#1072 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1070 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1103 …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" => "1113738" "type" => "task" ] "text" => "<p><strong>7.8.a)</strong> $ \sin\left(\frac{x}{2} - \frac{\pi}{6}\right) - 1 = 0; $</p><p><strong>б)</strong> $ \cos\left(\frac{x}{3} + \frac{\pi}{4}\right) - 1 = 0; $</p><p><strong>В)</strong> $ \operatorname{tg}\left(\pi + \frac{x}{3}\right) - 1 = 0; $</p><p><strong>Г)</strong> $ \operatorname{ctg}\left(\frac{\pi}{3} - 4x\right) = \sqrt{3}. $</p>" "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "2248" "height" => 462 "path" => "/media/algebra_10/Abylkasymova-u/0-07/8-1.webp?ts=1753263030" ] ] ] #original: array:7 [ "id" => 1290749 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1107} "task" => array:2 [ "refs" => "1113738" "type" => "task" ] "text" => "<p><strong>7.8.a)</strong> $ \sin\left(\frac{x}{2} - \frac{\pi}{6}\right) - 1 = 0; $</p><p><strong>б)</strong> $ \cos\left(\frac{x}{3} + \frac{\pi}{4}\right) - 1 = 0; $</p><p><strong>В)</strong> $ \operatorname{tg}\left(\pi + \frac{x}{3}\right) - 1 = 0; $</p><p><strong>Г)</strong> $ \operatorname{ctg}\left(\frac{\pi}{3} - 4x\right) = \sqrt{3}. $</p>" "img" => array:1 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "2248" "height" => 462 "path" => "/media/algebra_10/Abylkasymova-u/0-07/8-1.webp?ts=1753263030" ] ] ] #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 {#1102 #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" => 1291263 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1153 #items: array:1 [ 0 => App\Models\Edition {#1111 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1108 …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 {#1106 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1150 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1108 …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 {#1106 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1150 …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" => "1113738" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1613" "height" => 2962 "path" => "/media/algebra_10/Abylkasymova-u/1-07/8-1.webp?ts=1753263209" ] 1 => array:5 [ "name" => "8-2.jpg" "alt" => null "width" => "1143" "height" => 551 "path" => "/media/algebra_10/Abylkasymova-u/1-07/8-2.webp?ts=1753263209" ] ] ] #original: array:6 [ "id" => 1291263 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1153} "task" => array:2 [ "refs" => "1113738" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "8-1.jpg" "alt" => null "width" => "1613" "height" => 2962 "path" => "/media/algebra_10/Abylkasymova-u/1-07/8-1.webp?ts=1753263209" ] 1 => array:5 [ "name" => "8-2.jpg" "alt" => null "width" => "1143" "height" => 551 "path" => "/media/algebra_10/Abylkasymova-u/1-07/8-2.webp?ts=1753263209" ] ] ] #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 {#1151 #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" => 1549392 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1161 #items: array:1 [ 0 => App\Models\Edition {#1152 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1155 …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 {#1156 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1157 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1158 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1155 …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 {#1156 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1157 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1158 …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" => "1113738" "type" => "task" ] "text" => "<p><strong>а)</strong> $ \sin(\frac{x}{2} - \frac{\pi}{6}) - 1 = 0 $</p><p>Перенесем 1 в правую часть уравнения, чтобы получить стандартное тригонометрическое уравнение:</p><p>$ \sin(\frac{x}{2} - \frac{\pi}{6}) = 1 $</p><p>Это частный случай решения тригонометрического уравнения. Равенство $ \sin(t) = 1 $ истинно, когда аргумент синуса $ t $ равен $ \frac{\pi}{2} + 2\pi k $, где $ k $ — любое целое число ($ k \in \mathbb{Z} $).</p><p>Приравняем аргумент нашего синуса к этому выражению:</p><p>$ \frac{x}{2} - \frac{\pi}{6} = \frac{\pi}{2} + 2\pi k $</p><p>Теперь выразим $ x $. Сначала перенесем $ \frac{\pi}{6} $ в правую часть:</p><p>$ \frac{x}{2} = \frac{\pi}{2} + \frac{\pi}{6} + 2\pi k $</p><p>Приведем дроби к общему знаменателю 6:</p><p>$ \frac{x}{2} = \frac{3\pi}{6} + \frac{\pi}{6} + 2\pi k $</p><p>$ \frac{x}{2} = \frac{4\pi}{6} + 2\pi k $</p><p>Сократим дробь:</p><p>$ \frac{x}{2} = \frac{2\pi}{3} + 2\pi k $</p><p>Умножим обе части уравнения на 2, чтобы найти $ x $:</p><p>$ x = 2 \cdot (\frac{2\pi}{3} + 2\pi k) = \frac{4\pi}{3} + 4\pi k $</p><p><strong>Ответ:</strong> $ x = \frac{4\pi}{3} + 4\pi k, k \in \mathbb{Z} $.