Problem 78 - Precalculus with Limits

Question

In Exercises 65-84, convert the rectangular equation to polar form. Assume \(a>0\). \((x^2+y^2)^2=9(x^2-y^2)\)

Step-by-Step Solution

Answer

The polar form of the given equation is \(r^2 = 9(cos^2(θ) - sin^2(θ))\).

Step 1: Substitute \(x^2+y^2\) with \(r^2\)

Substitute \(x^2+y^2\) by \(r^2\) in the given equation. This results in \((r^2)^2 = 9(x^2 - y^2)\). Hence, the equation simplifies to \(r^4 = 9(x^2 - y^2)\).

Step 2: Substitute \(x\) and \(y\) with their polar form counterparts

Next, Substitute \(x\) by \(r cos(θ)\) and \(y\) by \(r sin(θ)\) in the equation obtained from step 1. This results in \(r^4 = 9((r cos(θ))^2 - (r sin(θ))^2)\).

Step 3: Simplify the equation

Now, simplify this equation. This results in \(r^4 = 9r^2(cos^2(θ) - sin^2(θ))\). Divide both sides by \(r^2\) to achieve the poetic form equation: \(r^2 = 9((cos^2(θ) - sin^2(θ))\).

Key Concepts

Rectangular Equation

A rectangular equation is an equation involving the coordinates

• \(x\) and \(y\) that describe the location of points in the Cartesian coordinate system, also known as the rectangular coordinate system.

These equations can take various forms, such as linear, quadratic, or more complex algebraic equations involving the variables \(x\) and \(y\).
Converting a rectangular equation into polar coordinates involves using the relationship between Cartesian and polar coordinates.
In this particular exercise, the given rectangular equation is \((x^2 + y^2)^2 = 9(x^2 - y^2)\).
The goal is to express this in terms of polar coordinates \(r\) and \(θ\).

Polar Form

The polar form of an equation describes the location of points in the plane using the polar coordinate system.

• Polar coordinates consist of a radius \(r\) from the origin and an angle \(θ\) from the positive x-axis.

The conversion from rectangular (Cartesian) coordinates to polar coordinates involves:

• \(x = r \cos(θ)\)
• \(y = r \sin(θ)\)
• \(x^2 + y^2 = r^2\)

In the step-by-step solution, these transformations are used to convert the rectangular equation into its polar equivalent. This helps visualize the equation in terms of circles and angles.

Trigonometric Substitution

Trigonometric substitution is a method used to simplify mathematical expressions and equations by replacing algebraic terms with their trigonometric counterparts.
This substitution is especially useful when dealing with equations involving circular functions due to the nature of polar coordinates.
A key part of the exercise is substituting \(x\) and \(y\) with their polar form equivalents \(r \cos(θ)\) and \(r \sin(θ)\) respectively.
By applying these substitutions, the expression becomes simpler and directly relates to polar coordinates:

• Substitute \(x = r \cos(θ)\) and \(y = r \sin(θ)\) to simplify terms of \(x^2\) and \(y^2\).

This transformation is key to achieving the final polar form equation.

Trigonometric Identities

Trigonometric identities are fundamental relationships between trigonometric functions that hold true for all values of the variables involved.
They are essential tools in simplifying and solving equations in trigonometry.
One identity frequently used in this exercise is:

• \(\cos^2(θ) - \sin^2(θ)\), also known as the cos double angle identity.

This identity allows the expression \(cos^2(θ) - sin^2(θ)\) to be directly used in the polar form.
Understanding these identities and how to apply them helps in resolving complex trigonometric equations and converting them to simpler forms.
These identities lay the groundwork for understanding the relationships between angles and ratios in any trigonometric context.