Question

I need help with this practice problem solving from the trig section in my ACT prep guide

Answer 1

Step 1

Find the polar coordinates of the rectangular coordinates

$(-\sqrt[]{3},1)$

To convert from the cartesian coordinates (x,y) to polar coordinates (r,θ), we use the following equation.

$r^2=x^2+y^2$$\theta=arc\tan (\frac{y}{x})$

Step 2

We have;

$(x,y)=(-\sqrt[]{3},1)$$\begin{gathered} r^2=(-\sqrt[]{3})^2+1^2 \\ r^2=3+1 \\ r=\sqrt[]{4} \\ r=2 \\ \end{gathered}$$\begin{gathered} \theta=arc\tan (\frac{y}{x}) \\ \theta=arc\tan (\frac{1}{-\sqrt[]{3}}) \\ \theta=\arctan (\frac{1}{-\sqrt[]{3}}\times\frac{\sqrt[]{3}}{\sqrt[]{3}}) \\ \theta=\arctan \frac{\sqrt[]{3}}{-3} \\ \theta=-\frac{\pi}{6}+\pi \\ \theta=\frac{5}{6}\pi \end{gathered}$

The answer therefore is;

$(2,\frac{5\pi}{6})$