How to find general solution of #costheta=sintheta-1#?

1 Answer
Dean R.
Jun 5, 2018

We conclude #theta = pi + 2pi k or theta = pi/2 + 2pi k quad# integer #k#

Explanation:

# cos theta = sin theta - 1 #

We have to square equations, so we may introduce extraneous roots, so it's necessary to check them at the end. We should always check, but here it's part of the problem proper.

#cos^2 theta = sin ^2 theta - 2 sin theta + 1 #

#1 - sin ^2 theta = sin ^2 theta - 2 sin theta + 1 #

#2 sin^2 theta - 2 sin theta = 0#

#2 sin theta (sin theta - 1 )= 0#

#sin theta = 0 or sin theta=1#

Because we squared the sign of the cosine is ambiguous from the first clause. But only one sign will work. These are really

#sin theta=0 and cos theta=-1 quad or quad sin theta=1#

We conclude #theta = pi + 2pi k or theta = pi/2 + 2pi k quad# integer #k#

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