Tangents with Polar Coordinates — Question 10

Question 10

Problem

For the circle $r=4\sin\theta$, find the tangent at its highest point using polar calculus.

Question 10 – 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 highest point occurs at $\theta=\pi/2$, giving $(0,4)$.

4. The slope numerator is $0$ and denominator $-4$, so the tangent is horizontal: $\boxed{y=4}$.