Sec theta = 2 to rectangular coordinates??

Feb 26, 2018

See below.

Explanation:

Identity.

$\textcolor{red}{\boldsymbol{\sec x = \frac{1}{\cos} x}}$

$\frac{1}{\cos} \left(\theta\right) = 2$

$\cos \left(\theta\right) = \frac{1}{2}$

$\theta = \arccos \left(\cos \left(\theta\right)\right) = \arccos \left(\frac{1}{2}\right) \implies \theta = \frac{\pi}{3} , \frac{5 \pi}{3}$

For:

$0 \le \theta \le 2 \pi$

From the above diagram, we can see that:

$x = r \cos \left(\theta\right)$

$y = r \sin \left(\theta\right)$

So coordinates will be:

$\left(r \cos \left(\frac{\pi}{3}\right) , r \sin \left(\frac{\pi}{3}\right)\right)$

$\left(r \cos \left(\frac{5 \pi}{3}\right) , r \sin \left(\frac{5 \pi}{3}\right)\right)$

Without knowing the radius we can't go any further than this.

If this is on a unit circle then the radius is $1$ and:

$\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$ , $\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$

$\cos \left(\frac{5 \pi}{3}\right) = \frac{1}{2}$ , $\sin \left(\frac{5 \pi}{3}\right) = - \frac{\sqrt{3}}{2}$

Coordinates:

$\left(\frac{1}{2} , \frac{\sqrt{3}}{2}\right)$ , $\left(\frac{1}{2} , - \frac{\sqrt{3}}{2}\right)$