Problem 46 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand Cartesian and Polar Systems

In the Cartesian system, the equation given is a horizontal line at a constant y-value of 5. In polar coordinates, each point is represented by an angle and a distance from the origin. We'll convert the Cartesian line equation to a polar form.

Step 2: Use Polar Coordinates Definition

In polar coordinates, a point is represented as \(r, \theta\), where \(r\) is the radius or distance from the origin, and \(\theta\) is the angle from the positive x-axis. The conversion formulas are: 1. \(x = r\cos(\theta)\)2. \(y = r\sin(\theta)\)

Step 3: Substitute for x and y in Polar Form

Given the equation \(y = 5\), substitute \(y = r\sin(\theta)\). This yields: \(r\sin(\theta) = 5\). Keep \(x = r\cos(\theta)\) in mind for understanding the relation between x, y, and r.

Step 4: Express the Equation in Polar Form

Now the polar form of the equation is \(r\sin(\theta) = 5\). This expresses the line \(y = 5\) in polar coordinates, meaning any point on this line must satisfy this relationship.

Key Concepts

Cartesian coordinates

In mathematics, Cartesian coordinates are one of the most common systems used to describe the position of points on a plane. This method utilizes two axes: the horizontal axis (x-axis) and the vertical axis (y-axis). Each point in this system is signified by an ordered pair \(x, y\).
For example, the point \(3, 5\) represents a point located 3 units along the x-axis and 5 units up the y-axis.
Cartesian coordinates are exceptionally intuitive when dealing with straight lines because they directly relate to linear equations like \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
In the original exercise, the equation given is \(y = 5\), representing a horizontal line crossing the y-axis at 5.

polar form conversion

Polar coordinates provide an alternative way of representing points by specifying their distance from the origin and the angle from the positive x-axis.
Instead of using x and y coordinates like in the Cartesian system, polar coordinates use \(r, \theta\), where \(r\) is the radius – the distance from the origin, and \(\theta\) is the angle.
The conversion between Cartesian and polar coordinates is accomplished using the formulas:

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

For the provided equation \(y = 5\), we substitute using \(y = r\sin(\theta)\), resulting in the equation \(r\sin(\theta) = 5\).
This expresses the same line in polar form, conveying the necessary angle and radius relationship across all points on the line.

trigonometric identities

Trigonometric identities are fundamental in the conversion between Cartesian and polar coordinates, as they relate to the angles and sides of right triangles.
The main trigonometric functions involved are sine ( ext{sin}), cosine ( ext{cos}), and tangent ( ext{tan}). For converting equations, we primarily use:

• \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\)
• \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
• \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)

In the equation conversion from Cartesian to polar, the identity \(y = r\sin(\theta)\) directly employs the sine function.
This simplification allows us to express the linear alignment of points in polar form effectively.

coordinate systems

Coordinate systems are crucial in mathematics and applied sciences for identifying the position of a point or an object in space.
The two most common systems are Cartesian and polar coordinates. Cartesian coordinates use a grid-like system with axes at right angles, suitable for algebraic descriptions of geometry.
Polar coordinates, on the other hand, are circular in nature, making them incredibly useful for addressing phenomena involving rotational symmetry or periodic functions, such as waves and circular paths.
Understanding how these systems relate, as demonstrated in the exercise by converting \(y = 5\) into polar form ( ext{\(r\sin(\theta) = 5\)}), showcases their versatility and the power of conversion between systems.
This knowledge empowers students to tackle diverse problems by choosing the most suitable coordinate framework.