Problem 47 - Algebra and Trigonometry

Question

Convert the polar equation to rectangular coordinates. $$ r=7 $$

Step-by-Step Solution

Answer

The rectangular form is \(x^2 + y^2 = 49\).

Step 1: Understand the Relationship Between Polar and Rectangular Coordinates

In polar coordinates, each point is determined by the distance from the origin \( r \) and an angle \( \theta \). Rectangular coordinates are based on \( x \) and \( y \) coordinates in a Cartesian plane. The relationships are given by: \( x = r \cos \theta \) and \( y = r \sin \theta \).

Step 2: Substitute the Given Polar Equation

Substitute the given polar equation \( r = 7 \) into the relationships for \( x \) and \( y \). We have: \( x = 7 \cos \theta \) and \( y = 7 \sin \theta \).

Step 3: Square and Add the Equations

Square the expressions for \( x \) and \( y \) to use the Pythagorean identity. This gives us: \( x^2 = (7 \cos \theta)^2 \) and \( y^2 = (7 \sin \theta)^2 \). Adding these gives: \( x^2 + y^2 = 49(\cos^2 \theta + \sin^2 \theta) \).

Step 4: Use the Pythagorean Identity

Recognize that \( \cos^2 \theta + \sin^2 \theta = 1 \), thus transforming the equation into: \( x^2 + y^2 = 49 \).

Step 5: Write the Result in Rectangular Form

The rectangular form of the equation is derived as \( x^2 + y^2 = 49 \), which represents a circle with a radius of 7 centered at the origin.

Key Concepts

Rectangular Coordinates

Rectangular coordinates refer to the pair of values \(x\) and \(y\) that locate a point on the Cartesian plane. This system is named rectangular because the axes divide the plane into rectangular grids.
The \(x\)-coordinate tells us the horizontal distance from the origin, while the \(y\)-coordinate specifies the vertical distance. Together, they provide a complete address for any point on the 2D grid.

• When interpreting geometric shapes like circles and lines, rectangular coordinates are very intuitive.
• They form the basis for most algebraic manipulation and graphing tasks.

In the given exercise, we transform polar information into this form to make it easier to handle in typical mathematical operations.

Polar Coordinates

Polar coordinates describe a point based on its distance from a fixed point (the origin) and an angle relative to a fixed direction (typically the positive \(x\)-axis). The two components of polar coordinates are \(r\) and \(\theta\):

• \(r\) is the radial distance from the origin.
• \(\theta\) is the angle measured counterclockwise from the positive \(x\)-axis.

This system is particularly useful for circular or rotational problems.

• For a constant \(r\), as in the given exercise \(r=7\), the points form a circle around the origin.
• If \(\theta\) varies while \(r\) remains constant, the tracing describes a full circle.

This concept is key when converting to rectangular coordinates for equations that naturally follow a circular path.

Trigonometric Identities

Trigonometric identities are mathematical equalities involving trigonometric functions that are true for all values of the involved variables. They are tools that enable elegant manipulation and simplification of equations and expressions.
The primary identity used in coordinate conversion is the Pythagorean identity:

• \(\cos^2\theta + \sin^2\theta = 1\)

This identity is crucial when transitioning equations from polar to rectangular form.
In the exercise, applying this identity allows us to easily convert \(x^2 + y^2 = r^2\), which neatly simplifies the description of a circle in rectangular coordinates.