# What is the slope of the polar curve f(theta) = theta - sectheta+thetasin^3theta  at theta = (7pi)/12?

Mar 21, 2018

$f ' \left(\frac{7 \pi}{12}\right) = - 13.8459$

#### Explanation:

Given:
$f \left(\theta\right) = \theta - \sec \theta + \theta {\sin}^{3} \theta$
Differentiating wrt $\theta$
$f ' \left(\theta\right) = 1 - \sec \theta \tan \theta + \theta \left(3 {\sin}^{2} \theta\right) \cos \theta + {\sin}^{3} \theta$
$= 1 - \frac{1}{\cos} \theta \sin \frac{\theta}{\cos} \theta + {\sin}^{3} \theta + 3 \theta {\sin}^{2} \theta \cos \theta$
$\theta = \frac{7 \pi}{12} = 1.8326$

$\frac{7 \pi}{12} = \pi - \frac{5 \pi}{12}$
$\cos \theta = \cos \left(\pi - \frac{5 \pi}{12}\right) = - \cos \left(\frac{5 \pi}{12}\right) = - 0.2588$
$\sin \theta = \sin \left(\pi - \frac{5 \pi}{12}\right) = \sin \left(\frac{5 \pi}{12}\right) = 0.9659$
Substituting the values in $f ' \left(\theta\right)$

$f ' \left(\frac{7 \pi}{12}\right) = 1 - \frac{1}{-} 0.2588 \times \frac{0.9659}{-} 0.2588 + {0.9659}^{3} + 3 \times 1.8326 \times {0.9659}^{2} \left(- 0.2588\right)$
$f ' \left(\frac{7 \pi}{12}\right) = - 13.8459$