Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Find a rectangular equation for each curve and describe the curve. $$x=2 \cos ^{2} t, y=2 \sin ^{2} t ; \text { for } t \text { in }\left[0, \frac{\pi}{2}\right]$$

Step-by-Step Solution

Answer

The curve is a line segment given by \(x + y = 2\).

Step 1: Recall Trigonometric Identity

First, recall the fundamental trigonometric identity: \[\cos^2 t + \sin^2 t = 1\].

Step 2: Express Trigonometric Expressions in Terms of x and y

Given that \(x = 2 \cos^2 t\) and \(y = 2 \sin^2 t\), rewrite these expressions:Divide both sides by 2, we have \( \cos^2 t = \frac{x}{2} \) and \( \sin^2 t = \frac{y}{2} \).

Step 3: Substitute into the Identity

Substitute the expressions for \( \cos^2 t \) and \( \sin^2 t \) into the identity from Step 1:\[\frac{x}{2} + \frac{y}{2} = 1\].

Step 4: Simplify the Equation

Multiply the entire equation by 2 to remove the fractions:\[x + y = 2\].

Step 5: Describe the Curve

The equation \(x + y = 2\) represents a straight line with a slope of -1 in the xy-plane. The curve is a segment of this line for the given parametric interval.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental in the study of trigonometry and are mathematical equations that involve trigonometric functions. These identities are true for all values of the included variables for which the functions are defined. A key example is the Pythagorean identity:

• \( \cos^2 t + \sin^2 t = 1 \)

This identity expresses the intrinsic relationship between the sine and cosine of an angle. In this exercise, this identity plays a pivotal role in transitioning from parametric equations to a rectangular equation. By substituting the segments \( \cos^2 t \) and \( \sin^2 t \) in terms of \( x \) and \( y \), we could recast the expressions in a way that reveals the geometric nature of the equation.

Parametric Equations

Parametric equations provide a method of expressing a geometric object, such as a curve, in terms of one or more variables called parameters. Instead of directly relating \( x \) and \( y \), each of these coordinates is expressed individually as a function of a parameter, often denoted as \( t \). In our original exercise, the parametric equations provided are:

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

By using these equations, we can describe motion or positions on the curve more flexibly. Within the interval \( t \text{ in } \left[0, \frac{\pi}{2}\right] \), these equations help us piece together the behavior of \( x \) and \( y \). Such equations are beneficial in many real-world applications because they allow us to model complex curves easily with simple equations. Moving to a rectangular form can simplify the understanding of the curve's overall shape.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. It allows us to represent geometric figures and establish the properties of shapes through algebraic equations. In the exercise, we've transformed the parametric equations into the rectangular form:

• \( x + y = 2 \)

This equation is a straightforward linear equation representing a line in the coordinate plane with a slope of \(-1\). The choice of a rectangular equation like \( x + y = 2 \) permits easy graph plotting and interpretation because it illustrates how \( x \) and \( y \) are interdependent. It offers a concise way to describe the geometric relationship simplified from the parametric form. This method of translating between parametric and rectangular forms highlights the power of coordinate geometry in problem-solving and geometry analysis.