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Консультация № 64806
29.11.2006, 19:49
0.00 руб.
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3
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как найти cos(18*), без использования скажем брадиса или калькулятора...распишите пожалуйста подробно...
P.s.:в решении плиз можно также вычислить и синус 18*
P.s.:в решении плиз можно также вычислить и синус 18*
Обсуждение
Неизвестный
29.11.2006, 19:59
общий
это ответ
Здравствуйте, Dushin Igor!
решение задачи приведено в справочнике А.Г.Цыпкина. Решение геометрическое.Рассматривается равнобедренный треугольник с углом при вершине 36 градусов, проводятся биссектрисы, рассматриваются подобные треурольники и из пропорции получают sin18=(√5 - 1)/4
решение задачи приведено в справочнике А.Г.Цыпкина. Решение геометрическое.Рассматривается равнобедренный треугольник с углом при вершине 36 градусов, проводятся биссектрисы, рассматриваются подобные треурольники и из пропорции получают sin18=(√5 - 1)/4
Неизвестный
29.11.2006, 20:33
общий
это ответ
Здравствуйте, Dushin Igor!
Предлагаю алгебраическое решение
x = sin18*, 0* < 18* < 90* => Область Возможных Решений 0<x<1
sin54* = cos36*
3x - 4x³ = 1 - 2x²
4x³- 2x² - 3x + 1 = 0
(x-1)(4x²+2x-1)=0
x=1 - не удовлетворяет ОВР
4x²+2x-1=0
D/4 = 1+4 = 5, x = (-1±√5)/4
Корень с минусом не уд ОВР => sin18* = (√5 - 1)/4.
cos18* = √(1-sin²x) = √(1 - (6 - 2√5)/16) = √(16 - 6 + 2√5) / 4 = ( √(10+2√5)) / 4
Удачи!
Предлагаю алгебраическое решение
x = sin18*, 0* < 18* < 90* => Область Возможных Решений 0<x<1
sin54* = cos36*
3x - 4x³ = 1 - 2x²
4x³- 2x² - 3x + 1 = 0
(x-1)(4x²+2x-1)=0
x=1 - не удовлетворяет ОВР
4x²+2x-1=0
D/4 = 1+4 = 5, x = (-1±√5)/4
Корень с минусом не уд ОВР => sin18* = (√5 - 1)/4.
cos18* = √(1-sin²x) = √(1 - (6 - 2√5)/16) = √(16 - 6 + 2√5) / 4 = ( √(10+2√5)) / 4
Удачи!
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