How do you solve cos (theta/2) - cos theta = 1 over the interval 0 to 2pi?

1 Answer
Nghi N.
May 21, 2016

pi, (2pi)/3, (4pi)/3

Explanation:

Use the trig identity to transform cos t to cos (t/2) -->
cos t = 2cos^2 (t/2) - 1.
The equation becomes:
cos (t/2) - 2cos^2 (t/2) + 1 = 1
cos (t/2) - 2cos^2 (t/2) = 0
Put cos (t/2) into common factor -->
cos (t/2)(1 - 2cos (t/2)) = 0
Trig table and unit circle -->
a. cos (t/2) = 0 --> There are 2 solution arcs:
(t/2) = pi/2 --> t = (2pi)/2 = pi
t/2 = (3pi)/2 --> t = (6pi)/2 = 3pi = pi
b. cos (t/2) = 1/2 --> t/2 = +- pi/3 -->
t/2 = pi/3 --> t = (2pi)/3
and t/2 = - pi/3 or t/2 = (5pi)/3 (co-terminal of -(pi/3))-->
t = (10pi)/3 = ((4pi)/3 + 2pi) = (4pi)/3

Answers for (0, 2pi):
pi, (2pi)/3, (4pi)/3
Checking these answers by calculator is advised.

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