Question

Solve the equation:$5{\sin }^{2}x-7\sin x\cos x+16{\cos }^{2}x=4$

Answer 1

$5{\sin }^{2}x-7\sin x\cos x+16{\cos }^{2}x=4$

To solve this equation; we use the trigonometric identity,${\sin }^{2}x+{\cos }^{2}x=1$

$\Rightarrow 5{\sin }^{2}x-7\sin x\cos x+16{\cos }^{2}x=4({\sin }^{2}x+{\cos }^{2}x)$

$\Rightarrow 5{\sin }^{2}x-7\sin x\cos x+16{\cos }^{2}x=4{\sin }^{2}x+4{\cos }^{2}x$

$\Rightarrow {\sin }^{2}x-7\sin x\cos x+12{\cos }^{2}x=0$

Dividing by ${\cos }^{2}x$ on both sides,we get

$\Rightarrow {\tan }^{2}x-7\tan x+12=0$

$\Rightarrow {\tan }^{2}x-3\tan x-4\tan x+12=0$

$\Rightarrow \tan x(\tan x-3)-4(\tan x-3)=0$

$\Rightarrow (\tan x-3)(\tan x-4)=0$

$\Rightarrow \tan x=3,4$

$\Rightarrow x=n\pi +{\tan }^{-1}3,n\pi +{\tan }^{-1}4$