</p><p><strong>б)</strong> $ \cos(\frac{x}{3} + \frac{\pi}{4}) - 1 = 0 $</p><p>Перенесем 1 в правую часть:</p><p>$ \cos(\frac{x}{3} + \frac{\pi}{4}) = 1 $</p><p>Это частный случай. Равенство $ \cos(t) = 1 $ истинно, когда аргумент косинуса $ t $ равен $ 2\pi k $, где $ k \in \mathbb{Z} $.</p><p>Приравниваем аргумент нашего косинуса к этому выражению:</p><p>$ \frac{x}{3} + \frac{\pi}{4} = 2\pi k $</p><p>Выразим $ x $. Сначала изолируем слагаемое с $ x $:</p><p>$ \frac{x}{3} = -\frac{\pi}{4} + 2\pi k $</p><p>Умножим обе части уравнения на 3:</p><p>$ x = 3 \cdot (-\frac{\pi}{4} + 2\pi k) = -\frac{3\pi}{4} + 6\pi k $</p><p><strong>Ответ:</strong> $ x = -\frac{3\pi}{4} + 6\pi k, k \in \mathbb{Z} $.</p><p><strong>в)</strong> $ \text{tg}(\pi + \frac{x}{3}) - 1 = 0 $</p><p>Перенесем 1 в правую часть:</p><p>$ \text{tg}(\pi + \frac{x}{3}) = 1 $</p><p>Используем свойство периодичности тангенса: $ \text{tg}(\alpha + \pi n) = \text{tg}(\alpha) $ для любого целого $ n $. В нашем случае $ n=1 $.</p><p>Уравнение упрощается до:</p><p>$ \text{tg}(\frac{x}{3}) = 1 $</p><p>Общее решение уравнения $ \text{tg}(t) = 1 $ имеет вид $ t = \text{arctg}(1) + \pi k $, где $ k \in \mathbb{Z} $.</p><p>Так как $ \text{arctg}(1) = \frac{\pi}{4} $, получаем:</p><p>$ \frac{x}{3} = \frac{\pi}{4} + \pi k $</p><p>Умножим обе части на 3, чтобы найти $ x $:</p><p>$ x = 3 \cdot (\frac{\pi}{4} + \pi k) = \frac{3\pi}{4} + 3\pi k $</p><p><strong>Ответ:</strong> $ x = \frac{3\pi}{4} + 3\pi k, k \in \mathbb{Z} $.</p><p><strong>г)</strong> $ \text{ctg}(\frac{\pi}{3} - 4x) = \sqrt{3} $</p><p>Это уравнение вида $ \text{ctg}(t) = a $. Его общее решение: $ t = \text{arcctg}(a) + \pi k $, где $ k \in \mathbb{Z} $.</p><p>В нашем случае $ t = \frac{\pi}{3} - 4x $ и $ a = \sqrt{3} $.</p><p>Значение арккотангенса: $ \text{arcctg}(\sqrt{3}) = \frac{\pi}{6} $.</p><p>Подставляем это значение в формулу решения:</p><p>$ \frac{\pi}{3} - 4x = \frac{\pi}{6} + \pi k $</p><p>Теперь выразим $ x $. Сначала изолируем $ -4x $:</p><p>$ -4x = \frac{\pi}{6} - \frac{\pi}{3} + \pi k $</p><p>Приведем дроби в правой части к общему знаменателю 6:</p><p>$ -4x = \frac{\pi}{6} - \frac{2\pi}{6} + \pi k $</p><p>$ -4x = -\frac{\pi}{6} + \pi k $</p><p>Умножим обе части на -1, чтобы изменить знак перед $ 4x $:</p><p>$ 4x = \frac{\pi}{6} - \pi k $</p><p>Разделим обе части на 4:</p><p>$ x = \frac{\pi}{24} - \frac{\pi k}{4} $</p><p>Поскольку $k$ может быть любым целым числом (положительным, отрицательным или нулем), то выражение $ -k $ также пробегает все целые числа. Поэтому ответ можно записать и в виде $ x = \frac{\pi}{24} + \frac{\pi k}{4} $, что является более стандартной формой. Оба варианта математически эквивалентны.</p><p><strong>Ответ:</strong> $ x = \frac{\pi}{24} - \frac{\pi k}{4}, k \in \mathbb{Z} $ (или $ x = \frac{\pi}{24} + \frac{\pi k}{4}, k \in \mathbb{Z} $).</p>" ] #original: array:6 [ "id" => 1549392 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1161} "task" => array:2 [ "refs" => "1113738" "type" => "task" ] "text" => "<p><strong>а)</strong> $ \sin(\frac{x}{2} - \frac{\pi}{6}) - 1 = 0 $</p><p>Перенесем 1 в правую часть уравнения, чтобы получить стандартное тригонометрическое уравнение:</p><p>$ \sin(\frac{x}{2} - \frac{\pi}{6}) = 1 $</p><p>Это частный случай решения тригонометрического уравнения. Равенство $ \sin(t) = 1 $ истинно, когда аргумент синуса $ t $ равен $ \frac{\pi}{2} + 2\pi k $, где $ k $ — любое целое число ($ k \in \mathbb{Z} $).</p><p>Приравняем аргумент нашего синуса к этому выражению:</p><p>$ \frac{x}{2} - \frac{\pi}{6} = \frac{\pi}{2} + 2\pi k $</p><p>Теперь выразим $ x $. Сначала перенесем $ \frac{\pi}{6} $ в правую часть:</p><p>$ \frac{x}{2} = \frac{\pi}{2} + \frac{\pi}{6} + 2\pi k $</p><p>Приведем дроби к общему знаменателю 6:</p><p>$ \frac{x}{2} = \frac{3\pi}{6} + \frac{\pi}{6} + 2\pi k $</p><p>$ \frac{x}{2} = \frac{4\pi}{6} + 2\pi k $</p><p>Сократим дробь:</p><p>$ \frac{x}{2} = \frac{2\pi}{3} + 2\pi k $</p><p>Умножим обе части уравнения на 2, чтобы найти $ x $:</p><p>$ x = 2 \cdot (\frac{2\pi}{3} + 2\pi k) = \frac{4\pi}{3} + 4\pi k $</p><p><strong>Ответ:</strong> $ x = \frac{4\pi}{3} + 4\pi k, k \in \mathbb{Z} $.</p><p><strong>б)</strong> $ \cos(\frac{x}{3} + \frac{\pi}{4}) - 1 = 0 $</p><p>Перенесем 1 в правую часть:</p><p>$ \cos(\frac{x}{3} + \frac{\pi}{4}) = 1 $</p><p>Это частный случай. Равенство $ \cos(t) = 1 $ истинно, когда аргумент косинуса $ t $ равен $ 2\pi k $, где $ k \in \mathbb{Z} $.