# What is the equation of the tangent line of r=cos(2theta-pi/4)/sintheta - sin(theta-pi/8) at theta=(-3pi)/8?

##### 1 Answer
Nov 10, 2016

The equation is:

$y = 0.183 \left(x - 0.797\right) - 1.924$

#### Explanation:

Here is a reference for the slope of a tangent with polar coordinates
The follow is the equation for $\frac{\mathrm{dy}}{\mathrm{dx}}$ taken from the reference:

dy/dx = (r'(theta)sin(theta) + r(theta)cos(theta))/(r'(theta)cos(theta) - r(theta)sin(theta)

The slope, m, of the tangent line is the above evaluted at $\theta = \frac{- 3 \pi}{8}$

m = (r'((-3pi)/8)sin((-3pi)/8) + r((-3pi)/8)cos((-3pi)/8))/(r'((-3pi)/8)cos((-3pi)/8) - r((-3pi)/8)sin((-3pi)/8)

Compute $r ' \left(\theta\right)$ and then evaluate it at $\theta = \frac{- 3 \pi}{8}$

$r ' \left(\setminus \theta\right) = \frac{d \left[\cos \frac{2 \setminus \theta - \setminus \frac{\pi}{4}}{\sin} \left(\setminus \theta\right) - \sin \left(\setminus \theta - \frac{\pi}{8}\right)\right]}{d \setminus \theta}$

Using the quotient rule we can simplify this:
$r ' \left(\setminus \theta\right) = \frac{\sin \left(\setminus \theta\right) \frac{d}{d \setminus \theta} \left[\cos \left(2 \setminus \theta - \setminus \frac{\pi}{4}\right)\right] - \cos \left(2 \setminus \theta - \setminus \frac{\pi}{4}\right) \frac{d}{d \setminus \theta} \left[\sin \left(\setminus \theta\right)\right]}{\sin {\left(\setminus \theta\right)}^{2}} - \frac{d}{d \setminus \theta} \left[\sin \left(\setminus \theta - \frac{\pi}{8}\right)\right]$

Using the derivatives of trig functions we can simplify this:
$r ' \left(\setminus \theta\right) = \frac{\sin \left(\setminus \theta\right) \left(- \sin \left(2 \setminus \theta - \setminus \frac{\pi}{4}\right) \setminus \cdot 2\right) - \cos \left(2 \setminus \theta - \setminus \frac{\pi}{4}\right) \left(\cos \left(\setminus \theta\right)\right)}{\sin {\left(\setminus \theta\right)}^{2}} - \cos \left(\setminus \theta - \frac{\pi}{8}\right)$

Now, evaluate it at $\setminus \theta = \frac{- 3 \setminus \pi}{8}$: (there is too much simplifying and the answer is irrational, so I am just going to give the decimal version) $r ' \left(\frac{- 3 \setminus \pi}{8}\right) \approx 0.448$

We need the value of $r \left(\frac{- 3 \pi}{8}\right)$:

$r \left(\frac{- 3 \pi}{8}\right) = \cos \frac{2 \left(\frac{- 3 \pi}{8}\right) - \setminus \frac{\pi}{4}}{\sin} \left(\frac{- 3 \pi}{8}\right) - \sin \left(\frac{- 3 \pi}{8} - \frac{\pi}{8}\right)$

$r \left(\frac{- 3 \pi}{8}\right) \approx 2.082$

Thus far we have

m = (0.448sin((-3pi)/8) + 2.082cos((-3pi)/8))/(0.448cos((-3pi)/8) - 2.082sin((-3pi)/8)

Evaluating the sines and cosines:

$m \approx 0.183$

Compute the $\left({x}_{1} , {y}_{1}\right)$ point:

${x}_{1} = r \cos \left(\theta\right) = 2.082 \cos \left(\frac{- 3 \pi}{8}\right) \approx 0.797$

${y}_{1} = r \sin \left(\theta\right) = 2.082 \sin \left(\frac{- 3 \pi}{8}\right) \approx - 1.924$

Use the point-slope form of the equation of a line:

$y = m \left(x - {x}_{1}\right) + {y}_{1}$

The equation is:

$y = 0.183 \left(x - 0.797\right) - 1.924$