Problem 9 - 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. $$ \left\\{\begin{array}{l} x=\cos (t) \\ y=\sin (t) \end{array} \text { for }-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}\right. $$

Step-by-Step Solution

Answer

The curve is a semicircle on the right side of the unit circle, oriented counterclockwise.

Step 1: Understand the Parametric Equations

The given parametric equations are \( x = \cos(t) \) and \( y = \sin(t) \), with \( -\frac{\pi}{2} \leq t \leq \frac{\pi}{2} \). These equations describe the coordinates \((x, y)\) on the unit circle as \(t\) varies.

Step 2: Convert to Cartesian Equation

Recognize that \(x = \cos(t)\) and \(y = \sin(t)\) describe a circle on the unit circle formula \( x^2 + y^2 = 1 \). This is because for any \(t\), \( \cos^2(t) + \sin^2(t) = 1 \). Here it describes a semicircle since \(t\) is restricted from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Step 3: Analyze the Interval of t

The parameter \(t\) ranges from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), covering half the unit circle. At \(t = -\frac{\pi}{2}\), the point is \((0, -1)\); at \(t = 0\), it is \((1, 0)\); and at \(t = \frac{\pi}{2}\), it is \((0, 1)\).

Step 4: Determine the Curve's Orientation

The orientation from \(t = -\frac{\pi}{2}\) to \(t = \frac{\pi}{2}\) shows that the curve moves counterclockwise from the bottom of the circle (\(0, -1\)) up to the top (\(0, 1\)).

Step 5: Plot the Curve by Hand

On a coordinate plane, draw a semicircle with a radius of 1 above the x-axis, starting at the point \((0, -1)\) and ending at \((0, 1)\). Indicate the counterclockwise orientation of the curve.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry, representing a circle with a radius of 1 centered at the origin of a coordinate plane. All points on this circle satisfy the equation \( x^2 + y^2 = 1 \). The simplicity of the unit circle makes it incredibly useful for understanding trigonometric functions and their properties.
As the parameter \( t \) varies, the coordinates \((x, y)\) trace out positions on the circle. In the context of the given parametric equations, \( x = \cos(t) \) and \( y = \sin(t) \), each value of \( t \) corresponds to a distinct point on the unit circle. This means that as \( t \) changes from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), the resulting points form a semicircle.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are crucial for understanding relationships within a circle. The unit circle provides a clear visual representation of these functions, where each angle \( t \) can be considered as an angle in standard position.

• \( \cos(t) \) represents the x-coordinate of a point on the unit circle.
• \( \sin(t) \) represents the y-coordinate of a point on the unit circle.

These relationships allow us to easily transform between angles and coordinates. The identities \( \cos^2(t) + \sin^2(t) = 1 \) reflect the definition of the unit circle. Such properties illustrate that regardless of the angle or parameter \( t \), the combination of the squared values of sine and cosine always equals 1.

Cartesian Coordinates

Cartesian coordinates are how we specify points on a 2-dimensional plane using an x-axis and a y-axis. When dealing with parametric equations, these coordinates are determined by expressing both x and y as functions of a parameter, often denoted as \( t \).
In the given problem, the equations \( x = \cos(t) \) and \( y = \sin(t) \) are used to define the x and y coordinates of points on the semicircle. Each particular value of \( t \) within the range \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) provides an exact point, or a pair of Cartesian coordinates, situated on the semicircle of the unit circle. By transitioning from parametric form to Cartesian form through these equations, we're effectively illustrating the geometric positions on this plane.

Curve Orientation

Curve orientation is an important aspect when dealing with parametric equations. It refers to the direction in which the curve is traced as the parameter, here \( t \), varies within its set interval.
For the given parametric equations, as we vary \( t \) from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\), the resulting path on the unit circle is traced counterclockwise. This is crucial for understanding the path's movement: starting from the lowest point \((0, -1)\), passing through the rightmost point \((1, 0)\) at \( t = 0 \), and ending at the topmost point \((0, 1)\). This counterclockwise movement helps in visualizing the flow of the curve and is essential when plotting or understanding the nature of the path.