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

Jul 26, 2018

Slope of tangent at $\frac{\pi}{4}$ is $1.09$

Explanation:

r= theta- sin(theta/3- (2 pi)/3) ; theta = pi/4

$\frac{\mathrm{dr}}{d \theta} = 1 - \cos \left(\frac{\theta}{3} - \frac{2 \pi}{3}\right) \cdot \frac{1}{3}$

$\frac{\mathrm{dr}}{d \theta} \left(\frac{\pi}{4}\right) = 1 - \cos \left(\frac{\frac{\pi}{4}}{3} - \frac{2 \pi}{3}\right) \cdot \frac{1}{3}$ or

$\frac{\mathrm{dr}}{d \theta} \left(\frac{\pi}{4}\right) = 1 - \cos \left(\frac{\pi}{12} - \frac{2 \pi}{3}\right) \cdot \frac{1}{3}$or

$\frac{\mathrm{dr}}{d \theta} \left(\frac{\pi}{4}\right) = 1 - \cos \left(- 1.8326\right) \cdot \frac{1}{3}$or

$= \frac{\mathrm{dr}}{d \theta} \left(\frac{\pi}{4}\right) = 1 - \left(- 0.2588\right) \cdot \frac{1}{3}$

$= \frac{\mathrm{dr}}{d \theta} \left(\frac{\pi}{4}\right) \approx 1.086$

Slope of tangent at $\frac{\pi}{4}$ is $1.09$ [Ans]