# What is the slope of the tangent line of r=2theta-3sin((13theta)/8-(5pi)/3) at theta=(7pi)/6?

$\textcolor{b l u e}{\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\left[\frac{7 \pi}{3} - 3 \sin \left(\frac{11 \pi}{48}\right)\right] \cos \left(\frac{7 \pi}{6}\right) + \left[2 - \left(\frac{39}{8}\right) \cos \left(\frac{11 \pi}{48}\right)\right] \cdot \sin \left(\frac{7 \pi}{6}\right)}{- \left[\frac{7 \pi}{3} - 3 \sin \left(\frac{11 \pi}{48}\right)\right] \sin \left(\frac{7 \pi}{6}\right) + \left[2 - \left(\frac{39}{8}\right) \cos \left(\frac{11 \pi}{48}\right)\right] \cos \left(\frac{7 \pi}{6}\right)}}$
SLOPE $\textcolor{b l u e}{m = \frac{\mathrm{dy}}{\mathrm{dx}} = - 0.92335731861741}$

#### Explanation:

The solution:

The given
$r = 2 \theta - 3 \sin \left(\frac{13 \theta}{8} - \frac{5 \pi}{3}\right)$ at $\theta = \frac{7 \pi}{6}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{r \cos \theta + r ' \sin \theta}{- r \sin \theta + r ' \cos \theta}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\left[2 \theta - 3 \sin \left(\frac{13 \theta}{8} - \frac{5 \pi}{3}\right)\right] \cos \theta + \left[2 - 3 \left(\frac{13}{8}\right) \cos \left(\frac{13 \theta}{8} - \frac{5 \pi}{3}\right)\right] \cdot \sin \theta}{- \left[2 \theta - 3 \sin \left(\frac{13 \theta}{8} - \frac{5 \pi}{3}\right)\right] \sin \theta + \left[2 - 3 \left(\frac{13}{8}\right) \cos \left(\frac{13 \theta}{8} - \frac{5 \pi}{3}\right)\right] \cos \theta}$

Evaluating $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\theta = \frac{7 \pi}{6}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\left[2 \left(\frac{7 \pi}{6}\right) - 3 \sin \left(\frac{13 \left(\frac{7 \pi}{6}\right)}{8} - \frac{5 \pi}{3}\right)\right] \cos \left(\frac{7 \pi}{6}\right) + \left[2 - 3 \left(\frac{13}{8}\right) \cos \left(\frac{13 \left(\frac{7 \pi}{6}\right)}{8} - \frac{5 \pi}{3}\right)\right] \cdot \sin \left(\frac{7 \pi}{6}\right)}{- \left[2 \left(\frac{7 \pi}{6}\right) - 3 \sin \left(\frac{13 \left(\frac{7 \pi}{6}\right)}{8} - \frac{5 \pi}{3}\right)\right] \sin \left(\frac{7 \pi}{6}\right) + \left[2 - 3 \left(\frac{13}{8}\right) \cos \left(\frac{13 \left(\frac{7 \pi}{6}\right)}{8} - \frac{5 \pi}{3}\right)\right] \cos \left(\frac{7 \pi}{6}\right)}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\left[\frac{7 \pi}{3} - 3 \sin \left(\frac{91 \pi}{48} - \frac{5 \pi}{3}\right)\right] \cos \left(\frac{7 \pi}{6}\right) + \left[2 - \left(\frac{39}{8}\right) \cos \left(\frac{91 \pi}{48} - \frac{5 \pi}{3}\right)\right] \cdot \sin \left(\frac{7 \pi}{6}\right)}{- \left[\frac{7 \pi}{3} - 3 \sin \left(\frac{91 \pi}{48} - \frac{5 \pi}{3}\right)\right] \sin \left(\frac{7 \pi}{6}\right) + \left[2 - \left(\frac{39}{8}\right) \cos \left(\frac{91 \pi}{48} - \frac{5 \pi}{3}\right)\right] \cos \left(\frac{7 \pi}{6}\right)}$
$\textcolor{b l u e}{\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\left[\frac{7 \pi}{3} - 3 \sin \left(\frac{11 \pi}{48}\right)\right] \cos \left(\frac{7 \pi}{6}\right) + \left[2 - \left(\frac{39}{8}\right) \cos \left(\frac{11 \pi}{48}\right)\right] \cdot \sin \left(\frac{7 \pi}{6}\right)}{- \left[\frac{7 \pi}{3} - 3 \sin \left(\frac{11 \pi}{48}\right)\right] \sin \left(\frac{7 \pi}{6}\right) + \left[2 - \left(\frac{39}{8}\right) \cos \left(\frac{11 \pi}{48}\right)\right] \cos \left(\frac{7 \pi}{6}\right)}}$

$\textcolor{b l u e}{\frac{\mathrm{dy}}{\mathrm{dx}} = - 0.92335731861741}$

$x = r \cos \theta = \left(2 \theta - 3 \sin \left(\frac{13 \theta}{8} - \frac{5 \pi}{3}\right)\right) \cdot \cos \theta$
$x = \left[\frac{7 \pi}{3} - 3 \sin \left(\frac{91 \pi}{48} - \frac{5 \pi}{3}\right)\right] \cos \left(\frac{7 \pi}{6}\right)$
$x = \left[\frac{7 \pi}{3} - 3 \sin \left(\frac{11 \pi}{48}\right)\right] \cos \left(\frac{7 \pi}{6}\right)$
$x = - 4.6352670975528$

$y = r \sin \theta = \left(2 \theta - 3 \sin \left(\frac{13 \theta}{8} - \frac{5 \pi}{3}\right)\right) \cdot \sin \theta$
$y = \left[\frac{7 \pi}{3} - 3 \sin \left(\frac{91 \pi}{48} - \frac{5 \pi}{3}\right)\right] \sin \left(\frac{7 \pi}{6}\right)$
$y = \left[\frac{7 \pi}{3} - 3 \sin \left(\frac{11 \pi}{48}\right)\right] \sin \left(\frac{7 \pi}{6}\right)$
$y = - 2.6761727065385$

Using Point-Slope Form:
Equation of the tangent line is

$y - {y}_{1} = m \left(x - {x}_{1}\right)$
$y - - 2.6761727065385 = - 0.92335731861741 \left(x - - 4.6352670975528\right)$

Check the graph:

God bless....I hope the explanation is useful.