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

Jun 27, 2018

$\frac{\left(1 + 2 \pi\right) \sqrt{2}}{2}$

#### Explanation:

$r = \theta \cos \left(\frac{7 \pi}{4} - 3 \theta\right)$

color(blue)(d/(d(theta)))r=color(blue)(d/(d(theta)))(thetacos((7pi)/4-3theta)

$\frac{\mathrm{dr}}{d \left(\theta\right)} = \cos \left(\frac{7 \pi}{4} - 3 \theta\right) + 3 \theta \sin \left(\frac{7 \pi}{4} - 3 \theta\right)$

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

$\cos \left(\frac{7 \pi}{4} + 2 \pi\right) - 2 \pi \sin \left(\frac{7 \pi}{4} + 2 \pi\right)$

$\cos \left(\frac{7 \pi}{4}\right) - 2 \pi \sin \left(\frac{7 \pi}{4}\right)$

$\frac{\sqrt{2}}{2} - 2 \pi \cdot - \frac{\sqrt{2}}{2}$

$\frac{\left(1 + 2 \pi\right) \sqrt{2}}{2}$