Question

Graph each pair of parametric equations. $$\begin{aligned} &x=\sin \theta\\\ &y=\sin \theta \end{aligned}$$

Short answer

The graph of the parametric equations \(x=\text{sin}\Theta\) and \(y=\text{sin}\Theta\) is a line segment from (-1, -1) to (1, 1) on the line \(y = x\).

Step-by-step solution

Understanding the parametric equations

We have two parametric equations, both involving the same trigonometric function but for different variables. The parametric equations are given by: 1. For the x-coordinate: \(x = \text{sin}\Theta \), 2. For the y-coordinate: \(y = \text{sin}\Theta \). Since both \(x\) and \(y\) are equal, we can infer that the graph will be a line where all points satisfy the condition \(x = y\).

Plotting the equations

To plot the parametric equations, select a range of values for \(\theta \). For \(\theta \) ranging from 0 to \(2\pi\), plug in these values into both the equations to find the corresponding x and y coordinates. Since \(x\) and \(y\) are equal, the resulting points will lie on the line \(y = x\). However, because \(\text{sin}\theta\) only ranges from -1 to 1, the points will be constrained between these values along the line \(y = x\).

Drawing the graph

Draw a set of axes, then draw the line \(y = x\) but only between the coordinates (-1, -1) and (1, 1), since this is the range of the sine function. The complete graph will be a line segment on the line \(y = x\) between these two points.

Key concepts

Trigonometric Parametric Equations

Trigonometric parametric equations are a form of representation where a pair of equations is used to express the coordinates of a point as functions of a separate parameter, often denoted as \theta (theta). These functions are typically trigonometric functions such as sine, cosine, or tangent.

When dealing with trigonometric parametric equations, it's crucial to understand the behaviors of the trigonometric functions involved. For instance, the sine function, which is represented as \text{sin}(\theta), varies between -1 and 1 as \theta changes. The equations: \( x = \text{sin}(\theta) \) and \( y = \text{sin}(\theta) \) mean that at every instance, the x-coordinate and y-coordinate of a point on the graph are both determined by the sine of the same angle, \theta.

Thus, when the exercise projects sin(\theta) in both the x and y dimensions equally, it implies that x will always be equal to y. This knowledge is crucial, as it simplifies the graphing process by reducing the graph to a specific, predictable form.

Plotting Parametric Functions

Plotting parametric functions requires the selection of a range for the parameter, in this case, \theta. Unlike regular Cartesian functions where y is directly dependent on x, parametric equations define both x and y based on a third variable. Here, both x and y are dependent on \theta.

To plot the equations given in our exercise, one would choose a range of values for \theta – commonly from 0 to \( 2\text{pi} \). Each value of \theta yields corresponding values of x and y based on the equations provided. In a typical scenario where the parametric equations differ, the resulting graph could be a curve or some complex shape. However, because our given equations define both x and y with the same function of \theta, points will always align along the line y = x, confined within the range that the sine function allows, from -1 to 1. This demonstrates the importance of understanding both the parameter range and the function's behavior when plotting parametric curves.

Sine Function Visualization

Visualizing the sine function is essential in interpreting trigonometric parametric equations because it provides insight into how the values of x and y coordinates are determined over the parameter's range. The sine curve oscillates between its maximum and minimum values (1 and -1, respectively) and has a period of \( 2\text{pi} \).

In the context of our exercise, since both x and y are functions of sin(\theta), their values will follow this oscillation. However, because they are described by the same sine function, the resulting visualization on a graph is a line, rather than the typical 'wave' shape of a sine curve. The graph of our equation would therefore be a diagonal line along y = x, but only visible within the interval where \theta leads to sine values between -1 and 1. Understanding this helps learners visualize how the trigonometric motion of the sine function is translated into a linear representation in this special case.