# What is the slope of the tangent line of r=theta-cos(-4theta+(2pi)/3) at theta=(7pi)/4?

Nov 8, 2017

$m = 0.0564$

#### Explanation:

The reference Tangents with Polar Coordinates gives us this formula for $\frac{\mathrm{dy}}{\mathrm{dx}}$:

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

Substitute $\theta - \cos \left(- 4 \theta + \frac{2 \pi}{3}\right)$ for every instance of r in equation [1]:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{\mathrm{dr}}{d \theta} \sin \left(\theta\right) + \left(\theta - \cos \left(- 4 \theta + \frac{2 \pi}{3}\right)\right) \cos \left(\theta\right)}{\frac{\mathrm{dr}}{d \theta} \cos \left(\theta\right) - \left(\theta - \cos \left(- 4 \theta + \frac{2 \pi}{3}\right)\right) \sin \left(\theta\right)} \text{ [2]}$

Substitute $1 - 4 \cos \left(\frac{\pi}{6} - 4 \theta\right)$ for every instance of (dr)/(d theta) in equation [2]:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\left(1 - 4 \cos \left(\frac{\pi}{6} - 4 \theta\right)\right) \sin \left(\theta\right) + \left(\theta - \cos \left(- 4 \theta + \frac{2 \pi}{3}\right)\right) \cos \left(\theta\right)}{\left(1 - 4 \cos \left(\frac{\pi}{6} - 4 \theta\right)\right) \cos \left(\theta\right) - \left(\theta - \cos \left(- 4 \theta + \frac{2 \pi}{3}\right)\right) \sin \left(\theta\right)} \text{ [3]}$

The slope, m, is equation [3] evaluated at $\theta = \frac{7 \pi}{4}$:

$m = \frac{\left(1 - 4 \cos \left(\frac{\pi}{6} - 7 \pi\right)\right) \sin \left(\frac{7 \pi}{4}\right) + \left(\frac{7 \pi}{4} - \cos \left(- 7 \pi + \frac{2 \pi}{3}\right)\right) \cos \left(\frac{7 \pi}{4}\right)}{\left(1 - 4 \cos \left(\frac{\pi}{6} - 7 \pi\right)\right) \cos \left(\frac{7 \pi}{4}\right) - \left(\frac{7 \pi}{4} - \cos \left(- 7 \pi + \frac{2 \pi}{3}\right)\right) \sin \left(\frac{7 \pi}{4}\right)}$

$m = 0.0564$

Here is a graph of r $0 \le \theta < 2 \pi$, the specified point, and the tangent line with the above slope: