Question

Sin^4 a(альфа) * cos^4 a(альфа) = 1/64
Найдите острый угол а(альфа)

Answer 1

$sin^4\alpha*cos^4\alpha=\frac{1}{64}, \alpha\ \textless \ 90$°
$(sin \alpha *cos \alpha )^4= \frac{1}{64},$
$sin\alpha*cos\alpha=+-\frac{1}{2\sqrt{2}},$ 
$sin\alpha*cos\alpha= \frac{ \sqrt{2} }{4},$
$\frac{1}{2}sin2\alpha= \frac{ \sqrt{2} }{4},$
$sin2\alpha=\frac{\sqrt{2}}{2},$
$2\alpha=(-1)^n* \frac{\pi}{4} +n\pi, n \in Z$
$\alpha=(-1)^n*\frac{\pi}{8}+\frac{n\pi}{2}, n \in Z;$
$\alpha=\frac{\pi}{8} = \frac{180}{8}=22,5$°.
Ответ: 22,5°.

Answer 2

Sin^4 α  · cos^4 α = 1/64
(2sin α  · cos α)^4 : 2^4 = 1/64
(2sin α  · cos α)^4 = 16/64
2sin α  · cos α = 1/√2
sin 2α = 1/√2
2α = π/4
α = π/8
Ответ: α = π/8