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

Apr 10, 2018

the slope of the tangent is $2.57$

#### Explanation:

$r = 2 \theta + 3 \cos \left(\frac{\theta}{2} - \frac{2 \pi}{3}\right)$

$\frac{\mathrm{dr}}{d \theta} = 2 - \frac{3}{2} \sin \left(\frac{\theta}{2} - \frac{2 \pi}{3}\right)$

At $\theta = \frac{3 \pi}{4}$,

$\frac{\mathrm{dr}}{d \theta} = 2 - \frac{3}{2} \sin \left(\frac{3 \pi}{8} - \frac{2 \pi}{3}\right)$

$\frac{\mathrm{dr}}{d \theta} = 2 - \frac{3}{2} \sin \left(- \frac{7 \pi}{8}\right)$

$\frac{\mathrm{dr}}{d \theta} = 2.57$

Therefore, the slope of the tangent is $2.57$