Question

Express the following Cartesian coordinates in polar coordinates in at least two different ways. $$(2,2)$$

Short answer

Question: Convert the Cartesian coordinates (2,2) into polar coordinates in at least two different ways. Answer: The two different ways to express the Cartesian coordinates (2,2) in polar coordinates are (2√2, 𝜋/4) and (2√2, 9𝜋/4).

Step-by-step solution

Calculate the radius

Using the relationship \(r = \sqrt{x^2 + y^2}\), we can calculate the radius for the given Cartesian coordinates, \((2,2)\): $$r = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$$

Calculate the angle for the first polar coordinate

Using the relationship \(\theta = \arctan\left(\frac{y}{x}\right)\), we can calculate the angle for the given Cartesian coordinates, \((2,2)\): $$ \theta = \arctan\left(\frac{2}{2}\right) = \arctan(1) = \frac{\pi}{4} $$ Our first polar coordinate is $$(r,\theta) = (2\sqrt{2},\frac{\pi}{4})$$.

Find another representation of the polar coordinates

To find another representation of the polar coordinates, we can add \(2\pi\) to the angle \(\theta\): $$ \theta_2 = \frac{\pi}{4} + 2\pi = \frac{\pi}{4} + \frac{8\pi}{4} = \frac{9\pi}{4} $$ Our second polar coordinate is $$(r,\theta) = (2\sqrt{2},\frac{9\pi}{4})$$. These are the two different ways we can express the Cartesian coordinates \((2,2)\) in polar coordinates: 1. \(\boldsymbol{(2\sqrt{2},\frac{\pi}{4})}\) 2. \(\boldsymbol{(2\sqrt{2},\frac{9\pi}{4})}\)

Key concepts

Cartesian Coordinates

Cartesian coordinates are a popular way to define a point in a plane using a pair of numerical values. These values are known as the x-coordinate and the y-coordinate, often presented as \((x, y)\). Each coordinate represents the position along the two perpendicular axes of a Cartesian plane.
In a two-dimensional plane, the x-axis is horizontal, while the y-axis is vertical. The point \((2, 2)\) means that it is located 2 units from the origin along the x-axis and 2 units along the y-axis.

Using Cartesian coordinates is intuitive for many, especially in visualizing positions on a graph. They are exceptionally useful in scenarios involving linear motion due to their simplicity. However, in cases involving rotational movement or waves, polar coordinates can be more advantageous.

Radius Calculation

In polar coordinates, the distance from the origin to the point is referred to as the radius, denoted by \(r\).
To transform a point from Cartesian coordinates to polar, we first calculate this radius using the formula: \[ r = \sqrt{x^2 + y^2} \]For \((2, 2)\), the radius calculates to: \[ r = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2} \]

• Ensure calculations are precise since any miscalculation can lead to an error in the transformation.
• The radius is always a non-negative value and represents the direct line distance to the point.

Remember, this step is crucial as it sets the foundation for defining the point's position in polar coordinates thoroughly.

Angle Calculation

The angle in polar coordinates indicates the direction of the point relative to the positive x-axis. It's denoted as \(\theta\) and can be calculated by taking the tangent inverse of the ratio between the y-coordinate and x-coordinate:
\[ \theta = \arctan\left(\frac{y}{x}\right) \]For the point \((2, 2)\), this becomes: \[ \theta = \arctan\left(\frac{2}{2}\right) = \arctan(1) = \frac{\pi}{4} \]

Keep these insights in mind when calculating the angle:

• Angles are typically measured in radians in mathematical contexts.
• If the point lies on the axes themselves, the angle might conform to specific common values like \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}\).

This angle, along with the radius, allows you to fully describe the point in polar format.

Coordinate Transformation

Coordinate transformation involves converting points from one type of coordinate system to another, in this case from Cartesian to polar coordinates. The previous calculations of radius and angle are the foundation of this transformation.
The point \((2, 2)\) has been converted using these steps to produce two different versions of polar coordinates.
For example, we have:

• First, \((r,\theta) = (2\sqrt{2},\frac{\pi}{4})\)
• Second, \((r,\theta) = (2\sqrt{2},\frac{9\pi}{4})\)

Key concepts to remember:

• Adding \(2\pi\) to the angle \(\theta\) results in an equivalent point because rotations of full cycles end up in the same position.
• Coordinate transformations require both accuracy and understanding of geometry concepts.

While challenging, conquering coordinate transformations opens the door to understanding motion and space in different ways, aligning mathematics with real-world scenarios.