Question

Find all solutions to 4tan(3x)=-4 between 0° and 360°​

Answer 1

Divide both sides by 4 to get

$\tan(3x)=-1$

Recalling the definition of tangent as ratio between sine and cosine, we have

$\tan(3x)=-1 \iff \dfrac{\sin(3x)}{\cos(3x)}=-1 \iff \sin(3x)=-\cos(3x)$

The sine and cosine of an angle are opposite only if the angle is

$\alpha = 135+180k,\quad k \in \mathbb{Z}$

So, we have

$3x=135+180k \iff x = 45+60k,\quad k \in \mathbb{Z}$

So, the solutions are

$\alpha \in \{45,\ 105,\ 165,\ 225,\ 285,\ 345\}$