Problem 83 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: \(4 = x^2 + y^2\)

Step 1: Solve for \(\sin 8t\) and \(\cos 8t\) in the given equations

To solve for \(\sin 8t\), divide both sides of \(x = 2 \sin 8t\) by 2: $$\sin 8t = \frac{x}{2}$$ Similarly, to solve for \(\cos 8t\), divide both sides of \(y = 2 \cos 8t\) by 2: $$\cos 8t = \frac{y}{2}$$

Step 2: Square both equations

Square both equations to prepare for applying the Pythagorean identity: $$(\sin 8t)^2 = \left(\frac{x}{2}\right)^2$$ $$(\cos 8t)^2 = \left(\frac{y}{2}\right)^2$$

Step 3: Add the squared equations

Now, apply the Pythagorean identity by adding both equations: $$(\sin 8t)^2 + (\cos 8t)^2 = \left(\frac{x}{2}\right)^2 + \left(\frac{y}{2}\right)^2$$ Since \(\sin^2 \theta + \cos^2 \theta = 1\), we have: $$1 = \left(\frac{x}{2}\right)^2 + \left(\frac{y}{2}\right)^2$$

Step 4: Simplify the equation

We can multiply the equation by 4 to remove the fractions: $$4(1) = 4\left(\left(\frac{x}{2}\right)^2 + \left(\frac{y}{2}\right)^2\right)$$ Which simplifies to: $$4 = x^2 + y^2$$ This is the final equation expressing the relationship between \(x\) and \(y\) without the parameter \(t\).

Key Concepts

Pythagorean Identity

In trigonometry, the Pythagorean Identity is a fundamental identity expressing the relationship between the sine and cosine of an angle. It states that for any angle \( \theta \):\[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity stems from the Pythagorean theorem and is essential in expressing one trigonometric function in terms of another. In the context of parametric equations, like the ones given in your problem, it helps in eliminating parameters by relating squared trigonometric functions to unity.

• The identity is used extensively in trigonometric proofs and simplifications, making it a cornerstone in understanding the behavior of sine and cosine functions.


• It provides an important relation that is consistent and holds true for all real numbers \( \theta \).


• This identity was crucial in transitioning from functions of \( t \) to an equation independent of \( t \), leading to \( x^2 + y^2 = 4 \).

Understanding how to apply the Pythagorean Identity is key when simplifying or transforming trigonometric expressions and is a powerful tool in the study of circular functions.

Eliminating Parameters

Eliminating parameters from parametric equations involves deriving a single equation in terms of \(x\) and \(y\), free of the parameter. This process makes the relationship between \(x\) and \(y\) more obvious and is typically achieved by using mathematical identities or substitutions.

• Start with the given parametric equations, which are expressions that define \(x\) and \(y\) in terms of a third variable or parameter, like \(t\).


• Solve one or both equations in terms of familiar functions or using functions like sine and cosine.


• Apply identities, such as the Pythagorean Identity, to relate these equations together, allowing the elimination of the parameter.

The provided solution effectively eliminates the parameter \(t\) by using trigonometric identities, leading to the squared terms \((\sin 8t)^2\) and \((\cos 8t)^2\). When added together, according to the Pythagorean Identity, they simplify to \(1\), providing a straightforward and elegant approach to eliminating \(t\). As a result, you express the parametric equations as \(x^2 + y^2 = 4\), showing the relation between \(x\) and \(y\) without the parameter.

Trigonometric Functions

Trigonometric functions, notably sine and cosine, are at the heart of this exercise and much of mathematics involving cycles and waves. They describe the relationship of angles within a right triangle and have numerous properties that make them vital in both theoretical and applied mathematics.Sine and cosine functions are periodic and repeat their values in regular intervals. For any angle \(\theta\),:- \( \sin(\theta) \) is the ratio of the opposite side to the hypotenuse in a right triangle.- \( \cos(\theta) \) is the ratio of the adjacent side to the hypotenuse in a right triangle.In parametric equations, these functions help describe motion along a path, with the parameter often representing time.

• They allow complex motions and curves to be described simply.


• Their properties, like periodicity and the Pythagorean Identity, assist in transitioning parametric equations to Cartesian forms.


• In the exercise, they were instrumental for expressing \(x\) and \(y\) in terms of \(\sin 8t\) and \(\cos 8t\), leading to their simplification.

Understanding trigonometric functions and how they apply to parametric equations is crucial for grasping how complex mathematical relationships can be simplified and solved.