Chapter 15.1, Problem 1MRE

To determine

Convert the rectangular coordinates into polar coordinates.

Answer to Problem 1MRE

  $\left(6,120°\right)$

Explanation of Solution

Given information:

Given: $\left(-3,3\sqrt{3}\right)$

Calculation:

Here, the polar coordinates of a point are an ordered pair $\left(r,\theta \right)$ .

Now, we will use the following formula for the conversion of rectangular coordinate into polar coordinate:

  $\begin{array}{l}r=\pm \sqrt{x^2+y^2};\cos \theta =\frac{x}{r},\sin \theta =\frac{y}{r}\\ puttingx=-3andy=3\sqrt{3}:\\ r=\pm \sqrt{{\left(-3\right)}^2+{\left(3\sqrt{3}\right)}^2}\\ r=\pm \sqrt{9+27}\\ r=\pm \sqrt{36}\\ r=\pm 6\end{array}$

Now, we have observed that the given rectangular coordinates lie in the second quadrant. So, the terminal side of $\theta$ passes through the second quadrant.

  $\begin{array}{l}Now,puttingr=6andy=3\sqrt{3}in:\\ \sin \theta =\frac{y}{r}\\ \sin \theta =\frac{3\sqrt{3}}{6}\\ \sin \theta =\frac{\sqrt{3}}{2}\\ \therefore \theta =120°\end{array}$

Hence, the polar coordinates are $\left(6,120°\right)$ .