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
Nghi N
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#

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