Problem 75 - Precalculus

Question

Convert (-2,-2) to polar coordinates. Solution: Label \(x\) and \(y.\) \(x=-2, y=-2\) Find \(r . \quad r=\sqrt{x^{2}+y^{2}}=\sqrt{4+4}=\sqrt{8}=2 \sqrt{2}\) Find \(\theta . \quad \tan \theta=\frac{-2}{-2}=1\) \(\theta=\tan ^{-1}(1)=\frac{\pi}{4}\) Write the point in polar coordinates. \(\quad\left(2 \sqrt{2}, \frac{\pi}{4}\right)\) This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The correct polar coordinates are \((2\sqrt{2}, \frac{5\pi}{4})\).

Step 1: Identify Cartesian Coordinates

Start with the given Cartesian coordinates \(x = -2\) and \(y = -2\).

Step 2: Calculate the Radius (r)

Calculate \( r \) using the formula \[ r = \sqrt{x^2 + y^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

Step 3: Calculate the Angle (\(\theta\))

Calculate \( \theta \) using the formula of the tangent function \[ \tan(\theta) = \frac{y}{x} = \frac{-2}{-2} = 1 \] \(\theta = \tan^{-1}(1)\) gives two possible angles, \( \frac{\pi}{4} \) or \( \frac{5\pi}{4} \), because the point is in the third quadrant where both \( x \) and \( y \) are negative, we choose \( \theta = \frac{5\pi}{4} \).

Step 4: Write the Polar Coordinates

Now, write the point in polar coordinates: \((r, \theta) = (2\sqrt{2}, \frac{5\pi}{4})\).

Key Concepts

Cartesian coordinates

Cartesian coordinates are a way of defining the position of a point in 2D space. They are represented by two values, typically denoted as \((x, y)\). These values indicate the point's horizontal and vertical distances from the origin, which is located at \((0, 0)\).
To illustrate, the Cartesian coordinates \((-2, -2)\) specify a point that is two units to the left along the x-axis and two units down along the y-axis. This representation is intuitive when visualizing a point on a grid or a graph since it directly corresponds to the x and y axes we're familiar with from algebra and geometry.
In this exercise, we're asked to convert these Cartesian coordinates into polar coordinates, a different method of expression using a distance and an angle.

radius calculation

The radius in polar coordinates, often noted as \(r\), is the distance from the origin to the point. To find this, we use the Pythagorean theorem, which helps us determine the distance between two points in 2D space.
The formula to calculate the radius is:

• \(r = \sqrt{x^2 + y^2}\)

Applying it to our given Cartesian coordinates \((-2, -2)\):

• Calculate each square: \((-2)^2 = 4\) and \((-2)^2 = 4\).
• Add them together: \(4 + 4 = 8\).
• Find the square root: \(\sqrt{8} = 2\sqrt{2}\).

Thus, the radius \(r = 2\sqrt{2}\), representing how far the point is from the origin.

angle determination

To convert to polar coordinates, we also need to calculate the angle, \(\theta\), which indicates the direction from the origin to the point. The tangent of this angle can be found using the ratio of the y-coordinate to the x-coordinate.
The formula is:

• \(\tan(\theta) = \frac{y}{x}\)

Using the point \((-2, -2)\):

• \(\tan(\theta) = \frac{-2}{-2} = 1\)

The arctangent or inverse tangent function \(\tan^{-1}(1)\) can yield two possible standard angles: \(\frac{\pi}{4}\) and \(\frac{5\pi}{4}\).
The correct angle needs to be chosen based on the quadrant where the point lies, which leads us to the next concept.

third quadrant determination

In the Cartesian plane, the third quadrant is where both x and y coordinates are negative. Since the point \((-2, -2)\) is in this quadrant, we must choose the angle that corresponds to this position.
From the possible angles \(\frac{\pi}{4}\) and \(\frac{5\pi}{4}\), only \(\frac{5\pi}{4}\) represents a direction pointing both left and downward from the origin, aligning with the third quadrant.
Choosing \(\frac{5\pi}{4}\) correctly accounts for the positioning of the point on the grid. Therefore, the final polar coordinates are correctly expressed as \((2\sqrt{2}, \frac{5\pi}{4})\), combining the radius and the appropriately chosen angle for the third quadrant.