Answer
The slope of the tangent line at $\theta = \pi/6 $ is equal to $\sqrt 3$
Work Step by Step
1. Write the formula for the slope of the tangent line:
$$\frac{dy}{dx} = \frac{\frac{dr}{d\theta}sin(\theta) + rcos(\theta)}{\frac{dr}{d\theta}cos(\theta) - rsin(\theta)} $$
2. Calculate $\frac{dr}{d\theta}$ and substitute the equation into the formula:
$\frac{dr}{d\theta} = \frac{d(2sin(\theta))}{d\theta} = 2cos(\theta)$
$\frac{dy}{dx} = \frac{(2cos(\theta)sin(\theta)) + (2sin(\theta))cos(\theta)}{2cos(\theta)cos(\theta) - (2sin(\theta))sin(\theta)}$
$\frac{dy}{dx} = \frac{2(2cos(\theta)sin(\theta)) }{2(cos^2(\theta) - sin^2(\theta))} = \frac{sin(2\theta)}{cos(2\theta)} = tan(2\theta)$
3. Calculate the slope at the given $\theta$ value:
$\frac{dy}{dx} = tan(2\frac {\pi} 6)= tan(\pi/3) = \sqrt 3$
