Problem 11 - 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. $$x=2 \sin t, \quad y=2 \cos t, \quad 0 \leq t \leq \pi$$

Step-by-Step Solution

Answer

The curve is the upper half of a circle: \(x^2 + y^2 = 4\).

Step 1: Understand the Parametric Equations

The pair of parametric equations given are \(x = 2 \sin t\) and \(y = 2 \cos t\). These equations relate the parameters \(t\) to the coordinates \(x\) and \(y\), respectively.

Step 2: Sketch the Parametric Curve

Notice that for parametric equations involving sine and cosine, the shape often resembles part of a circle or ellipse. To sketch the curve, examine values of \(t\) between 0 and \(\pi\). At \(t = 0\), \(x = 0\) and \(y = 2\). At \(t = \pi\), \(x = 0\) and \(y = -2\). This implies the curve is half of a circle of radius 2, moving from top to bottom in the first two quadrants.

Step 3: Eliminate the Parameter \(t\)

To eliminate \(t\), use the trigonometric identity \(\sin^2 t + \cos^2 t = 1\). From the parametric equations, \(\sin t = \frac{x}{2}\) and \(\cos t = \frac{y}{2}\). Substitute these into the identity: \(\left(\frac{x}{2}\right)^2 + \left(\frac{y}{2}\right)^2 = 1\), simplifying to \(\frac{x^2}{4} + \frac{y^2}{4} = 1\).

Step 4: Derive the Rectangular Equation

Multiply through by 4 to clear the fractions: \(x^2 + y^2 = 4\). This is the equation for a circle with radius 2 centered at the origin. However, since \(0 \leq t \leq \pi\), the curve is only the upper half of the circle from \((0, 2)\) to \((0, -2)\).

Key Concepts

Rectangular Coordinate Conversion

Converting from parametric equations to rectangular coordinate equations is a common task in mathematics, especially when dealing with curves such as circles and ellipses. The given parametric equations are \( x = 2 \sin t \) and \( y = 2 \cos t \). These relate a parameter \( t \) to the Cartesian coordinates \( x \) and \( y \). The goal is to express the relationship between \( x \) and \( y \) without explicitly involving \( t \).
To achieve this, we use trigonometric identities to eliminate \( t \). In this instance, the identity \( \sin^2 t + \cos^2 t = 1 \) is particularly useful. By isolating \( \sin t \) and \( \cos t \) in terms of \( x \) and \( y \), which gives \( \sin t = \frac{x}{2} \) and \( \cos t = \frac{y}{2} \), we can substitute into the identity:

• \( \left(\frac{x}{2}\right)^2 + \left(\frac{y}{2}\right)^2 = 1 \)
• \( \frac{x^2}{4} + \frac{y^2}{4} = 1 \)

To simplify, multiply every term by 4, yielding \( x^2 + y^2 = 4 \), the equation of a circle.

Circle Equation

The equation derived from the conversion process \( x^2 + y^2 = 4 \) is the equation of a circle. This equation represents all the points \( (x, y) \) that are 2 units from the origin whenever plotted on a Cartesian plane.
The general formula for a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h,k)\) is the center and \(r\) is the radius. In this situation, \(h = 0\), \(k = 0\), making the center at the origin, and the radius \(r = 2\) since \(2^2 = 4\).
However, it's crucial to remember that the original parametric equations defined \(t\) in the range \(0 \leq t \leq \pi\). This constraint means the equation only describes the upper half of the circle, running from the point \((0, 2)\) to \((0, -2)\).
Drawing this curve will display a semicircle above the x-axis, capturing a core aspect of combining circles and parametrics.

Trigonometric Identities

Trigonometric identities play an essential role in converting parametric equations to rectangular coordinates, as seen in the parametric equations \( x = 2 \sin t \) and \( y = 2 \cos t \). These identities form the backbone of transformation techniques.
The primary identity used in this exercise is \( \sin^2 t + \cos^2 t = 1 \). This identity is one of the Pythagorean identities, crucial for working with circular functions. It connects the squares of sine and cosine, reflecting the fundamental relationship within a right triangle.
When you replaced \( \sin t \) and \( \cos t \) with \( \frac{x}{2} \) and \( \frac{y}{2} \), respectively, you were effectively applying this identity to our problem. The identity allowed us to seamlessly eliminate the parameter \(t\) and rewrite the equations in a format that reveals the underlying circle equation.
Understanding these identities allows for flexible manipulation of equations, transforming them from one form to another while maintaining their inherent properties.