Question

Solve the equation cos (2 theta)=cos squared theta on the interval ( 0,2pi )

Answer 1

\[\cos(2\Theta)=\cos^2(\Theta)\]
First you want to use a trig identity to change the form of the left side. Let's use a double angle identity. \[\cos(2\Theta)=2\cos^2(\Theta)-1\] We can simply substitute this in for the left side because they are equal.\[2\cos^2(\Theta)-1=\cos^2(\Theta)\] We want everything on one side of the equation so let's subtract cosine of theta squared on both sides.\[2\cos^2(\Theta)-1-\cos^2(\Theta)=0\]\[\cos^2(\Theta)-1=0\] We need to isolate cosine, so add 1 on both sides and take the square root of both sides.\[\cos^2(\Theta)=1\]\[\cos(\Theta)=+1\]We're almost done. Now the only step is to find the angles at which the cosine equals plus or minus one. This is:\[x={0,\Pi}\]That is your answer. Does that make sense?