Math / Geometry

Questiona point are given. Find polar c
16. $(-2,2)$

Studdy Solution

STEP 1

What is this asking? Convert the point $(-2, 2)$ from Cartesian coordinates to polar coordinates. Watch out! Don't forget that the angle $\theta$ should be in the correct quadrant!

STEP 2

1. Calculate the radius $r$
2. Determine the angle $\theta$
3. Adjust $\theta$ for the correct quadrant

STEP 3

Alright, let's calculate the radius $r$ .
The formula for the radius in polar coordinates is:
$r = \sqrt{x^2 + y^2}$ where $x = -2$ and $y = 2$ .
So let's plug in those values:
$r = \sqrt{(-2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$ Boom! We've got our radius: $r = 2\sqrt{2}$ .

STEP 4

Next, we need to determine the angle $\theta$ .
The basic formula for $\theta$ is:
$\theta = \tan^{-1}\left(\frac{y}{x}\right)$ Substitute $x = -2$ and $y = 2$ :
$\theta = \tan^{-1}\left(\frac{2}{-2}\right) = \tan^{-1}(-1)$ Now, $\tan^{-1}(-1)$ gives us $-\frac{\pi}{4}$ radians.
But wait!
We need to make sure $\theta$ is in the correct quadrant.

STEP 5

The point $(-2, 2)$ is in the second quadrant, where the angle should be between $\frac{\pi}{2}$ and $\pi$ .
So, let's adjust the angle by adding $\pi$ to $-\frac{\pi}{4}$ :
$\theta = -\frac{\pi}{4} + \pi = \frac{3\pi}{4}$ Now, we've got the correct angle: $\theta = \frac{3\pi}{4}$ .

STEP 6

The polar coordinates for the point $(-2, 2)$ are $(2\sqrt{2}, \frac{3\pi}{4})$ .

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