Problem 16 - Precalculus: Mathematical for Calculus

Question

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. $$x=\cos 2 t, \quad y=\sin 2 t$$

Step-by-Step Solution

Answer

The curve is a unit circle, given by the equation \( x^2 + y^2 = 1 \).

Step 1: Understand the Parametric Equations

The given parametric equations are \( x = \cos 2t \) and \( y = \sin 2t \). These equations describe a curve where \( x \) and \( y \) are functions of the parameter \( t \). The parameter \( t \) typically varies over a specific interval, often \( 0 \leq t < 2\pi \), though it can be different depending on context.

Step 2: Sketch the Curve

To sketch the curve, note that \( x = \cos 2t \) and \( y = \sin 2t \) describe a transformation of the unit circle. Here, the parameter \( t \) is scaled by 2, meaning this is essentially a unit circle traced as \( t \) varies. This will sketch the full unit circle as \( t \) goes from \( 0 \) to \( \pi \). The full cycle of \( x \) and \( y \) is completed when \( t \) goes from \( 0 \) to \( \pi \), since each loop becomes one full rotation of a circle.

Step 3: Eliminate the Parameter

To find the rectangular equation, eliminate \( t \) by using trigonometric identities. We know that \( x = \cos 2t \) and \( y = \sin 2t \). Use the Pythagorean identity: \( \cos^2 2t + \sin^2 2t = 1 \). Substitute \( x \) and \( y \) into this identity: \( x^2 + y^2 = \cos^2 2t + \sin^2 2t = 1 \). The rectangular-coordinate equation is hence \( x^2 + y^2 = 1 \).

Step 4: Finalize the Sketch and Conclusion

The rectangular equation \( x^2 + y^2 = 1 \) confirms that the parametric curve is indeed a unit circle centered at the origin (0,0). Thus, the curve represented by the parametric equations is a standard circle with radius 1.

Key Concepts

Rectangular-Coordinate Equation

A rectangular-coordinate equation expresses a relationship between the Cartesian coordinates, usually written with the variables \( x \) and \( y \). In the context of parametric equations, we are often tasked with eliminating the parameter, in this case, \( t \), to find this relationship. For the given parametric equations, \( x = \cos 2t \) and \( y = \sin 2t \), we can eliminate \( t \) by using a trigonometric identity. By applying the identity \( \cos^2 2t + \sin^2 2t = 1 \), we quickly see that the direct substitution leads us to \( x^2 + y^2 = 1 \). This equation represents a circle centered at the origin with radius 1 in the rectangular-coordinate system. Thus, converting parametric equations to a rectangular-coordinate form helps visualize and comprehend the geometric shape they represent.

Trigonometric Identities

Trigonometric identities are crucial when dealing with parametric equations. They are formulas that involve angles and allow us to simplify or transform expressions. In this exercise, the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \) was instrumental in converting the parametric equations to a rectangular form. This identity holds for any angle \( \theta \), and since our parametric equations use \( \cos 2t \) and \( \sin 2t \), we can directly apply it. By recognizing this identity, we substitute the trigonometric expressions for \( x \) and \( y \). Therefore, it allows us to simplify the parametric representation to the equation of a unit circle, \( x^2 + y^2 = 1 \). This showcases the power of trigonometric identities in bridging parametric and Cartesian coordinate systems.

Unit Circle

The unit circle is a central concept in trigonometry and is defined as a circle with a radius of 1, centered at the origin of a coordinate plane. It serves as a fundamental tool for understanding characteristic angles and their sine and cosine values. When dealing with the parametric equations \( x = \cos 2t \) and \( y = \sin 2t \), the unit circle comes into play. These equations essentially trace the path around the unit circle as the parameter \( t \) varies. The scaling by 2 in \( \cos 2t \) and \( \sin 2t \) suggests a faster traversal around the circle, where the parameter \( t \) covers one full circle as it ranges from 0 to \( \pi \). Thus, understanding the unit circle helps visualize the parametric equations and interpret them correctly as forming a circle of radius 1.