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

1 Answer
May 16, 2018

0, pi/4, (3pi)/4, pi, (5pi)/4, (7pi)/4, 2pi

Explanation:

tan t - 2sin t.cos t = 0
sin t/(cos t) - 2sin t.cos t = 0
sin t - 2sin t.cos^2 t = 0
Condition cos t != 0
sin t( 1 - 2cos^2 t) = 0
Use trig identity: 1 - 2cos^2 t = - cos 2t
- sin t.cos 2t = 0
Either factor should be zero.
a. sin t = 0 --> t = kpi
For (0, 2pi) the answers are: t = 0; t = pi; and t = 2pi
b. cos 2t = 0 --> Unit circle gives 2 solutions for 2t:
2t = pi/2 + 2kpi, and 2t = (3pi)/2 + 2kpi
1. 2t = pi/2 + 2kpi
t = pi/4 + kpi
For (0, 2pi), the answers are:
t = pi/4, and t = pi/4 + pi = (5pi)/4
2. 2t = (3pi)/2 + 2kpi
t = (3pi)/4 + kpi
For (0, 2pi), the answers are:
t = (3pi)/4, and t = (3pi)/4 + pi = (7pi)/4