How to find the intersection of the graphs: $r^2=sin2\theta$ and $r^2=cos2\theta$

Question

I tried to graph the two graphs on paper and realized that I was unable to find the solutions visually so I did the following:

$2cos^2\theta-1=2sin\theta*cos2\theta$ after setting each of the equations equal to each other.

user id:229023 15k · Accepted Answer

Hint,

$$\sin (2x)-\cos(2x)=0$$

$$=\langle -1,1 \rangle \cdot \langle \cos (2x), \sin (2x) \rangle$$

$$=\sqrt{2} \cos (2x-\frac{3\pi}{4})=0$$

Or if you prefer we have,

$$\frac{\sin(2x)}{\cos (2x)}=\tan (2x)=1$$

user id:369983 1k · Accepted Answer

$(r^2)^2 + (r^2)^2 = \sin^2 2\theta + \cos^2 2\theta $

$2r^4=1$

$r=\dfrac{1}{2^{\frac{1}{4}}}$

2 Answers 2