Problem 83 - Precalculus

Question

Suppose that a complex number \(z\) lies on the circle \(x^{2}+y^{2}=\pi^{2} .\) If \(\cos \left(\frac{\theta}{2}\right)=\frac{1}{2}\) and \(\sin \theta<0,\) find the rectangular form of \(z\)

Step-by-Step Solution

Answer

The rectangular form of \(z\) is \(-\frac{\pi}{2} - \frac{\pi \sqrt{3}}{2} i\).

Step 1: Identify Parameters and Equations

The complex number \(z\) lies on the circle \(x^2 + y^2 = \pi^2\). Also, given that \(\cos\left(\frac{\theta}{2}\right) = \frac{1}{2}\) and \(\sin \theta < 0\), we need to determine the rectangular form of \(z = x + yi\).

Step 2: Determine \(\theta\) from \(\cos\left(\frac{\theta}{2}\right) = \frac{1}{2}\)

The equation \(\cos\left(\frac{\theta}{2}\right) = \frac{1}{2}\) implies that \(\frac{\theta}{2} = \frac{\pi}{3}\) or \(\frac{\theta}{2} = \frac{5\pi}{3}\). Thus, \(\theta = \frac{2\pi}{3}\) or \(\theta = \frac{10\pi}{3}\).

Step 3: Select Valid \(\theta\) Condition

Considering \(\sin \theta < 0\), we rule out \(\theta = \frac{2\pi}{3}\) because its sine is positive. Thus, \(\theta = \frac{10\pi}{3}\) which is equivalent to \(\theta = \frac{4\pi}{3}\) (in the first full rotation).

Step 4: Convert to Rectangular Coordinates

For \(\theta = \frac{4\pi}{3}\), use the unit circle to find \(\cos(\theta) = -\frac{1}{2}\) and \(\sin(\theta) = -\frac{\sqrt{3}}{2}\). Since \(z\) lies on the circle with radius \(\pi\), we have \(z = \pi \cos(\theta) + \pi \sin(\theta) i = -\frac{\pi}{2} - \frac{\pi \sqrt{3}}{2} i\).

Step 5: Verification and Conclusion

Verify if \(x^2 + y^2 = \pi^2\) holds: \((-\frac{\pi}{2})^2 + (-\frac{\pi \sqrt{3}}{2})^2 = \frac{\pi^2}{4} + \frac{3\pi^2}{4} = \pi^2\). The calculations are correct. Thus, the rectangular form of \(z\) is \(-\frac{\pi}{2} - \frac{\pi \sqrt{3}}{2} i\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to express complex numbers, mixing both real and imaginary numbers. When dealing with complex numbers such as \( z = x + yi \), we break it down into two parts:

• The real part, \(x\), which lies along the horizontal axis.
• The imaginary part, \(yi\), which extends along the vertical axis.

By combining these two components, we effectively plot a point in a 2D coordinate system known as the complex plane. Think of rectangular coordinates like addressing a location on a map using north (imaginary) and east (real) directions.
The real focus here is on how both components contribute to the positioning of a point on the circle described by an equation, such as \(x^2 + y^2 = \pi^2\). By determining the x and y values, we find the exact spot for \(z\) on this circle.

Unit Circle

The unit circle serves as a cornerstone in understanding trigonometric functions and complex numbers. It is a circle with a radius of 1, centered at the origin of a coordinate system. This makes it extremely handy for interpreting angles and their respective sine and cosine values.
For example, if you picture going counterclockwise around this circle starting from the positive x-axis, every point \((x, y)\) on the circle satisfies \(x^2 + y^2 = 1\). Here, \(x\) represents \(\cos(\theta)\) and \(y\) represents \(\sin(\theta)\), where \(\theta\) is the angle formed with the positive x-axis.
In this exercise, the knowledge of the unit circle helped identify trigometric values such as \(\cos\left(\frac{1}{2}\theta\right) = \frac{1}{2}\). Knowing where these values lie on the unit circle can guide us to map these onto circles of any radius, like here with radius \(\pi\).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. These identities often simplify complex problems, like converting trigonometric expressions into something more manageable.
Some foundational identities include:

• \(\sin^2(\theta) + \cos^2(\theta) = 1\)
• \(1 + \tan^2(\theta) = \sec^2(\theta)\)
• \(\cos(\theta) = \cos(-\theta)\)
• \(\sin(-\theta) = -\sin(\theta)\)

These identities provide a toolkit for solving equations by reducing them to simpler forms. In this exercise, using these identities allowed us to verify the resultant coordinates of the complex number \(z\). The transformation of \(\cos(\theta) = -\frac{1}{2}\) and \(\sin(\theta) = -\frac{\sqrt{3}}{2}\) comes naturally from these trigonometric standbys when working with angles such as \(\theta = \frac{4\pi}{3}\).