$r=a\sin \theta$ $r=a\cos \theta$ intersect at right angles

Question

In Stewart's Calculus book, he asks:

Show that the curves $r=a\sin \theta$, $r=a\cos \theta$ at right angles.

I do not understand the question.

It is clear that they meet for example when $\theta=\pi/4.$

Answer 1

You already found the intersection point. It seems thsoe curves should be interpreted in polar coordinates. The intersecting angle is usually defined as the angle between the tangents of the two curves at the intersection point.

All you have to do now is calculating the tangent line of each curve at the intersection point, and calculate the angle in between them.

Answer 2

The plot shows the case where $a=2$. We have two circles, one centre $(0,\frac{a}{2})$ radius $\frac{a}{2}$ and the other centre $(\frac{a}{2},0)$ radius $\frac{a}{2}$. They intersect at the origin and at $(\frac{a}{2},\frac{a}{2})$. It is easy to check that they are orthogonal.