Problem 92 - Algebra and Trigonometry

Question

Explain how to convert from rectangular coordinates to polar coordinates.

Step-by-Step Solution

Answer

(5, 0.93)

Step 1: Identify Rectangular Coordinates

Identify the given rectangular coordinates \( x, y \). These coordinates are typically in the form (x, y). For example, let's say the coordinates are (3, 4).

Step 2: Calculate the Radius (r)

To find the radius \(r\), use the formula: \[ r = \sqrt{x^2 + y^2} \]. For the given coordinates (3, 4), calculate \[ r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \].

Step 3: Calculate the Angle (θ)

To find the angle \( \theta \), use the formula: \[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \]. For the given coordinates (3, 4), calculate \[ \theta = \tan^{-1} \left( \frac{4}{3} \right) = 0.93 \text{ radians} \]. Note that the angle should be in radians unless specified otherwise.

Step 4: Express in Polar Coordinates

Combine the calculated radius and angle to express the rectangular coordinates in polar form as \( (r, \theta) \). For the given example, the polar coordinates would be \( (5, 0.93) \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to determine a point's position on a plane. They use an ordered pair of values (x, y).
The 'x' value represents the horizontal distance from the origin, while the 'y' value represents the vertical distance.
Example: The point (3, 4) means moving 3 units right and 4 units up from the origin.
Rectangular coordinates are very intuitive and are widely used in various fields such as physics, engineering, and computer graphics.

Polar Coordinates

Polar coordinates offer another way to locate points on a plane. They use two values: the radius (r) and the angle (θ).
The radius 'r' is the distance from the origin to the point, and the angle 'θ' is the direction from the positive x-axis to the point.
Example: A point with polar coordinates (5, 0.93) means it is 5 units away from the origin at an angle of 0.93 radians.
Polar coordinates are especially useful in scenarios involving circular or rotational motion, like in navigation and robotics.

Radius Calculation

The radius (r) is found using the Pythagorean theorem. It gives the straight-line distance from the origin to the point (x, y).
You can calculate it using the formula: \[ r = \sqrt{x^2 + y^2} \]
For example, for the point (3, 4), we calculate:\[ r = \sqrt{3^2 + 4^2} \] \[ r = \sqrt{9 + 16} = \sqrt{25} = 5 \]
The radius is important because it tells us how far the point is from the origin.

Angle Calculation

The angle (θ) provides the direction from the positive x-axis to the point (x, y).
We use the inverse tangent function to find the angle: \[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \]
For instance, for the point (3, 4), we find:\[ \theta = \tan^{-1} \left( \frac{4}{3} \right) \]
Using a calculator, this comes out to approximately 0.93 radians.
Note that angles are typically expressed in radians unless specified otherwise.

Coordinate Transformation

Transforming coordinates from rectangular to polar involves both the radius and the angle.
First, calculate the radius using \[ r = \sqrt{x^2 + y^2} \]
Then, find the angle with \[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \]
Finally, express the point in polar form as (r, θ).
For our example (3, 4), the transformation is:\[ r = 5 \] \( \theta = 0.93 \text{ radians} \)So, the polar coordinates are (5, 0.93).
This coordinate transformation is essential in many applications like electromagnetics and control systems.