Problem 64 - Algebra and Trigonometry

Question

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (-3,3) $$

Step-by-Step Solution

Answer

(3\sqrt{2}, \frac{3\pi}{4})

Step 1: Understand the given coordinates

The given rectangular coordinates are (-3, 3). This means the x-coordinate is -3 and the y-coordinate is 3.

Step 2: Calculate the radius

The radius or distance from the origin to the point (r) is given by the formula: \(r = \sqrt{x^2 + y^2}\). Therefore: \(r = \sqrt{(-3)^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}\).

Step 3: Calculate the angle

The angle (\theta) is given by \(\tan^{-1}( \frac{y}{x} )\). Therefore: \(\theta = \tan^{-1}( \frac{3}{-3} ) = \tan^{-1}( -1 )\). This angle is in the second quadrant, so: \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\).

Step 4: State the final polar coordinates

Hence the polar coordinates are: \((3\sqrt{2}, \frac{3\pi}{4})\).

Key Concepts

rectangular coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to locate a point in a 2D plane. We use two numbers to specify a point: the x-coordinate and the y-coordinate.
The x-coordinate tells us how far left or right the point is from the origin (0, 0).
The y-coordinate indicates how far up or down the point is from the origin.
For example, given the coordinates (-3, 3):

• -3 is the x-coordinate, which means the point is 3 units to the left of the y-axis.
• 3 is the y-coordinate, telling us the point is 3 units above the x-axis.

radius calculation

The radius (r) is the distance from the origin to the point. To find r, we use the Pythagorean theorem, which works perfectly for right-angled triangles.
The formula to calculate the radius is:
\( r = \sqrt{x^2 + y^2} \).
Here's how it works for the point (-3, 3):

• Square both the x and y values: \( (-3)^2 = 9 \) and \( 3^2 = 9 \).
• Add these squared values: \( 9 + 9 = 18 \).
• Take the square root of the sum: \( \sqrt{18} = 3\sqrt{2} \).

So, the radius r is \( 3\sqrt{2} \).

angle calculation

The angle (\theta) in polar coordinates represents the direction from the origin to the point and is usually measured in radians.
To find this angle, we use the arctangent function: \( \tan^{-1}\bigg(\frac{y}{x}\bigg) \).
For the point (-3, 3):

• Calculate the ratio \( \frac{3}{-3} = -1 \).
• Find the arctangent: \( \tan^{-1}(-1) = -\frac{\pi}{4} \).

Remember, because the point is in the second quadrant, we need to adjust our angle:
So, \( \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \).
The final angle \( \theta \) is \( \frac{3\pi}{4} \).

coordinate transformation

Coordinate transformation is the process of converting from one coordinate system to another.
In this case, we're converting from rectangular coordinates to polar coordinates.
The two key transformations involved are:

• Calculating the radius using \( r = \sqrt{x^2 + y^2} \).
• Calculating the angle using \( \tan^{-1}\big(\frac{y}{x}\big) \), then adjusting the angle based on the quadrant.

Finally, to find the polar coordinates, combine the radius and angle:
For the point (-3, 3), the polar coordinates are \( (3\sqrt{2}, \frac{3\pi}{4}) \).