Problem 27 - Contemporary Precalculus

Question

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}$$

Step-by-Step Solution

Answer

Question: Express the complex number \(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}\) in the form \(a + bi\). Answer: The complex number can be expressed as \(i\), where \(a = 0\) and \(b = 1\).

Step 1: Evaluate \(\cos \frac{\pi}{2}\) and \(\sin \frac{\pi}{2}\)

To obtain the real numbers \(a\) and \(b\), we first need to evaluate \(\cos \frac{\pi}{2}\) and \(i \sin \frac{\pi}{2}\). Remember that the sine and cosine of common angles like \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \(2\pi\) should be memorized for convenience. Using this knowledge, we have: $$\cos \frac{\pi}{2} = 0$$ $$\sin \frac{\pi}{2} = 1$$

Step 2: Replace with the evaluated values

Now that we have evaluated the sine and cosine, we can substitute these values back into the expression we obtained from Step 1: $$\cos \frac{\pi}{2} + i \sin \frac{\pi}{2} = 0 + i(1)$$

Step 3: Simplify the expression

Finally, we can simplify the expression to obtain the final form of the complex number: $$0 + i(1) = i$$ The complex number is expressed in the form \(a + bi\) as \(\boxed{i}\). Here, \(a = 0\) and \(b = 1\).

Key Concepts

Trigonometric form of complex numbers

Complex numbers can be written in different forms, one of which is their trigonometric or polar form. This form is very useful for certain mathematical operations, such as multiplication and division. In this context, a complex number in trigonometric form is expressed as \[a + bi = r(\cos \theta + i \sin \theta)\]where:

• \(r\) is the magnitude (or modulus) of the complex number, calculated by \(r = \sqrt{a^2 + b^2}\).
• \(\theta\) is the argument (or angle) of the complex number.

This angle \(\theta\) is measured from the positive real axis to the line representing the complex number in the complex plane. Trigonometric form is especially useful in converting between rectangular and polar coordinates. It simplifies calculations that involve powers and roots of complex numbers and highlights the geometric interpretation of those numbers.
Understanding this concept will also make it easier to comprehend how Euler's formula, \(e^{i\theta} = \cos \theta + i \sin \theta\), relates to complex numbers, as it shows the innate relationship between exponential functions and trigonometry in the complex plane.

Sine and cosine values

Sine and cosine are fundamental trigonometric functions that describe relationships in right triangles and circular motion. Here, they help us express angles and lengths of arcs in the unit circle. The values of sine and cosine functions at key angles \((0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi)\) are particularly important because they simplify the expressions on the unit circle.
For example, the value \(\cos \frac{\pi}{2}\) is 0, and \(\sin \frac{\pi}{2}\) is 1. This is directly used in expressing complex numbers like \(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}\). These values are crucial because they represent the limits and extents of wave functions and oscillations, such as the peaks and troughs of sine waves.
Remember that:

• The cosine function represents the horizontal distance from the center of the unit circle along the x-axis.
• The sine function represents the vertical distance along the y-axis.

Memorizing sine and cosine values for these standard angles allows you to solve complex number exercises more quickly, as these exercises often fall back on these fundamental trigonometric identities.

Imaginary numbers

Imaginary numbers are a concept in mathematics that extends our number system to include solutions to equations like \(x^2 + 1 = 0\), which don't have solutions in the real number system. Imaginary numbers are defined using \(i\), where \(i = \sqrt{-1}\). This creates a new category of numbers beyond the real numbers, namely complex numbers, which are written as \(a + bi\), where \(a\) and \(b\) are real numbers.
Imaginary numbers behave like real numbers, but they have unique properties:

• \(i^2 = -1\), which is fundamental to their definition.
• They allow for complex arithmetic, which is essential in fields like electrical engineering and quantum physics.

In real-world applications, imaginary numbers help model oscillatory systems like AC circuits. In mathematics, they help in solving polynomials whose equations would otherwise lack roots in the realm of real numbers. Understanding imaginary numbers and their properties allows for deeper insights into the behavior of waves and other phenomena in both theoretical and applied sciences.