# Math Help - Polarization online hw help

1. ## Polarization online hw help

r2 = sin(2θ)
0 ≤ θ < π

(a) Find the points on the given curve where the tangent line is horizontal.

(b) Find the points on the given curve where the tangent line is vertical.

2. Hello, fogel1497!

We are expected to know (or be able to derive) this formula:

. . $\frac{dy}{dx} \;=\;\frac{r\cos\theta + r'\sin\theta}{-r\sin\theta + r'\cos\theta}$ .[1]

Given: . $r^2 \:=\:\sin2\theta\quad 0 \leq \theta < 2\pi$
Differentiate implicitly: . $2r\!\cdot\!r' \:=\:2\cos2\theta \quad\Rightarrow\quad r' \:=\:\frac{\cos2\theta}{r}$ .[2]

Substitute [2] into [1]: . $\frac{dy}{dx} \;=\;\frac{r\cos\theta + \left(\frac{\cos2\theta}{r}\right)\sin\theta}{-r\sin\theta + \left(\frac{\cos2\theta}{r}\right)\cos\theta}$

Multiply top and bottom by $r\!:\;\;\frac{dy}{dx} \;=\;\frac{r^2\cos\theta + \sin\theta\cos2\theta}{-r^2\sin\theta + \cos\theta\cos2\theta}$

Since $r^2 = \sin2\theta$ we have: . $\frac{dy}{dx} \;=\;\frac{\sin2\theta\cos\theta + \sin\theta\cos2\theta}{-\sin2\theta\sin\theta + \cos\theta\cos2\theta}$ . $= \;\frac{\sin(2\theta + \theta)}{\cos(2\theta + \theta)}$

. . and we have: . $\frac{dy}{dx} \;=\;\frac{\sin3\theta}{\cos3\theta}$

(a) Find the points on the curve where the tangent is horizontal.
The tangent line is horizontal when $\frac{dy}{dx} \,=\,0$

So we have: . $\sin3\theta \:=\:0\quad\Rightarrow\quad 3\theta \:=\:0,\:\pi,\:2\pi,\:3\pi,\:4\pi,\:5\pi$

Horizontal tangents when: . $x\;=\;0,\:\frac{\pi}{3},\:\frac{2\pi}{3},\:\pi,\:\ frac{4\pi}{3},\:\frac{5\pi}{3}$

(b) Find the points on the curve where the tangent is vertical.
The tangents line is vertical when $\frac{dy}{dx}$ is undefined.

So we have: . $\cos3\theta \:=\:0 \quad\Rightarrow\quad 3\theta \;=\;\frac{\pi}{2},\:\frac{3\pi}{2},\:\frac{5\pi}{ 2},\:\frac{7\pi}{2},\:\frac{9\pi}{2},\:\frac{11\pi }{2}$

Vertical tangents when: . $x \;=\;\frac{\pi}{6},\:\frac{\pi}{2},\:\frac{5\pi}{6 },\:\frac{7\pi}{6},\:\frac{3\pi}{2},\:\frac{11\pi} {6}$

3. Thank you, now that you explained it i was able to complete the rest of my homework. i missed this day in class, this stuff is actually a lot easier then the trig sub and other stuff we were doing before. thanks a lot! +rep