Problem 88 - Calculus Early Transcendentals

Question

Eliminate the parameter to express the following parametric equations as a single equation in \(x\) and \(y.\) \(x=a \sin ^{n} t, y=b \cos ^{n} t,\) where \(a\) and \(b\) are real numbers and \(n\) is a positive integer

Step-by-Step Solution

Answer

Question: Given parametric equations \(x = a\sin^n(t)\) and \(y = b\cos^n(t)\), eliminate the parameter \(t\) and express the equation in terms of \(x\) and \(y\) only. Answer: \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)

Step 1: Express sine and cosine as functions of t

Recall the definitions of sine and cosine: $$ \sin^2(t) + \cos^2(t) = 1 $$ Raise the first equation to the power of \(2\), then divide by \(a^2\): $$ \frac{x^2}{a^2}=\sin^2(t) $$ Raise the second equation to the power of \(2\), then divide by \(b^2\): $$ \frac{y^2}{b^2}=\cos^2(t) $$

Step 2: Eliminate t using the Pythagorean identity

Add the equations in Step 1: $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=\sin^2(t)+\cos^2(t) $$ Since \(\sin^2(t)+\cos^2(t)=1\), the equation simplifies to: $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $$ Now, the parametric equations are expressed as a single equation in \(x\) and \(y\): $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $$

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry. It states that \( \sin^2(t) + \cos^2(t) = 1 \). This identity is derived from the Pythagorean theorem and is always true for any angle \(t\).
This identity is particularly useful for converting parametric equations into a single equation involving \(x\) and \(y\), as it allows us to substitute trigonometric expressions with a known constant—1.

• In parametric equations, the identity helps to simplify the process by eliminating the parameter \(t\).
• It allows us to relate \(x\) and \(y\) directly through their trigonometric definitions.

Understanding the Pythagorean identity is crucial when dealing with circles and ellipses, given their close relation to these trigonometric functions.

Eliminating Parameters

Eliminating parameters in parametric equations involves writing the relationship between \(x\) and \(y\) without using the parameter, such as \(t\).
This process hinges on known identities and algebraic manipulation. In this specific application, we utilize the derived equations from the parameter's definitions:

• Start by expressing each parameter in terms of fundamental trigonometric functions: \(x = a \sin^n(t)\) and \(y = b \cos^n(t)\).
• Transform these into squared identities: \(\sin^2(t) = \frac{x^2}{a^2}\) and \(\cos^2(t) = \frac{y^2}{b^2}\).
• Using the Pythagorean identity, add these two equations.

By following these steps, you reach an equation that expresses \(x\) and \(y\) together, without direct dependence on the parameter \(t\). This creates a clearer mathematical model of the relationships in question.

Equation in x and y

Transforming parametric equations to a single equation in \(x\) and \(y\) simplifies the analysis and representation of geometric shapes, like ellipses and circles.
When the parameter is successfully eliminated, as shown, the resulting equation often reveals the standard form of these shapes.
In the given exercise, after applying the Pythagorean identity, the resulting equation is:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]This equation describes an ellipse with semi-major axis \(a\) and semi-minor axis \(b\) when \(a eq b\), or a circle when \(a = b\).

• The ellipse is centered at the origin, displaying symmetry about both the x-axis and y-axis.
• This form is elegant and offers a compact summary of the geometric properties involved.

Converting parametric equations into this format is not only convenient but also insightful for visualizing and understanding geometric structures.