Problem 84 - Calculus Early Transcendentals

Question

Eliminate the parameter to express the following parametric equations as a single equation in \(x\) and \(y.\) $$x=\sin 8 t, y=2 \cos 8 t$$

Step-by-Step Solution

Answer

Question: Eliminate the parameter \(t\) in the given parametric equations to express them as a single equation in \(x\) and \(y\): \(x=\sin 8t\) and \(y=2\cos 8t\). Answer: After eliminating the parameter \(t\), the single equation in \(x\) and \(y\) is \(4x^2 + y^2 = 4\).

Step 1: Solve for \(\cos 8t\) and \(\sin 8t\)

From the given parametric equations, we have two expressions: $$x=\sin 8t$$ $$y=2\cos 8t$$ Now we can find \(\cos 8t\) and \(\sin 8t\) in terms of \(x\) and \(y\) by isolating these trigonometric terms: $$\cos 8t=\frac{y}{2}$$ $$\sin 8t=x$$

Step 2: Use Pythagorean identity to eliminate \(t\)

The Pythagorean identity states that \(\sin^2\theta + \cos^2\theta=1\). Applying this identity to the expressions we found in Step 1, we get: $$x^2+\left(\frac{y}{2}\right)^2=1^2$$

Step 3: Simplify the equation

Now let's simplify the equation by squaring the terms and rearranging to better represent the equation. We can first square each term: $$x^2 + \frac{y^2}{4} = 1$$ Then, we can multiply both sides by 4 to get rid of the fraction: $$4x^2 + y^2 = 4$$

Step 4: Write the final equation

We now have the single equation in \(x\) and \(y\) that corresponds to the given parametric equations: $$4x^2 + y^2 = 4$$ This is the final equation in \(x\) and \(y\) after eliminating the parameter \(t\) from the given parametric equations.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is fundamental in trigonometry. It states that for any angle \( \theta \), the square of the sine plus the square of the cosine equals 1. Mathematically, \( \sin^2\theta + \cos^2\theta = 1 \). This identity is deeply rooted in the properties of right triangles and the unit circle.

In our problem, we use this identity to eliminate the parameter \( t \). The parametric equations are set in terms of trigonometric functions like \( \sin 8t \) and \( \cos 8t \).

• Using \( x = \sin 8t \), we directly have \( x^2 = \sin^2 8t \).
• Using \( y = 2\cos 8t \), rearrange to \( \cos 8t = \frac{y}{2} \), leading to \( \left(\frac{y}{2}\right)^2 = \cos^2 8t \).

Combining these in our Pythagorean identity gives us \( x^2 + \left(\frac{y}{2}\right)^2 = 1 \), allowing us to link \( x \) and \( y \) without \( t \).

Trigonometric Functions

Trigonometric functions like sine and cosine are essential mathematical functions with applications ranging from physics to engineering. Sine, represented as \( \sin \theta \), measures the vertical component of the angle in a unit circle, while cosine, \( \cos \theta \), measures the horizontal component.

In this exercise, these functions express the relationship between the parameter \( t \) and coordinates \( (x, y) \). The parametric form \( x = \sin 8t \) and \( y = 2\cos 8t \) showcases how these functions dictate the motion or plot in a plane.

• \( x = \sin 8t \) tells us \( x \) varies due to the sine function, often used to model oscillating motion.
• \( y = 2\cos 8t \) stretches the cosine function to impact \( y \). The "2" indicates a scaling that affects the shape's height in a graph.

These parametric equations demonstrate how trigonometric functions are manipulated to create more complex forms and motions in mathematics.

Equation Simplification

Simplification of equations is a process to make equations clearer or more straightforward without changing their meaning. Let's see how this applies here.

Starting with the identity-derived equation \( x^2 + \left(\frac{y}{2}\right)^2 = 1 \), we aim to simplify this expression. Simplification helps in making the equation easier to read and apply:

• First, remove fractions by multiplying the whole equation by 4, resulting in \( 4x^2 + y^2 = 4 \).
• This step is crucial for clarity and ease of use when solving or plotting the equation, avoiding complex numerators or denominators.

The result, \( 4x^2 + y^2 = 4 \), is a simplified equation representing an ellipse when graphed. Simplifying is essential to understand the structure and properties of equations better, enabling further mathematical analysis or applications.