Problem 20 - College Trigonometry

Question

In Exercises \(1-20\), plot the set of parametric equations by hand. Be sure to indicate the orientation imparted on the curve by the parametrization. $$ \left\\{\begin{array}{l} x=\sin (t) \\ y=t \end{array} \text { for }-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}\right. $$

Step-by-Step Solution

Answer

The curve starts at \((-1, -\frac{\pi}{2})\) and ends at \((1, \frac{\pi}{2})\), with an upward orientation.

Step 1: Understanding Parametric Equations

Parametric equations express a set of related quantities as explicit functions of an independent variable, often denoted as \( t \), which is called the parameter. Here, we are given the parametric equations \( x = \sin(t) \) and \( y = t \) for \(-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}\).

Step 2: Identify Range of Values

Since \( x = \sin(t) \), the range of \( x \) is determined from the range of the \( t \) values. Given \(-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}\), we know \( \sin(t) \) will range from \(-1\) to \(1\) because the sine function oscillates between these values over one period.

Step 3: Plot Points for Specific Values of t

Choose values for \( t \) within the given range and calculate corresponding \( x \) using \( x = \sin(t) \). For example, for \( t = -\frac{\pi}{2}, 0, \frac{\pi}{2} \): - For \( t = -\frac{\pi}{2}, x = \sin(-\frac{\pi}{2}) = -1, y = -\frac{\pi}{2} \).- For \( t = 0, x = \sin(0) = 0, y = 0 \).- For \( t = \frac{\pi}{2}, x = \sin(\frac{\pi}{2}) = 1, y = \frac{\pi}{2} \).Plot these points on the coordinate plane.

Step 4: Determine Curve's Shape and Orientation

The curve is a sine curve that extends vertically, shown by varying \( y = t \) while \( x = \sin(t) \). The points provided and continuity of sine give the orientation from bottom to top (as \( t \) increases from \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \)).

Step 5: Sketch the Curve

Draw a smooth curve passing through the plotted points. The curve starts from \((-1, -\frac{\pi}{2})\) and ends at \((1, \frac{\pi}{2})\). Arrowheads on the curve should indicate the increasing direction of \(t\), which is upwards.

Key Concepts

Sine Function

The sine function is one of the fundamental functions in trigonometry. It relates the angle in a right triangle to the ratio of the length of the opposite side over the hypotenuse. In the context of parametric equations, we use sine to represent oscillations and periodic behavior.

When you input a variable, such as an angle or a parameter like \( t \), into the sine function, you get a value between

• -1
• and 1.

This property creates a smooth waveform that oscillates. For our specific parametric equation, where \( x = \sin(t) \), as \( t \) ranges from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), the sine function will produce values of \( x \) that also range from \(-1\) to \(1\).

These values, when paired with corresponding \( y = t \) outputs, provide specific points we can plot on a coordinate plane. Remember that the essence of the sine function is its periodic nature and predictable pattern, which makes it a favorite for generating waves and cycles in mathematics and physics.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points and graph equations. It consists of two perpendicular axes:

• The horizontal axis, usually called the \( x \)-axis.
• The vertical axis, known as the \( y \)-axis.

Intersecting at a point called the origin (0, 0), these axes divide the plane into four quadrants.

In parametric equations like the one in our exercise, the coordinate plane allows us to visualize relations between \( x \) and \( y \). By assigning values to a parameter \( t \), we place corresponding points onto this plane. It’s the tool that transforms our abstract equations into visual representations.

To effectively use the coordinate plane, make sure you understand the scales and increments on your axes. Being able to visualize where your curve should lie based on calculated points is essential for accurate plotting.

Plotting Points

Plotting points is the process of placing them onto the coordinate plane based on their \( (x, y) \) values. When dealing with parametric equations, this means selecting specific values of the parameter \( t \) to find corresponding \( x \) and \( y \) values.

Start with a range of values that makes sense for your problem, here from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). Calculate the \( x \) and \( y \) for each chosen \( t \), then mark these points on the coordinate plane.

• For example, at \( t = 0 \), \( x = \sin(0) = 0 \) and \( y = 0 \).
• These values translate directly to the point (0, 0).

By repeating this process for other values of \( t \) within the range, we construct a series of dots that represent our parametric curve.

Once you've plotted enough points, connect them with a smooth curve to represent the overall shape. The precision of your curve depends on how many points you use. More points often mean a more accurate depiction of the function you are graphing.

Graph Orientation

Graph orientation refers to the direction and order in which points are plotted along the curve. With parametric equations, this is determined by how \( t \) progresses over its interval.

In our exercise, as \( t \) moves from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), we notice the curve moves from bottom to top. This is showcased by the increasing values of \( y \) corresponding to \( t \) while \( x \) oscillates within its range.

It’s important to show orientation on your graph. As you connect plotted points, you should add arrowheads indicating this direction of movement. This makes understanding easier, showing how the curve flows naturally as \( t \) increases.

• For example, you might see the arrows moving upwards along the curve from the point \((-1, -\frac{\pi}{2})\) to \((1, \frac{\pi}{2})\).

Properly indicating graph orientation not only clarifies the curve’s direction, but it's also crucial for comprehending the behavior of the parametric equations over the parameter's range.