</p><p>Приравниваем аргумент нашего косинуса к этому выражению:</p><p>$ \frac{x}{3} + \frac{\pi}{4} = 2\pi k $</p><p>Выразим $ x $. Сначала изолируем слагаемое с $ x $:</p><p>$ \frac{x}{3} = -\frac{\pi}{4} + 2\pi k $</p><p>Умножим обе части уравнения на 3:</p><p>$ x = 3 \cdot (-\frac{\pi}{4} + 2\pi k) = -\frac{3\pi}{4} + 6\pi k $</p><p><strong>Ответ:</strong> $ x = -\frac{3\pi}{4} + 6\pi k, k \in \mathbb{Z} $.</p><p><strong>в)</strong> $ \text{tg}(\pi + \frac{x}{3}) - 1 = 0 $</p><p>Перенесем 1 в правую часть:</p><p>$ \text{tg}(\pi + \frac{x}{3}) = 1 $</p><p>Используем свойство периодичности тангенса: $ \text{tg}(\alpha + \pi n) = \text{tg}(\alpha) $ для любого целого $ n $. В нашем случае $ n=1 $.</p><p>Уравнение упрощается до:</p><p>$ \text{tg}(\frac{x}{3}) = 1 $</p><p>Общее решение уравнения $ \text{tg}(t) = 1 $ имеет вид $ t = \text{arctg}(1) + \pi k $, где $ k \in \mathbb{Z} $.</p><p>Так как $ \text{arctg}(1) = \frac{\pi}{4} $, получаем:</p><p>$ \frac{x}{3} = \frac{\pi}{4} + \pi k $</p><p>Умножим обе части на 3, чтобы найти $ x $:</p><p>$ x = 3 \cdot (\frac{\pi}{4} + \pi k) = \frac{3\pi}{4} + 3\pi k $</p><p><strong>Ответ:</strong> $ x = \frac{3\pi}{4} + 3\pi k, k \in \mathbb{Z} $.</p><p><strong>г)</strong> $ \text{ctg}(\frac{\pi}{3} - 4x) = \sqrt{3} $</p><p>Это уравнение вида $ \text{ctg}(t) = a $. Его общее решение: $ t = \text{arcctg}(a) + \pi k $, где $ k \in \mathbb{Z} $.</p><p>В нашем случае $ t = \frac{\pi}{3} - 4x $ и $ a = \sqrt{3} $.</p><p>Значение арккотангенса: $ \text{arcctg}(\sqrt{3}) = \frac{\pi}{6} $.</p><p>Подставляем это значение в формулу решения:</p><p>$ \frac{\pi}{3} - 4x = \frac{\pi}{6} + \pi k $</p><p>Теперь выразим $ x $. Сначала изолируем $ -4x $:</p><p>$ -4x = \frac{\pi}{6} - \frac{\pi}{3} + \pi k $</p><p>Приведем дроби в правой части к общему знаменателю 6:</p><p>$ -4x = \frac{\pi}{6} - \frac{2\pi}{6} + \pi k $</p><p>$ -4x = -\frac{\pi}{6} + \pi k $</p><p>Умножим обе части на -1, чтобы изменить знак перед $ 4x $:</p><p>$ 4x = \frac{\pi}{6} - \pi k $</p><p>Разделим обе части на 4:</p><p>$ x = \frac{\pi}{24} - \frac{\pi k}{4} $</p><p>Поскольку $k$ может быть любым целым числом (положительным, отрицательным или нулем), то выражение $ -k $ также пробегает все целые числа. Поэтому ответ можно записать и в виде $ x = \frac{\pi}{24} + \frac{\pi k}{4} $, что является более стандартной формой. Оба варианта математически эквивалентны.</p><p><strong>Ответ:</strong> $ x = \frac{\pi}{24} - \frac{\pi k}{4}, k \in \mathbb{Z} $ (или $ x = \frac{\pi}{24} + \frac{\pi k}{4}, 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" => "1113739" "type" => "task" ] "previous" => array:2 [ "refs" => "1113737" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Book {#1115} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1361 #items: array:1 [ 0 => App\Models\BookPage {#333 #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" => 1098391 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "54" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/page-54" "field_display_title" => "54" "field_folder" => "1" "field_image_name" => "54" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1023 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1026 #items: array:1 [ 0 => App\Models\Book {#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" => 4293 "created_at" => "2026-04-10 13:58:26" "updated_at" => null 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Illuminate\Database\Eloquent\Collection {#1159 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1160 …2} "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1162 …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" => 4293 "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 {#1096 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1089 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1069 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1098 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1100 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1074 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1090 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1083 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1079 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1081 …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 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "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 {#1086 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1159 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1160 …2} "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1162 …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 {#1163 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1164 #items: array:1 [ 0 => App\Models\Element {#1165 #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" => 1261816 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1166 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1261816 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1166 …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" => "1098392" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098390" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1173 #items: array:10 [ 0 => App\Models\Task {#1174 #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" => 1113731 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "54" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/07-1" "field_display_title" => "7.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1175 …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 {#1177 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1178 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1184 …2} "content" => Illuminate\Database\Eloquent\Collection {#1222 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1241 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113731 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "54" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/07-1" "field_display_title" => "7.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1175 …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 {#1177 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1178 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1184 …2} "content" => Illuminate\Database\Eloquent\Collection {#1222 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1241 …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 {#1242 #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" => 1113732 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "54" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/07-2" "field_display_title" => "7.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1243 …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 {#1244 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1245 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1246 …2} "content" => Illuminate\Database\Eloquent\Collection {#1247 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1254 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113732 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "54" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/07-2" "field_display_title" => "7.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1243 …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 {#1244 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1245 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1246 …2} "content" => Illuminate\Database\Eloquent\Collection {#1247 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1254 …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 {#1255 #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" => 1113733 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "54" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/07-3" "field_display_title" => "7.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1256 …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 {#1257 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1258 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1259 …2} "content" => Illuminate\Database\Eloquent\Collection {#1260 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1267 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113733 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "54" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/07-3" "field_display_title" => "7.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1256 …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 {#1257 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1258 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1259 …2} "content" => Illuminate\Database\Eloquent\Collection {#1260 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1267 …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 {#1268 #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" => 1113734 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "54" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/07-4" "field_display_title" => "7.