Problem 22 - 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. $$x=\cos ^{3} t, \quad y=\sin ^{3} t, \quad 0 \leq t \leq 2 \pi$$

Step-by-Step Solution

Answer

The curve is an astroid, expressed as \(x^{2/3} + y^{2/3} = 1\) in rectangular coordinates.

Step 1: Understand the Parametric Equations

The given parametric equations are \(x = \cos^3 t\) and \(y = \sin^3 t\). These define a path in the xy-plane as the parameter \(t\) varies from 0 to \(2\pi\). The equations involve trigonometric functions, suggesting the curve has some kind of cyclical or periodic nature.

Step 2: Sketch the Curve

The parametric equations describe a path traced out by the point \((x, y)\) as the parameter \(t\) changes. We compute a few key points by plugging in values of \(t\):- For \(t = 0\), \(x = 1\), \(y = 0\), giving the point \((1, 0)\).- For \(t = \frac{\pi}{2}\), \(x = 0\), \(y = 1\), giving the point \((0, 1)\).- For \(t = \pi\), \(x = -1\), \(y = 0\), giving the point \((-1, 0)\).- For \(t = \frac{3\pi}{2}\), \(x = 0\), \(y = -1\), giving the point \((0, -1)\).- For \(t = 2\pi\), \(x = 1\), \(y = 0\), back to \((1, 0)\).These points indicate the curve is closed and forms a shape similar to a distorted circle, specifically a kind of astroid.

Step 3: Eliminate the Parameter

To find a rectangular-coordinate equation, express \(x\) and \(y\) in terms of \(\cos t\) and \(\sin t\):- Start with \(x = \cos^3 t\) and take the cube root: \(\cos t = x^{1/3}\).- Similarly, from \(y = \sin^3 t\), find \(\sin t = y^{1/3}\).- Use the Pythagorean identity \(\cos^2 t + \sin^2 t = 1\), and substitute:\[(x^{1/3})^2 + (y^{1/3})^2 = 1\]- Simplify to get the rectangular-coordinate equation:\[x^{2/3} + y^{2/3} = 1\].

Step 4: Conclusion

The given parametric equations trace out a path that repeats every \(2\pi\): it forms a closed loop known as an astroid.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to represent points on a plane using an ordered pair of numbers, typically \((x, y)\). They are also commonly referred to as Cartesian coordinates. Think of them as a grid system on a map, where you pick a point by telling how far to move horizontally (x-axis) and vertically (y-axis).
Points in this system can be derived from parametric equations, which define both the x and y coordinates in terms of another variable, commonly known as a parameter, like \(t\). Converting parametric equations to rectangular coordinates involves eliminating this parameter to find a relationship between \(x\) and \(y\) alone.
In our exercise, this transformation was achieved through the use of a mathematical identity, resulting in the equation \(x^{2/3} + y^{2/3} = 1\). This equation defines a specific closed curve known as an "astroid," a term derived from how its shape resembles a star.

Trigonometric Functions

Trigonometric functions are mathematical functions related to the angles of triangles and oscillations. These functions, such as sine, cosine, and tangent, are fundamental in various areas of mathematics and physics. They describe the relationship between the angles and lengths of a right triangle.
In our problem, the trigonometric functions \(\cos t\) and \(\sin t\) were used in the parametric equations \(x = \cos^3 t\) and \(y = \sin^3 t\). These functions have cyclical properties, meaning they repeat their values in a regular and predictable pattern over intervals.
This periodic nature makes them highly useful for modeling phenomena that repeat over time, such as waves or circular motion. For a complete cycle of these functions, the parameter \(t\) spans from \(0\) to \(2\pi\), covering all possible combinations of \(x\) and \(y\) before looping back to the start.
Understanding trigonometric functions and their behavior is crucial for transitioning between parametric equations and rectangular representations.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry. It states that for any angle \(t\), \(\cos^2 t + \sin^2 t = 1\). This identity is rooted in the Pythagorean theorem, which relates the lengths of the sides of a right triangle.
In the context of our parametric equations, this identity served as a bridge to convert parameters into rectangular coordinates. By expressing \(x\) and \(y\) through the cube roots of trigonometric functions, this identity allowed us to substitute and simplify their squares.
The transformation led to a new form, \((x^{1/3})^2 + (y^{1/3})^2 = 1\), which was further simplified to the equation \(x^{2/3} + y^{2/3} = 1\).
The use of the Pythagorean identity is crucial in problems involving circles or ellipses, where it helps describe the relationship between angles and radii.