Problem 52 - Algebra and Trigonometry

Question

Convert each rectangular equation to a polar equation that expresses r in terms of \(\theta\). $$ y=3 $$

Step-by-Step Solution

Answer

The converted polar equation expressing r in terms of \(\theta\) is \(r = \frac{3}{\sin(\theta)}\).

Step 1: Convert the Coordinate

We start by substituting \(y\) with its polar coordinate equivalent, which is \(r \cdot \sin(\theta)\). So the equation becomes \(r \cdot \sin(\theta) = 3\).

Step 2: Isolate Variable 'r'

To express 'r' in terms of \(\theta\), divide both sides by \(\sin(\theta)\). Our final equation is \(r = \frac{3}{\sin(\theta)}\).

Key Concepts

Rectangular to Polar Conversion

When dealing with different forms of equations, you might encounter the need to switch between rectangular and polar forms. Rectangular equations use Cartesian coordinates, which are based on horizontal and vertical axes

• - These coordinates are represented as \( (x, y) \).
• - Polar equations, however, use points defined by a distance from the origin and an angle from the horizontal axis. They are expressed as \( (r, \theta) \).

The conversion between these two systems is essential in many mathematical applications, especially in trigonometry and calculus.
To perform rectangular to polar conversion:

• - For the x-coordinate, use: \( x = r \cdot \cos(\theta) \).
• - For the y-coordinate, use: \( y = r \cdot \sin(\theta) \).

In our example of \( y=3 \), substituting \( y \) with \( r \cdot \sin(\theta) \) is a key step to expressing the equation in polar form.
This illustrates how understanding conversions can help in solving complex mathematical problems by manipulating equations into more useful forms for analysis.

Polar Coordinates

Polar coordinates offer a different perspective on graphing and analyzing points on a plane. Instead of using the traditional grid-like setup, polar coordinates define a point's position based on:

• - the radius \( r \), which is the distance from the origin.
• - the angle \( \theta \), which is measured from the positive x-axis.

This way of defining points can be particularly useful in situations involving rotations and circular motion.
Understanding polar coordinates:

• - The angle \( \theta \) moves counterclockwise, making it a positive angle.
• - \( r \) can be negative, indicating a direction opposite to the angle \( \theta \).

In our practice problem, \( r = \frac{3}{\sin(\theta)} \) shows how a simple line in rectangular coordinates transforms into a curve or spiral in polar coordinates. This helps visualize how certain equations relate to physical phenomena, like waves or rotations.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are vital in converting between rectangular and polar coordinates. These functions allow us to translate the properties of an angle into usable mathematical expressions:

• - \( \sin(\theta) \) represents the ratio of the opposite side to the hypotenuse in a right triangle.
• - \( \cos(\theta) \) similarly represents the ratio of the adjacent side to the hypotenuse.

In our exercise, the equation \( r = \frac{3}{\sin(\theta)} \) emerges from isolating \( r \) by dividing both sides by the trigonometric function \( \sin(\theta) \). This manipulation is crucial because it links a fixed \( y \)-coordinate in rectangular form with varying \( r \) coordinates in polar form.
Understanding these trig functions not only assists in conversions but also facilitates the exploration of wave functions, oscillations, and other periodic processes.