[Math] Solve $\frac{2}{\sin x \cos x}=1+3\tan x$

algebra-precalculustrigonometry

Solve this trigonometric equation given that $0\leq x\leq180$

$\frac{2}{\sin x \cos x}=1+3\tan x$

My attempt,

I've tried by changing to $\frac{4}{\sin 2x}=1+3\tan x$, but it gets complicated and I'm stuck. Hope someone can help me out.

Best Answer

Note that $\tan x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{1}{\sin x \cos x}$ (this is something I've filed away from seeing it in a lot of "verify this trig identity" type problems) .

So your equation can be written $2(\tan x + \cot x) = 1 + 3 \tan x$, or

$$2\left(\tan x + \frac{1}{\tan x}\right) = 1 + 3 \tan x.$$

Now that everything is in terms of tangent, you should be able to get it into the form of an equation that's quadratic in $\tan x$ (i.e., has multiples of $\tan^2 x,\, \tan x,$ and a number) and solve; the values of $\tan x$ are nice integers, although you will need inverse trig to get one of the $x$-values.

Related Question