Giải phương trình

Question

Giải các phương trình :
a) $\sin ^{2}2x+\cos ^{2}3x=1$
b) $\sin ^{6}x+\cos ^{6}x=\frac{1}{4}$

score 5 · Answer 1

b) Ta có:
$\sin^6x+\cos^6x=\frac{1}{4}$
$\Leftrightarrow (\sin^2x+\cos^2x)^3-3\sin^2x\cos^2x(\sin^2x+\cos^2x)=\frac{1}{4}$
$\Leftrightarrow \frac{3}{4}\sin^22x=\frac{3}{4}$
$\Leftrightarrow \sin^22x=1$
$\Leftrightarrow \cos4x=-1$
$\Leftrightarrow 4x=\pi+2k\pi, k\in\mathbb{Z}$
$\Leftrightarrow x=\frac{\pi}{4}+\frac{k}{2}\pi, k\in\mathbb{Z}$

score 4 · Answer 2

Bình chọn tăng 4 Bình chọn giảm

a) Ta có:
$\sin^2 2x+\cos^23x=1$
$\Leftrightarrow \cos^23x=\cos^22x$
$\Leftrightarrow \cos6x=\cos4x$
$\Leftrightarrow \left [ \begin{array}{l} 6x=4x+2k \pi \\ 6x=-4x+2k \pi \end{array} \right. (k\in\mathbb{Z})$
$\Leftrightarrow \left [ \begin{array}{l} x=k\pi\\ x=\frac{k}{5}\pi \end{array} \right. (k\in\mathbb{Z})$