Tangents with Polar Coordinates — Question 5

Question 5

Problem

Find the tangent directions at the pole for the rose $r=\sin3\theta$.

Question 5 – 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 pole occurs when $3\theta=k\pi$, so the tangent rays satisfy $\theta=k\pi/3$.

4. The distinct unoriented lines are $$\boxed{\theta=0,\quad\theta=\pi/3,\quad\theta=2\pi/3}.$$