What is the slope of the tangent line of r=2theta+3cos((theta)/2-(4pi)/3) at theta=(5pi)/4?

Dec 23, 2015

Equation of tangent line

$y = - 3.248347 x - 18.107412$

Explanation:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{\mathrm{dy}}{d \theta}}{\frac{\mathrm{dx}}{d \theta}}$

$= \frac{\frac{d}{d \theta} \left(r \sin \theta\right)}{\frac{d}{d \theta} \left(r \cos \theta\right)}$

$= \frac{r \cos \theta + \frac{\mathrm{dr}}{d \theta} \sin \theta}{- r \sin \theta + \frac{\mathrm{dr}}{d \theta} \cos \theta}$

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

$= 2 - \frac{3}{2} \sin \left(\frac{\theta}{2} - \frac{4 \pi}{3}\right)$

${\frac{\mathrm{dr}}{d \theta}}_{| \theta = \frac{5 \pi}{4}} = 2 - \frac{3}{2} \sin \left(\frac{\frac{5 \pi}{4}}{2} - \frac{4 \pi}{3}\right)$

$= 2 - \frac{3}{2} \sin \left(\frac{- 17 \pi}{24}\right)$

$= 2 + \frac{3}{2} \sin \left(\frac{7 \pi}{24}\right)$

${\frac{\mathrm{dy}}{\mathrm{dx}}}_{| \theta = \frac{5 \pi}{4}} = \frac{\left(2 \left(\frac{5 \pi}{4}\right) + 3 \cos \left(\frac{\frac{5 \pi}{4}}{2} - \frac{4 \pi}{3}\right)\right) \cos \frac{5 \pi}{4} + \frac{\mathrm{dr}}{d \theta} {\setminus}_{| \theta = \frac{5 \pi}{4}} \sin \frac{5 \pi}{4}}{- \left(2 \left(\frac{5 \pi}{4}\right) + 3 \cos \left(\frac{\frac{5 \pi}{4}}{2} - \frac{4 \pi}{3}\right)\right) \sin \frac{5 \pi}{4} + \frac{\mathrm{dr}}{d \theta} {\setminus}_{| \theta = \frac{5 \pi}{4}} \cos \frac{5 \pi}{4}}$

$= \frac{\left(\frac{5 \pi}{2} + 3 \cos \left(\frac{- 17 \pi}{24}\right)\right) \left(- \frac{1}{\sqrt{2}}\right) + \frac{\mathrm{dr}}{d \theta} {\setminus}_{| \theta = \frac{5 \pi}{4}} \left(- \frac{1}{\sqrt{2}}\right)}{- \left(\frac{5 \pi}{2} + 3 \cos \left(\frac{- 17 \pi}{24}\right)\right) \left(- \frac{1}{\sqrt{2}}\right) + \frac{\mathrm{dr}}{d \theta} {\setminus}_{| \theta = \frac{5 \pi}{4}} \left(- \frac{1}{\sqrt{2}}\right)}$

$= \frac{\frac{5 \pi}{2} + 3 \cos \left(\frac{17 \pi}{24}\right) + \frac{\mathrm{dr}}{d \theta} {\setminus}_{| \theta = \frac{5 \pi}{4}}}{- \frac{5 \pi}{2} - 3 \cos \left(\frac{17 \pi}{24}\right) + \frac{\mathrm{dr}}{d \theta} {\setminus}_{| \theta = \frac{5 \pi}{4}}}$

$= \frac{\frac{5 \pi}{2} + 3 \cos \left(\frac{17 \pi}{24}\right) + \left(2 + \frac{3}{2} \sin \left(\frac{7 \pi}{24}\right)\right)}{- \frac{5 \pi}{2} - 3 \cos \left(\frac{17 \pi}{24}\right) + \left(2 + \frac{3}{2} \sin \left(\frac{7 \pi}{24}\right)\right)}$

$= \frac{4 + 5 \pi - 6 \cos \left(\frac{7 \pi}{24}\right) + 3 \sin \left(\frac{7 \pi}{24}\right)}{4 - 5 \pi + 6 \cos \left(\frac{7 \pi}{24}\right) + 3 \sin \left(\frac{7 \pi}{24}\right)}$

$\approx - 3.24835$