Question

The rectangular coordinates of a point are given. Find polar coordinates of the point. Express θ in radians.

Answer 1

Answer:

$(10,\frac{\pi}{3})$

Explanation:

Given the rectangular coordinates of a point as;

$\begin{gathered} (-5,-5\sqrt[]{3}) \\ \text{where }x=-5,y=-5\sqrt[]{3} \end{gathered}$

A polar coordinate is generally given in the form;

$\begin{gathered} (r,\theta) \\ \text{where }r=\sqrt[]{x^2+y^2} \\ \theta=\tan ^{-1}(\frac{y}{x}) \end{gathered}$

Let's go ahead and substitute the given values into the equation for determining r and evaluate;

$\begin{gathered} r=\sqrt[]{(-5)^2+(-5\sqrt[]{3})^2} \\ r=\sqrt[]{25+25(3)} \\ r=\sqrt[]{25+75} \\ r=\sqrt[]{100} \\ r=10 \end{gathered}$

Let's also substitute the given values into the equation for determining theta and evaluate;

$\begin{gathered} \theta=\tan ^{-1}(\frac{-5\sqrt[]{3}}{-5}) \\ \theta=\tan ^{-1}(\sqrt[]{3}) \\ \theta=\frac{\pi}{3} \end{gathered}$

Therefore, the polar coordinates of the given point are;

$(10,\frac{\pi}{3})$