Question

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

Answer 1

`(10,(\pi)/(3))`

Step-by-step 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)((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=(\pi)/(3) \end{gathered}`

Therefore, the polar coordinates of the given point are;

`(10,(\pi)/(3))`