Problem 9 - A Graphical Approach to Precalculus with Limits

Question

Find a rectangular equation for each curve and describe the curve. $$x=3 \sin t, y=3 \cos t ; \text { for } t \text { in }[-\pi, \pi]$$

Step-by-Step Solution

Answer

The curve is a circle with radius 3, and its equation is \(x^2 + y^2 = 9\).

Step 1: Identify Parametric Equations

The given parametric equations are \(x = 3\sin t\) and \(y = 3\cos t\). These equations describe a parametric curve, where \(t\) varies between \(-\pi\) and \(\pi\).

Step 2: Use Trigonometric Identity

Recall the trigonometric identity \(\sin^2 t + \cos^2 t = 1\). We will use this identity to eliminate the parameter \(t\) and find a relationship between \(x\) and \(y\).

Step 3: Express \(\sin t\) and \(\cos t\) in terms of \(x\) and \(y\).

From \(x = 3\sin t\), express \(\sin t\) as \(\sin t = \frac{x}{3}\). Similarly, from \(y = 3\cos t\), express \(\cos t\) as \(\cos t = \frac{y}{3}\).

Step 4: Substitute into Trigonometric Identity

Substitute the expressions for \(\sin t\) and \(\cos t\) into the identity: \((\frac{x}{3})^2 + (\frac{y}{3})^2 = 1\).

Step 5: Simplify the Equation

Simplify \(\frac{x^2}{9} + \frac{y^2}{9} = 1\) by multiplying the entire equation by 9, which gives: \(x^2 + y^2 = 9\).

Step 6: Identify the Curve

The simplified equation \(x^2 + y^2 = 9\) represents a circle centered at the origin with a radius of 3.

Key Concepts

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where the functions are defined. In this exercise, we utilized the trigonometric identity \( \sin^2 t + \cos^2 t = 1 \). It is one of the most fundamental identities in trigonometry. By expressing the parametric equations \( x = 3\sin t \) and \( y = 3\cos t \), we can use this identity to connect the parametric expressions to form a rectangular equation.
Using this identity, we replace \( \sin t \) and \( \cos t \) with their expressions in terms of \( x \) and \( y \), which are \( \sin t = \frac{x}{3} \) and \( \cos t = \frac{y}{3} \). Substituting these into \( \sin^2 t + \cos^2 t = 1 \), we derive an equation that relates \( x \) and \( y \) directly, effectively eliminating the parameter \( t \). This process is key in translating parametric equations into familiar geometric shapes on the coordinate plane.

Rectangular Equation

A rectangular equation is expressed solely in terms of the Cartesian coordinates \( x \) and \( y \). Unlike parametric equations, which describe curves using a separate parameter such as \( t \), a rectangular equation removes the parameter by relating \( x \) directly to \( y \). In the exercise above, we started with the parametric equations \( x = 3\sin t \) and \( y = 3\cos t \).

By using the trigonometric identity to substitute \( \sin t \) and \( \cos t \), we obtained the rectangular equation \( \frac{x^2}{9} + \frac{y^2}{9} = 1 \). After simplifying, we arrived at the final form \( x^2 + y^2 = 9 \). This transformation allows us to identify the shape of the curve easily in standard terms, without needing the parameter \( t \). Understanding how to convert parametric to rectangular form is a useful skill for analyzing and working with curves in a more intuitive and graphical manner.

Circle Equation

The equation \( x^2 + y^2 = 9 \) represents a circle in the coordinate plane. This is known as the standard form of a circle equation, \( x^2 + y^2 = r^2 \), where \( r \) is the radius of the circle. From the simplified rectangular equation, we can see that the radius squared \( r^2 \) equals 9. Therefore, the radius \( r \) is 3.
Features of this circle include:

• Center: Located at the origin \( (0, 0) \).
• Radius: 3 units.

This circle is the geometric place of all points \( (x, y) \) that are exactly 3 units away from the center. Analyzing circle equations like this one is fundamental in geometry, as circles frequently appear in various mathematical contexts and applications. Recognizing the standard form and understanding its components, such as the center and radius, aid in graphing and visualizing the related curves effectively.