# How to find all solutions to the following equations in the interval 0 ≤ θ < 2π?

## Find all solutions to the following equations in the interval 0 ≤ θ < 2π. Give the exact value of the solution(s) where possible. If not possible, round your answer to 4 decimal places. tan θ − 2 cos θ sin θ = 0

May 16, 2018

$0 , \frac{\pi}{4} , \frac{3 \pi}{4} , \pi , \frac{5 \pi}{4} , \frac{7 \pi}{4} , 2 \pi$

#### Explanation:

tan t - 2sin t.cos t = 0
$\sin \frac{t}{\cos t} - 2 \sin t . \cos t = 0$
$\sin t - 2 \sin t . {\cos}^{2} t = 0$
Condition $\cos t \ne 0$
$\sin t \left(1 - 2 {\cos}^{2} t\right) = 0$
Use trig identity: $1 - 2 {\cos}^{2} t = - \cos 2 t$
$- \sin t . \cos 2 t = 0$
Either factor should be zero.
a. $\sin t = 0$ --> $t = k \pi$
$F \mathmr{and} \left(0 , 2 \pi\right)$ the answers are: t = 0; t = pi; and $t = 2 \pi$
b. $\cos 2 t = 0$ --> Unit circle gives 2 solutions for 2t:
$2 t = \frac{\pi}{2} + 2 k \pi$, and $2 t = \frac{3 \pi}{2} + 2 k \pi$
1. $2 t = \frac{\pi}{2} + 2 k \pi$
$t = \frac{\pi}{4} + k \pi$
For $\left(0 , 2 \pi\right)$, the answers are:
$t = \frac{\pi}{4}$, and $t = \frac{\pi}{4} + \pi = \frac{5 \pi}{4}$
2. $2 t = \frac{3 \pi}{2} + 2 k \pi$
$t = \frac{3 \pi}{4} + k \pi$
For $\left(0 , 2 \pi\right)$, the answers are:
$t = \frac{3 \pi}{4}$, and $t = \frac{3 \pi}{4} + \pi = \frac{7 \pi}{4}$