Problem 22 - College Trigonometry

Question

Find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=2 \operatorname{cis}\left(\frac{\pi}{6}\right) $$

Step-by-Step Solution

Answer

The rectangular form is \(\sqrt{3} + i\).

Step 1: Identify the given polar form

The complex number is given as \( z = 2 \operatorname{cis} \left( \frac{\pi}{6} \right) \). This is in polar form where the modulus \( r = 2 \) and the argument \( \theta = \frac{\pi}{6} \). Our goal is to convert this into rectangular form \( a + bi \).

Step 2: Write the formula for conversion

Recall the conversion formula from polar to rectangular form: \( z = r \left( \cos \theta + i \sin \theta \right) \). We will substitute the given values of \( r \) and \( \theta \) into this formula.

Step 3: Substitute values

Substitute \( r = 2 \) and \( \theta = \frac{\pi}{6} \) into the formula: \[z = 2 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right).\]

Step 4: Calculate \(\cos \frac{\pi}{6}\) and \(\sin \frac{\pi}{6}\)

Use the known exact values: \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \) and \( \sin \frac{\pi}{6} = \frac{1}{2} \).

Step 5: Plug in the trigonometric values

Insert the trigonometric values into the expression: \[z = 2 \left( \frac{\sqrt{3}}{2} + i \frac{1}{2} \right).\]

Step 6: Simplify the expression

Distribute the 2 across the terms: \[z = 2 \times \frac{\sqrt{3}}{2} + 2 \times i \frac{1}{2},\] which simplifies to \( z = \sqrt{3} + i \).

Step 7: Final Step: Write in rectangular form

Thus, the rectangular form of the complex number is \( \sqrt{3} + i \).

Key Concepts

Polar to Rectangular Conversion

Converting complex numbers from the polar to the rectangular form can seem daunting at first, but it's quite simple when you use the correct steps. This conversion involves using a specific formula that relies on the modulus and the argument of the complex number in polar form. In this case, the complex number is given as \( z = 2 \operatorname{cis} \left( \frac{\pi}{6} \right) \).

The modulus \( r \) is 2, and the argument \( \theta \) is \( \frac{\pi}{6} \). For the conversion, you'll use the formula:

• \( z = r ( \cos \theta + i \sin \theta ) \)

Substitute \( r \) and \( \theta \) into the formula to convert the complex number to its rectangular form \( a + bi \).

After plugging in the values, we find \( z = 2 ( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} ) \), which simplifies to \( z = \sqrt{3} + i \). This straightforward approach transforms the complex number into the familiar terms we're used to seeing in mathematics.

Complex Numbers

Complex numbers, like the one in this problem, are numbers that have both a real part and an imaginary part. They are expressed in the form \( a + bi \), where \( a \) is the real component and \( bi \) is the imaginary component. The imaginary unit \( i \) is equal to the square root of -1.

In polar form, complex numbers are represented by their magnitude, or modulus, and their direction, or argument. The polar form is useful for understanding the geometric interpretation of complex numbers, as it represents them as points in the plane.

• The magnitude, \( r \), represents the distance from the origin to the point.
• The argument, \( \theta \), is the angle the line makes with the positive real axis.

Understanding complex numbers in both rectangular and polar forms helps with their manipulation and use in various mathematical contexts.

Trigonometric Identities

Trigonometric identities play a crucial role when converting a complex number from polar to rectangular forms. These identities help simplify and compute the trigonometric functions \( \cos \) and \( \sin \) precisely. In this context, the expression \( 2(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}) \) relies on the exact values of the cosine and sine of specific angles.

For \( \frac{\pi}{6} \):

• \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \)
• \( \sin \frac{\pi}{6} = \frac{1}{2} \)

These values are derived from the unit circle and are fundamental in trigonometry. By mastering these identities, you can effortlessly shift between forms and solve complex number problems efficiently. Understanding trigonometric identities not only aids in numerical conversion but also paves the way for more advanced mathematical explorations.