Problem 44 - Algebra and Trigonometry

Question

Convert the equation to polar form. $$ y=5 $$

Step-by-Step Solution

Answer

The polar form of \( y = 5 \) is \( r \sin \theta = 5 \).

Step 1: Understand the given equation

The given equation is in Cartesian coordinates and is a horizontal line: \( y=5 \). This line is parallel to the x-axis.

Step 2: Recall the conversion formulas

In polar coordinates, any point is represented by \((r, \theta)\), where \(r\) is the distance from the origin, and \(\theta\) is the angle from the positive x-axis. The conversion formulas are: \( x = r \cos \theta \) and \( y = r \sin \theta \).

Step 3: Substitute and simplify the equation

Substitute \( y = r \sin \theta \) in the equation \( y = 5 \), resulting in \( r \sin \theta = 5 \).

Step 4: Express the equation in polar form

The equation \( r \sin \theta = 5 \) is the polar form of \( y = 5 \). This represents all points where their projection on the y-axis equals 5, in terms of polar coordinates.

Key Concepts

Conversion from Cartesian to Polar

When working with coordinates, it is important to understand the difference between Cartesian and polar systems. Cartesian coordinates use horizontal and vertical axes to describe a point's location with

• **x-coordinate** indicates horizontal position, and
• **y-coordinate** indicates vertical position.

For example, the equation \(y = 5\) represents all points that lie on a straight, horizontal line parallel to the x-axis at a y value of 5.
Polar coordinates, by contrast, specify a point with:

• **r**, the distance from the origin, and
• **\(\theta\)**, the angle from the positive x-axis.

To convert from Cartesian to polar, use the formulas:

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

Substituting \(y = r \sin \theta\) allows for converting Cartesian equations like \(y = 5\) into polar form.

Polar Equations

Understanding polar equations is crucial when dealing with curves and lines in polar coordinates. In the polar form of an equation, such as \(r \sin \theta = 5\), each point \((r, \theta)\) satisfies the relationship between the two elements: radius and angle.
This specific equation represents all points where the component along the y-axis, known as the projection, equals 5. It’s a direct translation of the Cartesian equation \(y = 5\) into the polar system.
With polar equations, visualization can be different but keeps the relationships consistent:

• Any \(r\) satisfying \(r \sin \theta = 5\) will be on the same y-level.
• In contrast to straight horizontal lines in Cartesian coordinates, in polar, it involves circular paths centered around the origin.

Navigating between polar and Cartesian systems allows deeper insights into geometrical and trigonometric interpretations of mathematical equations.

Trigonometric Functions

Trigonometric functions like sine and cosine play a crucial role in easing the transition between coordinate systems. They relate angles and lengths, forming the backbone of the conversion formulas used between Cartesian and polar.

• **Sine** is utilized to calculate the y-component in polar coordinates: \(y = r \sin \theta\), explaining its use in our equation \(r \sin \theta = 5\).
• **Cosine** relates to the x-component since \(x = r \cos \theta\).

When converting equations or analyzing their polar forms, understanding these trigonometric relationships is essential.
For instance, knowing that \(\sin \theta = \frac{y}{r}\) helps explain how \(r \sin \theta\) equals a consistent value such as 5. This showcases the constant nature of y-level lines in polar context.
Getting comfortable with trigonometric functions allows smoother calculations and a deeper appreciation of how angles affect distances in polar coordinates.