Problem 16 - Algebra and Trigonometry

Question

\(3-24=A\) 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. $$ X=\sin ^{2} t, \quad y=\cos t $$

Step-by-Step Solution

Answer

The curve is a leftward opening parabola: \( x = 1 - y^2 \).

Step 1: Understanding the Parametric Equations

We are given the parametric equations \( x = \sin^2 t \) and \( y = \cos t \). We need to sketch the curve these represent and convert them into a rectangular equation.

Step 2: Solve for the Common Trigonometric Identity

Recall the Pythagorean identity: \( \sin^2 t + \cos^2 t = 1 \). We'll use this identity to eliminate the parameter \( t \).

Step 3: Express \( \sin^2 t \) in terms of \( y \)

Since \( y = \cos t \), then \( \cos^2 t = y^2 \). Substitute this into the Pythagorean identity to get \( \sin^2 t = 1 - y^2 \).

Step 4: Substitute \( \sin^2 t \) into the Expression for \( x \)

From \( x = \sin^2 t \), substitute \( \sin^2 t = 1 - y^2 \) to get \( x = 1 - y^2 \). This is the rectangular coordinate equation.

Step 5: Identify the Nature of the Curve

The equation \( x = 1 - y^2 \) represents a parabola opening to the left, with vertex at (1,0).

Step 6: Sketch the Curve

Plot the parabola \( x = 1 - y^2 \). It opens to the left from vertex (1,0), indicating that for every point \((x,y)\), \(x\) is one minus the square of \(y\).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that often comes in handy when dealing with parametric equations involving sine and cosine. The identity is expressed as \( \sin^2 t + \cos^2 t = 1 \). It emerges from the properties of a right triangle, where the sum of the squares of the sine and cosine of an angle is always one. This identity is powerful because it allows us to relate sine and cosine to each other without directly involving the angle. For instance, if you have \( y = \cos t \), you can find \( \cos^2 t \) as \( y^2 \). Then with the Pythagorean Identity, you can derive \( \sin^2 t = 1 - y^2 \). By substituting this back into the expression for \( x \), we can eliminate the parameter and convert parametric equations to a rectangular form. This process not only helps in solving exercises but also aids in visualizing the relationship between trigonometric functions.

Rectangular Coordinate Equation

Transforming parametric equations into a rectangular coordinate equation involves removing the parameter to express the equation in terms of \( x \) and \( y \) only. This makes the graph of the equation simpler and often reveals the type of curve represented. For the given parametric equations \( x = \sin^2 t \) and \( y = \cos t \), we express \( \sin^2 t \) in terms of \( y \) by using the Pythagorean Identity. Since \( \cos^2 t = y^2 \), we find that \( \sin^2 t = 1 - y^2 \). Substituting \( \sin^2 t = 1 - y^2 \) into \( x = \sin^2 t \) gives us the rectangular equation: \( x = 1 - y^2 \). This equation signifies that for every point on the curve, \( x \) is calculated by subtracting the square of \( y \) from one. Converting to rectangular form helps in identifying the nature of the curve and facilitates sketching its graph.

Parabola

A parabola is a symmetrical open plane curve, which is an important type of curve in geometry, often represented by quadratic equations. The equation derived from the original parametric equations is \( x = 1 - y^2 \). This is a form of a parabola, specifically opening to the left. The standard form of a parabola opening sideways is \( x = a(y - k)^2 + h \), with vertex \((h,k)\). In our equation, \( x = 1 - y^2 \), we can see it matches the standard form with \( h = 1 \), \( k = 0 \), and \( a = -1 \), making \((1,0)\) the vertex.This parabola opens to the left because as \( y^2 \) becomes larger, \( x \) decreases since it is being subtracted from 1. By recognizing the equation of a parabola, students can easily identify the curve type, locate its vertex, and determine its direction of opening when sketching or analyzing it.