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1269 …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 {#1270 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1271 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1272 …2} "content" => Illuminate\Database\Eloquent\Collection {#1273 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1280 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113734 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "54" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/07-4" "field_display_title" => "7.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1269 …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 {#1270 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1271 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1272 …2} "content" => Illuminate\Database\Eloquent\Collection {#1273 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1280 …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 {#1281 #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" => 1113735 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "54" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/07-5" "field_display_title" => "7.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1282 …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 {#1283 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1284 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1285 …2} "content" => Illuminate\Database\Eloquent\Collection {#1286 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1293 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113735 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "54" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/07-5" "field_display_title" => "7.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1282 …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 {#1283 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1284 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1285 …2} "content" => Illuminate\Database\Eloquent\Collection {#1286 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1293 …2} "page" => array:2 [ …2] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1294 #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" => 1113736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "54" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/07-6" "field_display_title" => "7.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1295 …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 {#1296 …2} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1297 …2} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1298 …2} "content" => Illuminate\Database\Eloquent\Collection {#1299 …2} "next" => array:2 [ …2] "previous" => array:2 [ …2] "book" => Illuminate\Database\Eloquent\Collection {#1306 …2} "page" => array:2 [ …2] ] #original: array:24 [ "id" => 1113736 "created_at" => "2026-04-10 13:58:26" "updated_at" => null …21 ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1307 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …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] } 7 => App\Models\Task {#1320 #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] } 8 => App\Models\Task {#1333 #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] } 9 => App\Models\Task {#1346 #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: [] 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№7.8 (с. 54)
Условие. №7.8 (с. 54)
скриншот условия
7.8.a) $ \sin\left(\frac{x}{2} - \frac{\pi}{6}\right) - 1 = 0; $
б) $ \cos\left(\frac{x}{3} + \frac{\pi}{4}\right) - 1 = 0; $
В) $ \operatorname{tg}\left(\pi + \frac{x}{3}\right) - 1 = 0; $
Г) $ \operatorname{ctg}\left(\frac{\pi}{3} - 4x\right) = \sqrt{3}. $
Решение. №7.8 (с. 54)
Решение 2. №7.8 (с. 54)
а) $ \sin(\frac{x}{2} - \frac{\pi}{6}) - 1 = 0 $
Перенесем 1 в правую часть уравнения, чтобы получить стандартное тригонометрическое уравнение:
$ \sin(\frac{x}{2} - \frac{\pi}{6}) = 1 $
Это частный случай решения тригонометрического уравнения. Равенство $ \sin(t) = 1 $ истинно, когда аргумент синуса $ t $ равен $ \frac{\pi}{2} + 2\pi k $, где $ k $ — любое целое число ($ k \in \mathbb{Z} $).
Приравняем аргумент нашего синуса к этому выражению:
$ \frac{x}{2} - \frac{\pi}{6} = \frac{\pi}{2} + 2\pi k $
Теперь выразим $ x $. Сначала перенесем $ \frac{\pi}{6} $ в правую часть:
$ \frac{x}{2} = \frac{\pi}{2} + \frac{\pi}{6} + 2\pi k $
Приведем дроби к общему знаменателю 6:
$ \frac{x}{2} = \frac{3\pi}{6} + \frac{\pi}{6} + 2\pi k $
$ \frac{x}{2} = \frac{4\pi}{6} + 2\pi k $
Сократим дробь:
$ \frac{x}{2} = \frac{2\pi}{3} + 2\pi k $
Умножим обе части уравнения на 2, чтобы найти $ x $:
$ x = 2 \cdot (\frac{2\pi}{3} + 2\pi k) = \frac{4\pi}{3} + 4\pi k $
Ответ: $ x = \frac{4\pi}{3} + 4\pi k, k \in \mathbb{Z} $.
