Find the polar coordinates of the points whose cartesian coordinates are
`(-2, -2)`

Answer

To find the polar coordinates of the point whose Cartesian coordinates are \((-2, -2)\), we will follow these steps:

### Step 1: Identify the Cartesian coordinates
The given Cartesian coordinates are:
\[ (x, y) = (-2, -2) \]

### Step 2: Calculate the radius \( r \)
The formula for the radius \( r \) in polar coordinates is given by:
\[ r = \sqrt{x^2 + y^2} \]
Substituting the values of \( x \) and \( y \):
\[ r = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \]
We can simplify \( \sqrt{8} \):
\[ r = \sqrt{4 \cdot 2} = 2\sqrt{2} \]

### Step 3: Calculate the angle \( \theta \)
To find the angle \( \theta \), we use the tangent function:
\[ \tan \theta = \frac{y}{x} \]
Substituting the values of \( y \) and \( x \):
\[ \tan \theta = \frac{-2}{-2} = 1 \]
The angle \( \theta \) for which \( \tan \theta = 1 \) is:
\[ \theta = \frac{\pi}{4} \]
However, since both \( x \) and \( y \) are negative, the point \((-2, -2)\) lies in the third quadrant. In the third quadrant, we need to add \( \pi \) to the angle:
\[ \theta = \pi + \frac{\pi}{4} = \frac{4\pi}{4} + \frac{\pi}{4} = \frac{5\pi}{4} \]

### Step 4: Write the polar coordinates
Now that we have both \( r \) and \( \theta \), we can express the polar coordinates as:
\[ (r, \theta) = (2\sqrt{2}, \frac{5\pi}{4}) \]

Thus, the polar coordinates of the point \((-2, -2)\) are:
\[ \boxed{(2\sqrt{2}, \frac{5\pi}{4})} \]

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