Problem 33 - Precalculus

Question

A point in polar coordinates is given. Convert the point to rectangular coordinates. $$(-2.5,1.1)$$

Step-by-Step Solution

Answer

The conversion of the provided polar coordinates (-2.5, 1.1) results in the rectangular coordinates (x,y) after using the equations \(x = r cos(θ)\) and \(y = r sin(θ)\).

Step 1: Understand polar to rectangular conversion

Polar coordinates are converted to rectangular coordinates using the relationships: \(x = r cos(θ)\) and \(y = r sin(θ)\). Here, \(r\) is the radius and \(θ\) is the angle of the vector. These formulas result from the trigonometric relationships within a right triangle.

Step 2: Implement conversion for x-coordinate

We have \(r = -2.5\) and \(θ = 1.1\) radians. Substitute these values into the equation for x: \(x = r cos(θ) = -2.5 cos(1.1)\). Calculate this to get the x-coordinate.

Step 3: Implement conversion for y-coordinate

Similarly, substitute \(r\) and \(θ\) into the equation for y: \(y = r sin(θ) = -2.5 sin(1.1)\). Calculate this to get the y-coordinate.

Key Concepts

Understanding Polar Coordinates

Polar coordinates are a way to describe a point in a plane using a distance and an angle. This system is different from the more common Cartesian or rectangular coordinates, which use an x-y grid.
In polar coordinates, a point is represented as \((r, \theta)\), where:

• \(r\) is the radial distance from the origin (the center of the grid) to the point.
• \(\theta\) is the angle measured from the positive x-axis to the line connecting the origin to the point.

One important thing to remember is that angles can be measured in either degrees or radians. In most mathematical contexts, radians are preferred.
This system is particularly useful in scenarios involving rotation or circular motion. Converting between polar and rectangular coordinates helps us better understand and visualize these points.

Deciphering Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, express a point in terms of its horizontal and vertical distances from the origin. This method uses the familiar x-y grid format.
The rectangular coordinate system can represent any point by its \(x\) and \(y\) values:

• \(x\) is the horizontal distance from the origin.
• \(y\) is the vertical distance from the origin.

Whether dealing with graphs, functions, or geometry, rectangular coordinates are often the default mode of representation.
Understanding how to switch between rectangular and polar coordinates broadens your ability to solve different types of problems, especially those involving complex patterns or movements.

Exploring Trigonometric Relationships

Trigonometry bridges the gap between polar and rectangular coordinates. The key trigonometric relationships used for conversion come from the basics of a right triangle:

• The cosine function \(\cos(\theta)\) gives the adjacent side over the hypotenuse.
• The sine function \(\sin(\theta)\) gives the opposite side over the hypotenuse.

For conversion:

• To find the x-coordinate (rectangular): \(x = r \cos(\theta)\)
• To find the y-coordinate (rectangular): \(y = r \sin(\theta)\)

These equations arise from projecting the point onto the x and y axes using the unit circle as a reference.
By understanding these relationships, you can easily translate between polar and rectangular forms, enhancing your problem-solving toolkit.