How do you solve #cos2theta=costheta#?

1 Answer
Alan P.
Jun 25, 2015

#theta epsilon {0^o, 120^o, 240^o}# (if the range is restricted to #[0,2pi)#)

Explanation:

Limiting the range of #theta epsilon [0,2pi)#

Since #cos(2theta)= 2cos^2(theta)-1#
After re-arranging #cos(2theta)=cos(theta)# into the form:
#color(white)("XXXX")##cos(2theta)-cos(theta)=0#
we can write
#color(white)("XXXX")##2cos^2(theta)-cos(theta)-1 = 0#
which factors as
#color(white)("XXXX")##(2cos(theta)+1)*(cos(theta)-1) = 0#

So the equation holds if
#color(white)("XXXX")##cos(theta)= (-1/2)#
#color(white)("XXXX")##color(white)("XXXX")##rarr theta = 120^o or 240^o#
or if
#color(white)("XXXX")##cos(theta)=1#
#color(white)("XXXX")##color(white)("XXXX")##rarr theta = 0^o#

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