Question

convert polar equation to rectangular coordinates: cos2θ = 1

Answer 1

Use the fact that\[\cos 2 \theta = \cos^2 \theta - \sin ^2 \theta=\frac{x^2}{r^2}-\frac{y^2}{r^2}=\frac{x^2-y^2}{r^2}\]But since\[x=r \cos \theta , y= r \sin \theta \rightarrow x^2+y^2=r^2\]so\[\cos(2 \theta) = \frac{x^2-y^2}{x^2+y^2}=1 \rightarrow x^2-y^2=x^2+y^2\]\[2y^2=0 \rightarrow y=0\]