# What is the slope of the polar curve f(theta) = theta^2 - sectheta  at theta = (3pi)/4?

Apr 14, 2017

The slope:

$m = \frac{9 {\pi}^{2} + 32 \sqrt{2} - 24 \pi}{24 \pi + 9 {\pi}^{2}}$

$m \approx 0.357$

#### Explanation:

From the reference Tangents with Polar Coordinates we obtain the equation

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{\mathrm{dr}}{d \theta} \sin \left(\theta\right) + r \cos \left(\theta\right)}{\frac{\mathrm{dr}}{d \theta} \cos \left(\theta\right) - r \sin \left(\theta\right)} \text{ [1]}$

We are given $r = f \left(\theta\right) = {\theta}^{2} - \sec \left(\theta\right)$; substitute this into equation [1]:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{\mathrm{dr}}{d \theta} \sin \left(\theta\right) + \left({\theta}^{2} - \sec \left(\theta\right)\right) \cos \left(\theta\right)}{\frac{\mathrm{dr}}{d \theta} \cos \left(\theta\right) - \left({\theta}^{2} - \sec \left(\theta\right)\right) \sin \left(\theta\right)} \text{ [2]}$

Compute $\frac{\mathrm{dr}}{d \theta}$:

$\frac{\mathrm{dr}}{d \theta} = 2 \theta - \tan \left(\theta\right) \sec \left(\theta\right)$

Substitute this into equation [2]:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\left(2 \theta - \tan \left(\theta\right) \sec \left(\theta\right)\right) \sin \left(\theta\right) + \left({\theta}^{2} - \sec \left(\theta\right)\right) \cos \left(\theta\right)}{\left(2 \theta - \tan \left(\theta\right) \sec \left(\theta\right)\right) \cos \left(\theta\right) - \left({\theta}^{2} - \sec \left(\theta\right)\right) \sin \left(\theta\right)} \text{ [3]}$

The slope, m, of the tangent line at $\theta = \frac{3 \pi}{4}$ is the above equation evaluated at $\theta = \frac{3 \pi}{4}$

$m = \frac{\left(2 \frac{3 \pi}{4} - \tan \left(\frac{3 \pi}{4}\right) \sec \left(\frac{3 \pi}{4}\right)\right) \sin \left(\frac{3 \pi}{4}\right) + \left({\left(\frac{3 \pi}{4}\right)}^{2} - \sec \left(\frac{3 \pi}{4}\right)\right) \cos \left(\frac{3 \pi}{4}\right)}{\left(2 \frac{3 \pi}{4} - \tan \left(\frac{3 \pi}{4}\right) \sec \left(\frac{3 \pi}{4}\right)\right) \cos \left(\frac{3 \pi}{4}\right) - \left({\left(\frac{3 \pi}{4}\right)}^{2} - \sec \left(\frac{3 \pi}{4}\right)\right) \sin \left(\frac{3 \pi}{4}\right)} \text{ [4]}$

$m = \frac{9 {\pi}^{2} + 32 \sqrt{2} - 24 \pi}{24 \pi + 9 {\pi}^{2}}$

The slope $m \approx 0.357$