Tangents with Polar Coordinates — Question 1

Question 1

Problem

Find the tangent line to $r=2\cos\theta$ at $\theta=\pi/4$.

Question 1 – 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. The point is $(1,1)$.

4. With $r'=-2\sin\theta$, the polar slope formula gives $0$, so the tangent is $\boxed{y=1}$.