Question

 

Find the polar coordinates of a point with the Cartesian coordinates

 

$$(x,y)=\left(-4\sqrt{2},4\sqrt{2}\right)\text{.}$$

Answer 1

Solution

Let's start by plotting the point $(x,y)=(-4\sqrt{2},4\sqrt{2})$ on a graph and using the Pythagorean theorem to find the corresponding radius of the point. We get that

$$\begin{array}{cc}{r}^2& =x^2+y^2\\ \\ r^2& =(-4\sqrt{2}{)}^2+(4\sqrt{2}{)}^2\\ \\ r^2& =32+32\\ \\ r^2& =64\\ \\ r& =8\text{.}\end{array}$$

Now that we know the radius, we can find the angle using any of the three trigonometric relationships. Keep in mind that there may be more than one solution when solving for $\theta$ and we will need to consider the quadrant that our $(x,y)$ point is in to decide which solution to use.

Using the sine function, we get that

$$\sin \left(\theta \right)=\frac{y}{r}=\frac{4\sqrt{2}}{8}=\frac{\sqrt{2}}{2}\text{.}$$

Since $y=\sqrt{2}/2$ corresponds to a common angle on the unit circle, we can find an exact value for $\theta \text{.}$ The two angles that have a sine value of $\sqrt{2}/2$ on the unit circle are

$$\theta =\frac{\pi }{4}\text{and}\theta =\frac{3\pi }{4}\text{.}$$

Since the point $(x,y)=(-4\sqrt{2},4\sqrt{2})$ is located in Quadrant II, we can determine that $\theta =3\pi /4\text{.}$ Thus, the polar coordinates are

$$(r,\theta )=\left(8,\frac{3\pi }{4}\right)\text{.}$$