Polar Coordinates — Question 9

Question 9

Problem

Find the smallest positive angle between the rays representing $(3,17\pi/12)$ and $(2,-\pi/4)$.

Question 9 – Solution

Solution

1. Use the polar–Cartesian relationships $$x=r\cos\theta,\qquad y=r\sin\theta,\qquad r^2=x^2+y^2.$$ Equivalent polar coordinates satisfy $$(r,\theta)=(r,\theta+2k\pi)=(-r,\theta+(2k+1)\pi).$$

2. Apply the identity that matches the requested conversion, symmetry test, or intersection, and then check the resulting point or curve in the original polar equation.

3. Reduce angles to $17\pi/12$ and $21\pi/12$.

4. Their smaller difference is $\boxed{\pi/3}$.