Problem 26 - Calculus III

Question

Eliminate the parameter \(t\), write the equation in Cartesian coordinates, then sketch the graphs of the vector-valued functions. $$ \mathbf{r}(t)=\langle 3 \sin t, 3 \cos t\rangle $$

Step-by-Step Solution

Answer

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

Step 1: Identify Parametric Equations

Given the vector-valued function \( \mathbf{r}(t) = \langle 3\sin t, 3\cos t \rangle \), identify the parametric equations for \( x \) and \( y \): \( x = 3 \sin t \) and \( y = 3 \cos t \).

Step 2: Square and Add the Equations

Square both equations: \( x^2 = (3 \sin t)^2 = 9 \sin^2 t \) and \( y^2 = (3 \cos t)^2 = 9 \cos^2 t \). Then add them: \( x^2 + y^2 = 9 \sin^2 t + 9 \cos^2 t \).

Step 3: Use Trigonometric Identity

Apply the Pythagorean identity \( \sin^2 t + \cos^2 t = 1 \) to the sum: \( x^2 + y^2 = 9(\sin^2 t + \cos^2 t) = 9 \).

Step 4: Write the Cartesian Equation

The resulting equation, after applying the identity, is \( x^2 + y^2 = 9 \). This equation represents a circle centered at the origin with a radius of 3.

Step 5: Sketch the Graph

Using the Cartesian equation \( x^2 + y^2 = 9 \), sketch the graph. It is a circle with center at the origin \((0,0)\) and radius 3. Draw it on the Cartesian plane, ensuring to correctly mark the radii along the axes.

Key Concepts

Parametric Equations

When we deal with vector-valued functions like \( \mathbf{r}(t) = \langle 3 \sin t, 3 \cos t \rangle \), we're often describing a path traced on a plane, based on a parameter. Here, the parameter is \( t \), which is typically related to time. The function gives us two parametric equations:

• \( x = 3 \sin t \)
• \( y = 3 \cos t \)

Parametric equations break down complex motion or curves by describing them in terms of a third variable, often not directly related to any single component of the movement. Changing \( t \) covers how \( x \) and \( y \) change over the path, making sure we capture the full trajectory described by the function. Understanding these equations is crucial because they help transform a dynamic process into manageable algebra that can be manipulated or visualized in standard coordinate systems.

Cartesian Coordinates

Once you have parametric equations, changing them into a Cartesian form makes it easier to understand the overall shape. Cartesian coordinates imply a direct relationship between \( x \) and \( y \) without an additional parameter. For the function given:- Start with \( x = 3 \sin t \)- And \( y = 3 \cos t \).Next, use algebraic manipulation by squaring and adding both equations:

• \( x^2 = 9 \sin^2 t \)
• \( y^2 = 9 \cos^2 t \)

Then, combine them: \( x^2 + y^2 = 9 \sin^2 t + 9 \cos^2 t \). Recognize that the sum of \( \sin^2 t \) and \( \cos^2 t \) is always 1, which simplifies everything to a neat expression we understand as \( x^2 + y^2 = 9 \). This is the equation of a circle centered at the origin \((0,0)\) with a radius of 3, showing that our original vector-valued function indeed repesents a circular path in the Cartesian plane.

Trigonometric Identities

The transition from parametric equations to Cartesian form often requires us to rely heavily on trigonometric identities. Here, to simplify the sum of squares of \( x \) and \( y \), we used:- The Pythagorean identity: \( \sin^2 t + \cos^2 t = 1 \).This identity is fundamental, capturing how the squares of the sine and cosine of an angle always add up to one, regardless of the angle. Let’s break down the steps of application:1. You find \( x^2 = 9 \sin^2 t \).2. Similarly, \( y^2 = 9 \cos^2 t \).3. Adding these gives \( x^2 + y^2 = 9 ( \sin^2 t + \cos^2 t ) \).4. Substituting the identity, we arrive at \( x^2 + y^2 = 9 \).Such trigonometric identities are handy tools in many mathematical transformations and applications. They help simplify expressions and reveal relationships, allowing us to switch between different forms of equations seamlessly. Familiarity with these identities is indispensable when working with curves and circles in physics or engineering.