Tangents with Polar Coordinates — Question 7

Question 7

Problem

For $r=e^\theta$, show that the angle between the radius vector and tangent is constant.

Question 7 – Solution

Solution

1. Write the polar curve parametrically as $$x(\theta)=r(\theta)\cos\theta, \qquad y(\theta)=r(\theta)\sin\theta.$$ Differentiation gives $$\frac{dx}{d\theta}=r'\cos\theta-r\sin\theta, \qquad \frac{dy}{d\theta}=r'\sin\theta+r\cos\theta.$$

2. Wherever $dx/d\theta\ne0$, compute $$\frac{dy}{dx}= \frac{r'\sin\theta+r\cos\theta} {r'\cos\theta-r\sin\theta}.$$ Test numerator and denominator separately when locating horizontal or vertical tangents.

3. In polar components, tangent has radial component $r'=r$ and transverse component $r$.

4. Thus it makes angle $\arctan(r/r)=\boxed{\pi/4}$ with the radius vector.