Problem 85 - A Graphical Approach to Precalculus with Limits

Question

Give the least positive radian measure of \(\theta\) if \(r>0\) and (a) \(r\) cis \(\theta\) has real part equal to 0 (b) \(r\) cis \(\theta\) has imaginary part equal to \(0 .\)

Step-by-Step Solution

Answer

\( \theta = \frac{\pi}{2} \) for zero real part; \( \theta = \pi \) for zero imaginary part.

Step 1: Understanding the Problem

We need to find the smallest positive angle \(\theta\) in radians for complex numbers expressed in polar form \(r\text{cis}\theta = r(\cos\theta + i\sin\theta)\), where the real part or the imaginary part is zero.

Step 2: Real Part Equal to Zero

For the real part to equal zero, we need \(\cos\theta = 0\). The cosine function equals zero at \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\). The smallest positive solution is \(\theta = \frac{\pi}{2}\).

Step 3: Imaginary Part Equal to Zero

For the imaginary part to be zero, we need \(\sin\theta = 0\). The sine function equals zero at \(\theta = 0, \pi, 2\pi\). The smallest positive solution is \(\theta = \pi\).

Step 4: Conclusion

Hence, the least positive radian measure of \(\theta\) is \(\frac{\pi}{2}\) when the real part is zero and \(\pi\) when the imaginary part is zero.

Key Concepts

Polar Coordinates

Polar coordinates offer a different approach to describing the position of a point in the plane. Unlike Cartesian coordinates, which use an \(x, y\) format, polar coordinates use \(r\) and \(\theta\). Here, \(r\) represents the distance from the origin, while \(\theta\) is the angle from the positive \(x\)-axis. This system is particularly useful for problems involving angles and circles.
When converting between polar and Cartesian coordinates, it's essential to remember that: \[ \begin{align*} x &= r \cos \theta, \ y &= r \sin \theta \end{align*} \] In polar form, a complex number is represented as \(r \, \text{cis} \, \theta = r(\cos \theta + i\sin \theta)\), simplifying many trigonometric calculations.
Understanding polar coordinates can provide insights into the geometry of complex numbers and offers a powerful tool for tackling problems in precalculus.

Complex Numbers

Complex numbers, represented as \(a + bi\), where \(a\) is the real part and \(b\) the imaginary part, are crucial in mathematics. These numbers extend the real number line to a complex plane. In the complex plane, each number is a point or vector, with the imaginary unit \(i\) satisfying \(i^2 = -1\).
Polar representation of complex numbers, \(r(\cos \theta + i\sin \theta)\), also known as \(r \text{cis} \theta\), is incredibly insightful, especially with Euler's formula: \(e^{i\theta} = \cos \theta + i \sin \theta\). This representation shows the link between complex numbers and circles, where \(r\) is the magnitude (distance from the origin) and \(\theta\) is the argument (angle from the positive \(x\)-axis).
In exercises involving complex numbers, it's essential to consider both their magnitude and direction to fully understand their geometry and how they interact in mathematical operations.

Trigonometric Functions

Trigonometric functions, including sine and cosine, play a vital role in connecting geometry and algebra. These functions relate the angles of a triangle to the ratios of its sides. With the unit circle, a circle with radius one centered at the origin, trigonometric functions become visual tools.
On the unit circle, \(\cos \theta\) is the \(x\)-coordinate of a point on the circle, while \(\sin \theta\) is the \(y\)-coordinate. As \(\theta\) changes, these values oscillate between -1 and 1. Key angles such as \(0, \frac{\pi}{2}, \pi, 3\pi/2\) exhibit unique characteristics that make them crucial in exercises.
For example, \(\cos \theta = 0\) at \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}\), and \(\sin \theta = 0\) at \(\theta = 0, \pi, 2\pi\). Understanding these relationships helps solve problems like determining when the real or imaginary part of a complex number is zero.