Problem 16 - Algebra and Trigonometry

Question

Use point plotting to graph the plane curve described by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of \( t.\) \(x=-\sin t, y=-\cos t ; 0 \leq t<2 \pi\)

Step-by-Step Solution

Answer

The plane curve described by the given parametric equations \( x=-\sin t \), \( y=-\cos t \) is a unit circle centered at the origin. Including the direction of the parameter \( t \) gives a counterclockwise orientation because we are decreasing the values of \( x \) and \( y \) as \( t \) increases.

Step 1: Understanding the parametric equations

First, note that the wonky sine curve: \( x=-\sin t \) and the cosine curve: \( y=-\cos t \) are given as the parametric equations. These equations give us the x and y coordinates of points on the curve for a given value of parameter t.

Step 2: Generate the points

Generate the points that will be plotted. A simple way to do this is by substituting the range of \( t \) values into the parametric equations. As \( 0 \leq t < 2 \pi \), we could use some frequently encountered angles in this range. Let's choose \( t=0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) and \( 2\pi \). Substituting these values of \( t \) in the parametric equations will yield corresponding \( x \) and \( y \) values.

Step 3: Plot the points

Plot the points in a two-dimensional plane. Remember that each point is given by \( (x, y) \) where \( x=-\sin t \) and \( y=-\cos t \). Label each point corresponding to the \( t \) value it was generated from. This will help in showing the orientation later.

Step 4: Draw the curve

Draw smooth curve through the points that were plotted. Start from the point which corresponds to \( t=0 \) and end at the point which corresponds to \( t=2 \pi \). Ensure that the curve passes through each point in order of the increasing \( t \) value.

Step 5: Show the orientation

Lastly, draw arrows on the curve to indicate the direction it is traced as \( t \) increases. The arrow should start from the point corresponding to \( t=0 \) and direct towards the point corresponding to \( t=2 \pi \). This gives the orientation of the curve.

Key Concepts

Curve Orientation

Understanding the orientation of a curve is essential when we're dealing with parametric equations. The orientation refers to the direction in which a curve is traced as the parameter, commonly denoted as t, increases. Imagine the parameter t as a timestamp on the journey of a point traveling along the curve. As t advances, we track the ordered pair \(x(t), y(t)\) to reveal the path of the curve.

For the given parametric equations \(x=-\sin t\) and \(y=-\cos t\), the curve will start at the point where t is at its minimum in the given interval, which is zero. The endpoint corresponds to t = 2\pi. By marking the curve with arrows, we indicate the path from start to end, aligning with the increments in t. This arrowed path reflects the curve's orientation, and it's a crucial aspect of understanding the dynamic nature of parametric plots.

Plotting Parametric Curves

Plotting parametric curves differs from graphing functions in Cartesian coordinates, where each x value is paired with a single y value. Parametric curves allow for more complex shapes since each coordinate is determined by a separate equation as a function of an independent parameter t.

To plot the curve described by \(x=-\sin t\) and \(y=-\cos t\), you start by selecting values of t from within the given range, which in this case is \(0 \leq t < 2\pi\). Once you have chosen the values, calculate the corresponding x and y for each t and mark these points on a coordinate plane. Gradually, connect these points in the order of increasing t to form the curve. This process gives a visual representation of the trajectory defined by the parametric equations. To aid in understanding, label each plotted point with its corresponding t value. This will clarify how the curve progresses over time.

Sine and Cosine Parametric Equations

Parametric equations using sine and cosine functions are frequent in the study of trigonometric curves. These functions can describe circular or elliptical paths when combined as parametric equations. For instance, the given parametric equations \(x=-\sin t\) and \(y=-\cos t\) are reminiscent of a unit circle with a negative orientation.

The sine and cosine functions usually vary between -1 and 1. As they sweep through their range, the resulting parametric plot creates a circular path for the combination of x(t) and y(t). Here, the negative signs essentially mirror the typical unit circle about the origin. To better visualize these trigonometric parametric equations, identify characteristic values of t that correspond to critical points on the unit circle, such as \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\). Due to periodicity, we expect the curve to close after one full cycle, \(t=2\pi\), returning to its starting point.