Graph each plane curve defined by the parametric equations for t in [0, 2π] Then find a rectangular equation for the plane curve. See Example 3.


x = 4 sin t , y = 3 cos t

Verified step by step guidance

1. Identify the given parametric equations: \(x = 4 \sin t\) and \(y = 3 \cos t\), where \(t\) ranges from \$0$ to \(2\pi\).
2. Recall the Pythagorean identity: \(\sin^2 t + \cos^2 t = 1\). This identity will help us eliminate the parameter \(t\) to find a rectangular equation.
3. Express \(\sin t\) and \(\cos t\) in terms of \(x\) and \(y\): from \(x = 4 \sin t\), we get \(\sin t = \frac{x}{4}\); from \(y = 3 \cos t\), we get \(\cos t = \frac{y}{3}\).
4. Substitute these expressions into the Pythagorean identity: \(\left(\frac{x}{4}\right)^2 + \left(\frac{y}{3}\right)^2 = 1\).
5. Recognize that this equation represents an ellipse in rectangular coordinates, which is the rectangular form of the given parametric curve.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. In this problem, x and y are given in terms of t, allowing the tracing of the curve as t varies over the interval [0, 2π]. Understanding how to interpret and plot these equations is essential for visualizing the curve.

Eliminating the Parameter to Find a Rectangular Equation

To convert parametric equations into a single rectangular equation involving only x and y, the parameter t must be eliminated. This often involves using trigonometric identities, such as sin²t + cos²t = 1, to relate x and y directly. This step simplifies the curve's description and aids in further analysis.

Graphing Ellipses

The given parametric equations describe an ellipse because x and y are scaled sine and cosine functions with different amplitudes. Recognizing this helps in sketching the curve accurately, knowing the ellipse's axes lengths correspond to the coefficients of sin t and cos t, here 4 and 3 respectively.