б) $ \cos(\frac{x}{3} + \frac{\pi}{4}) - 1 = 0 $
Перенесем 1 в правую часть:
$ \cos(\frac{x}{3} + \frac{\pi}{4}) = 1 $
Это частный случай. Равенство $ \cos(t) = 1 $ истинно, когда аргумент косинуса $ t $ равен $ 2\pi k $, где $ k \in \mathbb{Z} $.
Приравниваем аргумент нашего косинуса к этому выражению:
$ \frac{x}{3} + \frac{\pi}{4} = 2\pi k $
Выразим $ x $. Сначала изолируем слагаемое с $ x $:
$ \frac{x}{3} = -\frac{\pi}{4} + 2\pi k $
Умножим обе части уравнения на 3:
$ x = 3 \cdot (-\frac{\pi}{4} + 2\pi k) = -\frac{3\pi}{4} + 6\pi k $
Ответ: $ x = -\frac{3\pi}{4} + 6\pi k, k \in \mathbb{Z} $.
в) $ \text{tg}(\pi + \frac{x}{3}) - 1 = 0 $
Перенесем 1 в правую часть:
$ \text{tg}(\pi + \frac{x}{3}) = 1 $
Используем свойство периодичности тангенса: $ \text{tg}(\alpha + \pi n) = \text{tg}(\alpha) $ для любого целого $ n $. В нашем случае $ n=1 $.
Уравнение упрощается до:
$ \text{tg}(\frac{x}{3}) = 1 $
Общее решение уравнения $ \text{tg}(t) = 1 $ имеет вид $ t = \text{arctg}(1) + \pi k $, где $ k \in \mathbb{Z} $.
Так как $ \text{arctg}(1) = \frac{\pi}{4} $, получаем:
$ \frac{x}{3} = \frac{\pi}{4} + \pi k $
Умножим обе части на 3, чтобы найти $ x $:
$ x = 3 \cdot (\frac{\pi}{4} + \pi k) = \frac{3\pi}{4} + 3\pi k $
Ответ: $ x = \frac{3\pi}{4} + 3\pi k, k \in \mathbb{Z} $.
г) $ \text{ctg}(\frac{\pi}{3} - 4x) = \sqrt{3} $
Это уравнение вида $ \text{ctg}(t) = a $. Его общее решение: $ t = \text{arcctg}(a) + \pi k $, где $ k \in \mathbb{Z} $.
В нашем случае $ t = \frac{\pi}{3} - 4x $ и $ a = \sqrt{3} $.
Значение арккотангенса: $ \text{arcctg}(\sqrt{3}) = \frac{\pi}{6} $.
Подставляем это значение в формулу решения:
$ \frac{\pi}{3} - 4x = \frac{\pi}{6} + \pi k $
Теперь выразим $ x $. Сначала изолируем $ -4x $:
$ -4x = \frac{\pi}{6} - \frac{\pi}{3} + \pi k $
Приведем дроби в правой части к общему знаменателю 6:
$ -4x = \frac{\pi}{6} - \frac{2\pi}{6} + \pi k $
$ -4x = -\frac{\pi}{6} + \pi k $
Умножим обе части на -1, чтобы изменить знак перед $ 4x $:
$ 4x = \frac{\pi}{6} - \pi k $
Разделим обе части на 4:
$ x = \frac{\pi}{24} - \frac{\pi k}{4} $
Поскольку $k$ может быть любым целым числом (положительным, отрицательным или нулем), то выражение $ -k $ также пробегает все целые числа. Поэтому ответ можно записать и в виде $ x = \frac{\pi}{24} + \frac{\pi k}{4} $, что является более стандартной формой. Оба варианта математически эквивалентны.
Ответ: $ x = \frac{\pi}{24} - \frac{\pi k}{4}, k \in \mathbb{Z} $ (или $ x = \frac{\pi}{24} + \frac{\pi k}{4}, k \in \mathbb{Z} $).
Другие задания:
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