Question

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

Short answer

Question: Eliminate the parameter t from the parametric equations x = 2sin(8t) and y = 2cos(8t). Answer: x^2 + y^2 = 4

Step-by-step solution

Write the parametric equations

We are given the parametric equations: $$x = 2\sin 8t$$ $$y = 2\cos 8t$$

Solve for \(\sin 8t\) and \(\cos 8t\)

Dividing both equations by 2, we get: $$\sin 8t = \frac{x}{2}$$ $$\cos 8t = \frac{y}{2}$$

Apply the Pythagorean identity

Using the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\), we can eliminate the parameter \(\displaystyle t\): $$\left(\frac{x}{2}\right)^2 + \left(\frac{y}{2}\right)^2 = 1^2$$ which simplifies to: $$\frac{x^2}{4} + \frac{y^2}{4} = 1$$

Express the equation in terms of \(x\) and \(y\)

Multiplying both sides of the equation by 4, we obtain the single equation in \(\displaystyle x\) and \(\displaystyle y\): $$x^2 + y^2 = 4$$ The elimination of the parameter \(\displaystyle t\) is complete, and the single equation relating \(\displaystyle x\) and \(\displaystyle y\) is \(\displaystyle x^2 + y^2 = 4\).

Key concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that relate trigonometric functions like sine, cosine, and tangent. These identities play a crucial role in simplifying expressions and solving trigonometric equations.

In the context of parametric equations, these identities help us simplify and manipulate expressions to eliminate parameters. A commonly used identity is the Pythagorean identity:

• \( \sin^2 t + \cos^2 t = 1 \)

This identity is useful because it relates the squares of sine and cosine to 1, giving us the ability to form equations without direct reference to the angle \( t \).

When dealing with parametric equations, identifying the trigonometric identity that applies can simplify the process of expressing equations in terms of \( x \) and \( y \). Simply put, these identities are essential tools in bridging the relationships between these functions.

Elimination of Parameters

The elimination of parameters is a method used to convert parametric equations into a single equation in terms of \( x \) and \( y \). Parametric equations describe a curve by expressing the coordinates as functions of a variable, often called a parameter.

To eliminate the parameter, a common technique involves expressing the trigonometric functions in terms of \( x \) and \( y \), and then using relevant identities, like the Pythagorean identity.

In our example, we had:

• \( x = 2\sin 8t \)
• \( y = 2\cos 8t \)

By dividing both sides of each equation by 2, we expressed \( \sin 8t \) and \( \cos 8t \) as \( \frac{x}{2} \) and \( \frac{y}{2} \) respectively. Combining these using the Pythagorean identity allowed us to eliminate \( t \), leading to a clean equation where \( x \) and \( y \) solely define the relationship.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry. It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In algebraic terms, for a triangle with sides \( a \), \( b \), and hypotenuse \( c \), the theorem is expressed as:

• \( a^2 + b^2 = c^2 \)

In the realm of trigonometry, this idea extends to form the Pythagorean identities. Specifically, \( \sin^2 \theta + \cos^2 \theta = 1 \) derives from the theorem by considering the unit circle, where the radius is always 1.

During the process of eliminating parameters from parametric equations like \( x = 2\sin 8t \) and \( y = 2\cos 8t \), the Pythagorean theorem aids in combining \( \sin \,t \) and \( \cos \,t \) into a single, simplified equation, resulting in a geometric representation that elegantly captures the relationship between \( x \) and \( y \). This not only shows the interconnectedness of algebra and geometry but also unifies the concept across disciplines.