Problem 33 - Algebra and Trigonometry

Question

Eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that \(-\infty< t <\infty.\)) \(x=1+3 \cos t, y=-1+2 \sin t ; 0 \leq t \leq \pi\)

Step-by-Step Solution

Answer

The rectangular equation for the given parametric equations is \((x-1)^2/9 + (y+1)^2/4 = 1\). It represents an ellipse centered at (1, -1) with semimajor axis of length 3 along the x-axis and semiminor axis of length 2 along the y-axis. The orientation of the curve goes counterclockwise from the quadrant I to II.

Step 1: Solve parametric equations in terms of \(t\)

Rearrange two given parametric equations to solve for \(t\), giving \(\cos t = (x-1)/3\) and \(\sin t = (y+1)/2\).

Step 2: Eliminate \(t\)

Given that \(\sin^2 t + \cos^2 t = 1\), eliminate \(t\) to give a rectangular equation. Substitute \(\cos t = (x-1)/3\) and \(\sin t = (y+1)/2\) into the equation, and after simplifying we get \((x-1)^2/9 + (y+1)^2/4 = 1\).

Step 3: Identify the curve and the orientation

The curve is an ellipse centered at (1, -1), with semimajor axis of length 3 along the x-axis and semiminor axis of length 2 along the y-axis. Due to \(0 \leq t \leq \pi\), the orientation of the curve is counterclockwise and limited to the first and second quadrants because the cosine is positive and the sine ranges into negative.

Step 4: Sketch the curve

Start by sketching an ellipse centered at (1, -1) with axes of lengths 3 and 2. Draw a counterclockwise orientation, indicating the direction of increasing values of \(t\). It starts from the rightmost point when \(t=0\) and ends at the leftmost point when \(t=\pi\). Use arrows to indicate the orientation.

Key Concepts

Rectangular Equation

Converting parametric equations into a rectangular equation involves eliminating the parameter, typically denoted by \(t\), from the equations. Parametric equations represent a set of related pairs \((x(t), y(t))\), where \(t\) traces our curve as it varies over a certain range. By removing \(t\), we obtain a rectangular equation in \(x\) and \(y\), directly relating them without an explicit parameter.
This process helps in identifying the type of curve represented. In our example, we start with

• \(x = 1 + 3 \cos t\)
• \(y = -1 + 2 \sin t\)

We simplify these into trigonometric identities, ultimately arriving at the ellipse equation: \[\frac{(x-1)^2}{9} + \frac{(y+1)^2}{4} = 1.\]
Having a rectangular equation makes graphing the curves and understanding their properties easier, as it is more straightforward to then identify forms such as ellipses or circles.

Ellipse Sketch

Once the rectangular equation is established, the next step is to sketch the corresponding ellipse. An ellipse is defined as the set of points such that the sum of the distances to two given points (foci) is constant.
For our example, the rectangular equation \[\frac{(x-1)^2}{9} + \frac{(y+1)^2}{4} = 1\]
indicates an ellipse with the center at \((1, -1)\). The denominators \(9\) and \(4\) under the squared terms specify the lengths of the axes:

• The semimajor axis length is \(3\), extended along the \(x\)-axis.
• The semiminor axis length is \(2\), running along the \(y\)-axis.

To sketch, begin by plotting the center, then extend the axes symmetrically based on these lengths. Note where each major and minor axis intersects the ellipse's perimeter to ensure accuracy.

Parameter Elimination

Parameter elimination is a crucial step for transforming parametric equations into a more understandable form. This involves solving each parametric equation for the parameter \(t\), and then substituting these into a trigonometric identity. Here’s how we proceeded:
From the given equations

• \(\cos t = \frac{x-1}{3}\)
• \(\sin t = \frac{y+1}{2}\)

Using the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\), we substitute to eliminate \(t\).
This yields:\[\left(\frac{x-1}{3}\right)^2 + \left(\frac{y+1}{2}\right)^2 = 1.\]
This step simplifies the understanding of the properties of the curve and leads directly to the sketching phase.

Trigonometric Identities

Trigonometric identities play a significant role in parameter elimination. These identities are mathematical equations involving trigonometric functions that are true for any angle. One common identity is the Pythagorean identity:
\[\sin^2 t + \cos^2 t = 1\]
In our parametric scenario, we took advantage of this identity to eliminate \(t\) by substituting expressions for \(\cos t\) and \(\sin t\).

• Through substitution, \(\cos t = \frac{x-1}{3}\) and \(\sin t = \frac{y+1}{2}\),
• The squared terms are then placed into the identity.

Solving results in a rectangular equation that defines the curve in terms of \(x\) and \(y\). Such identities are valuable in linking trigonometric strategies to straightforward algebra, providing insights not only into the shapes involved but also their orientation and behaviors at specific parameter intervals.