solve x algebraically not graphically
3 cot (90 + x) = tan x sin x
Thanx
$\displaystyle 3\cot (90+x) = \tan x \sin x$
$\displaystyle 3\frac{\cos (90 + x)}{\sin (90 + x)} = \frac{\sin x}{\cos x} \sin x$
$\displaystyle 3\frac{-\sin x}{\cos x} = \frac{\sin^2 x}{\cos x}$
$\displaystyle 3\frac{-\sin x}{\not{\cos x}} = \frac{\sin^2 x}{\not{\cos x}}$
$\displaystyle -3\sin x = \sin^2 x$
.........$\displaystyle \sin x = 0$ works
We'll try to get another answer,
$\displaystyle -3\not{\sin x} = \sin^{\not 2} x$
$\displaystyle -3 = \sin x$ There's no solution here.
Then we only use $\displaystyle \sin x = 0$.
Hence, $\displaystyle x = k\pi$
Hello, Nico!
A slightly different approach . . .
$\displaystyle 3\cot(90 + x) \:=\: \tan x\sin x$
Since $\displaystyle \cot(x + 90) \:=\:-\tan x$, we have:
. . $\displaystyle -3\tan x \:=\:\tan x\sin x \quad\Rightarrow\quad \tan x\sin x + 3\tan x \:=\:0$
Factor: .$\displaystyle \tan x(\sin x + 3) \:=\:0$
But .$\displaystyle \sin x + 3 \:=\:0\quad\Rightarrow\quad \sin x \:=\:-3\;\text{ has no solutions}$
Therefore: .$\displaystyle \tan x \:=\:0\quad\Rightarrow\quad x \:=\:\pi n\:\text{ for any integer }n$