Problem 64 - Calculus Early Transcendentals: Pearson New International Edition

Question

If \(b=a\), the equations in Problem 63 are $$ \begin{aligned} &x=2 a \cos t-a \cos 2 t \\ &y=2 a \sin t-a \sin 2 t \end{aligned} $$ Find a Cartesian equation of the epicycloid by eliminating the parameter \(t\) between the equations.

Step-by-Step Solution

Answer

The Cartesian equation is derived as \( y^2 = 4ax(1 - \sqrt{1 - \frac{x}{a}}) \).

Step 1: Use Trigonometric Identities

Begin by recalling the double angle trigonometric identities: \( \cos 2t = 2\cos^2 t - 1 \) and \( \sin 2t = 2\sin t \cos t \). These identities will help simplify the elimination process of the parameter \( t \).

Step 2: Express in Simplified Form

Substitute the identities into the original parametric equations: \( x = 2a\cos t - a(2\cos^2 t - 1) = a(1 - 2\cos^2 t) + 2a\cos t \) and \( y = 2a\sin t - a(2\sin t \cos t) = 2a\sin t(1 - \cos t) \).

Step 3: Find Simplified Expressions for \( x \) and \( y \)

Upon simplifying the substitutions: \( x = a(1 + 2\cos t - 2\cos^2 t) \) and \( y = 2a\sin t(1 - \cos t) \).

Step 4: Apply Trigonometric Identity

Recognize that \( 1 + 2\cos t - 2\cos^2 t = \sin^2 t + \cos^2 t = 1 \). Thus \( x = a \sin^2 t \).

Step 5: Substitute into Other Expression and Eliminate \( t \)

As \( x = a\sin^2 t \), solve for \( \sin t \) giving \( \sin t = \sqrt{\frac{x}{a}} \). Substitute into the expression for \( y = 2a\sin t(1 - \cos t) \).

Step 6: Eliminate Parameter with Identity

Realizing \( \sin^2 t = 1 - \cos^2 t \) from \( x = a\sin^2 t \), it follows that \( \cos t = \sqrt{1 - \frac{x}{a}} \) or it's negative equivalent. Substituting in simplifies \( y = 2a\sin t(1 - \sqrt{1 - \frac{x}{a}}) \).

Step 7: Derive Cartesian Equation

From here deduce \( y = 2\sqrt{a\cdot x}(1 - \sqrt{1 - \frac{x}{a}}) \), tracing back to original equality settings.

Key Concepts

Parametric Equations

Parametric equations are a way to represent geometric shapes using variables, called parameters. These equations express the coordinates of points on a curve as functions of one or more parameters, typically labeled as \(t\). In the context of an epicycloid, the parametric equations provide a way to describe the location of points on the curve based on the changing angle \(t\).\\For our specific problem, we have:

• \(x = 2a \cos t - a \cos 2t\)
• \(y = 2a \sin t - a \sin 2t\)

The goal is to reshape these parametric equations into a form that does not rely on the parameter \(t\). This leads us to explore the concept of a Cartesian equation, which uses only \(x\) and \(y\) pairs.

Cartesian Equation

A Cartesian equation relates the coordinates \(x\) and \(y\) in a single equation, eliminating the parameter. This transformation is quite useful as it simplifies understanding and plotting of the curve, using only the coordinate plane.\\In our exercise, we aim to find such a Cartesian equation for the epicycloid by removing \(t\) from our parametric equations. By applying trigonometric identities and algebra, we rewrite the relationships into a single Cartesian form. For the epicycloid:

• The expressions simplify eventually to show: \(x = a \sin^2 t\)
• From here, substituting back into the equations yields a sole relation in terms of \(x\) and \(y\).

This form is convenient for graphing and analyzing the curve without worrying about different parameters.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold for any angle. They serve as powerful tools in mathematics to simplify expressions and solve equations. In this exercise, we rely on some well-known identities:

• \(\cos 2t = 2 \cos^2 t - 1\)
• \(\sin 2t = 2 \sin t \cos t\)

Using these identities allows us to rewrite the parametric equations to make the elimination of \(t\) more manageable. \\Applying these identities strategically reduces complexity, ultimately guiding us to the Cartesian equation. For example, simplifying the given expressions for \(x\) and \(y\) shows us interconnections leading to their transformation into a Cartesian form.

Elimination of Parameters

Elimination of the parameter \(t\) is a key process in converting parametric equations to a Cartesian form. This involves manipulating expressions to remove the parameter entirely.\\In this case, the steps involve using trigonometric identities and algebraic manipulation:

• Identify possible substitutions, like \(\sin^2 t\) and \(\cos^2 t\)
• Use these relationships to express one variable solely in terms of the other
• Substitute and rearrange to find a direct relationship

By eliminating \(t\), we are left with a clear connection between \(x\) and \(y\), resulting in a Cartesian equation. This allows for analyzing the curve without any parametric complexity, achieving an equation solely in terms of \(x\) and \(y\).