Problem 82 - Algebra and Trigonometry

Question

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ r=\sin \theta-\cos \theta $$

Step-by-Step Solution

Answer

The equation is already in polar coordinates: \(r = \sin \theta - \cos \theta\).

Step 1: Understand the relationship between rectangular and polar coordinates

In polar coordinates, the relationship between rectangular coordinates \(x, y\) and polar coordinates \(r, \theta\) is given by the equations \(x = r \cos \theta\) and \(y = r \sin \theta\).

Step 2: Rewrite the equation in terms of polar coordinates

Given the equation \(r = \sin \theta - \cos \theta\), observe that it is already in terms of the polar coordinates \(r\) and \(\theta\). Hence, we don't need any additional manipulation.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are also known as Cartesian coordinates. In this system, a point is represented by an ordered pair \(x, y\). Here, \(x\) represents the horizontal distance from the origin, while \(y\) represents the vertical distance.

This coordinate system is familiar because it is typically seen in everyday graphing and plotting situations.

• The \(x\)-axis runs horizontally.
• The \(y\)-axis runs vertically.

Points are located by moving along these axes. For instance, the point \( (3, 4) \) means 3 units to the right of the origin and 4 units up.

Understanding rectangular coordinates is essential for transitioning to polar coordinates, where distances and angles become the primary factors.

Polar Coordinates

Polar coordinates offer a different way of describing positions on a plane. Instead of using horizontal and vertical distances, polar coordinates use a radial distance and an angular displacement.

A point in polar coordinates \(r, \theta\) is defined by:

• \(r\) - The distance from the point to the origin.
• \( \theta \) - The angle measured counterclockwise from the positive \(x\)-axis to the line segment connecting the point to the origin.

This system is particularly useful in contexts where symmetry around a central point or angle measurements are involved.
For example, the point \(r = 5, \theta = \frac{\pi}{4}\) indicates that the point is 5 units away from the origin, at an angle \frac{\pi}{4}\ (or 45 degrees) from the positive \(x\)-axis.

Coordinate Transformation

Transforming between rectangular and polar coordinates involves special formulas that connect these two systems. The main relationships are: \[ x = r \cos \theta \] \[ y = r \sin \theta \]

To go from rectangular to polar coordinates, use: \[ r = \sqrt{x^2 + y^2} \] \[ \theta = \tan^{-1} \left(\frac{y}{x}\right) \]

For example, converting point \ (3, 4) \ from rectangular to polar coordinates involves:
1. Calculating \ r = \sqrt{3^2 + 4^2} = 5 \
2. Finding \theta = \tan^{-1} \left(\frac{4}{3}\right) \ Approx 53.13 degrees

Converting helps visualize and solve problems that may be simpler in one system over the other, such as those involving rotations.

Trigonometric Functions

Trigonometric functions are fundamental for connecting rectangular and polar coordinates. \ \ The primary functions include:

• Sine \( \sin \theta \)
• Cosine \( \cos \theta \)
• Tangent \( \tan \theta \)

These functions express relationships involving angles and sides of right triangles: \[ \ \sin \theta = \frac{opposite}{hypotenuse} \] \[ \ \cos \theta = \frac{adjacent}{hypotenuse} \]

When moving from rectangular coordinates \ (x, y) \ to polar coordinates \ (r, \theta) \, recall that:

• \ x = r \cos \theta
• \ y = r \sin \theta

Understanding these trigonometric connections makes the conversion between coordinate systems much more straightforward.