Problem 33 - College Trigonometry

Question

Find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=8 \operatorname{cis}\left(\frac{\pi}{12}\right) $$

Step-by-Step Solution

Answer

The rectangular form is approximately \(7.7272 + i 2.0704\).

Step 1: Understanding 'cis' Notation

The given complex number is expressed in polar form as \( z = r \text{cis}(\theta) \). The term "cis" represents \( \cos(\theta) + i\sin(\theta) \). Here, \( r = 8 \) and \( \theta = \frac{\pi}{12} \).

Step 2: Express the Complex Number Using Trigonometric Functions

Rewrite the complex number using trigonometric identities: \( z = 8(\cos(\frac{\pi}{12}) + i \sin(\frac{\pi}{12})) \).

Step 3: Calculate \( \cos(\frac{\pi}{12}) \) and \( \sin(\frac{\pi}{12}) \)

Use the half-angle identities to find the exact values:\[\cos(\frac{\pi}{12}) = \sqrt{\frac{1 + \cos(\frac{\pi}{6})}{2}} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}} = \sqrt{\frac{2 + \sqrt{3}}{4}} \approx 0.9659\]\[\sin(\frac{\pi}{12}) = \sqrt{\frac{1 - \cos(\frac{\pi}{6})}{2}} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}} = \sqrt{\frac{2 - \sqrt{3}}{4}}\approx 0.2588\]

Step 4: Multiply to Find the Rectangular Form

Substitute the exact values of \( \cos(\frac{\pi}{12}) \) and \( \sin(\frac{\pi}{12}) \) into the expression: \[z = 8(\sqrt{\frac{2 + \sqrt{3}}{4}} + i \sqrt{\frac{2 - \sqrt{3}}{4}})\]This simplifies to: \[z = 8 \times 0.9659 + i(8 \times 0.2588)\]\[z \approx 7.7272 + i(2.0704)\]

Step 5: Write the Rectangular Form

The rectangular form of the given complex number is approximately \( z \approx 7.7272 + i 2.0704 \).

Key Concepts

Polar and Rectangular Conversions

When working with complex numbers, we often see them in two different forms: polar and rectangular. The polar form is handy for multiplication and division, while the rectangular form is great for addition and subtraction. Here, the problem asks us to convert a complex number from polar form, written as \(z = r \text{cis}(\theta)\), into its rectangular form, which looks like \(z = a + bi\).

In the given exercise, \(r = 8\) and \(\theta = \frac{\pi}{12}\). To convert from polar to rectangular, we use the relationship \(\text{cis}(\theta) = \cos(\theta) + i\sin(\theta)\). This means you'll be finding the cosine and sine of the angle \(\theta\) to get the real and imaginary parts of the rectangular form (\(a\) and \(b\), respectively).

So, the task is to find \(a = r \cos(\theta)\) and \(b = r \sin(\theta)\) to redraft the polar into rectangular. This set of conversions is a basic yet crucial skill in dealing with complex numbers, allowing for easier manipulation and calculation.

Trigonometric Identities

Trigonometric identities can feel like our best friends when it comes to solving complex number problems. They allow us to simplify expressions and evaluate trigonometric functions more easily.

In this exercise, you specifically need to find \(\cos(\frac{\pi}{12})\) and \(\sin(\frac{\pi}{12})\). These specific values aren't commonly memorized, so we turn to trigonometric identities to help us. Using identities such as the double angle or half-angle formulas gives us exact values without reaching for numeric approximations right away.

By applying the half-angle identities, such as \(\cos(\frac{\theta}{2}) = \sqrt{\frac{1+\cos(\theta)}{2}}\) and \(\sin(\frac{\theta}{2}) = \sqrt{\frac{1-\cos(\theta)}{2}}\), we break down these tricky angles into something more straightforward. These identities are pivotal for handling problems involving trigonometric functions at uncommon angles.

Complex Numbers

Complex numbers combine real numbers and imaginary numbers into a new type that can be expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. These numbers are represented on a plane where the horizontal axis represents the real numbers and the vertical axis the imaginary.

In this scenario, the given complex number happens initially to be in polar form, \(z = 8 \text{cis}(\frac{\pi}{12})\). This representation in polar coordinates allows us to consider the angle and magnitude: 8 as the magnitude and \(\frac{\pi}{12}\) as the angle from the positive x-axis.

Converting to rectangular form smoothly uses trigonometric functions to identify the precise horizontal and vertical components, providing a direct juxtaposition of real and imaginary parts. Understanding this relationship deepens the insight into how complex numbers behave in various scenarios.

Half-Angle Identities

Half-angle identities are specific trigonometric identities used to find exact values of trigonometric functions at half of a given angle. For example, if you know the angle \(\theta\), you can find \(\sin(\frac{\theta}{2})\) and \(\cos(\frac{\theta}{2})\) accurately.

In the exercise, these identities are put into practice. You start with a known angle of \(\frac{\pi}{6}\) and, using the half-angle formula, find the cosine and sine of \(\frac{\pi}{12}\). This involves some square root manipulation and algebra, but the identities provide a solid pathway to get precise values.

For \(\cos(\frac{\pi}{12})\), you apply the identity \(\cos(\frac{\theta}{2}) = \sqrt{\frac{1+\cos(\theta)}{2}}\). For \(\sin(\frac{\pi}{12})\), it’s \(\sin(\frac{\theta}{2}) = \sqrt{\frac{1-\cos(\theta)}{2}}\). These identities are valuable not only in theoretical math but also throughout applied sciences and engineering where precise trigonometric calculations are crucial.