Problem 26 - College Trigonometry

Question

Find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=\sqrt{6} \operatorname{cis}\left(\frac{3 \pi}{4}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Given Form

The given complex number is in polar form, expressed as \( z = r \operatorname{cis}(\theta) \), where \( r = \sqrt{6} \) and \( \theta = \frac{3\pi}{4} \). "Cis" stands for \( \cos \theta + i \sin \theta \).

Step 2: Apply the Cis Representation

Convert the polar form to rectangular form using \( \operatorname{cis}(\theta) = \cos(\theta) + i \sin(\theta) \). Thus, \( z = \sqrt{6} (\cos(\frac{3\pi}{4}) + i \sin(\frac{3\pi}{4})) \).

Step 3: Calculate Trigonometric Values

For \( \theta = \frac{3\pi}{4} \), the trigonometric values are \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \) and \( \sin(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2} \).

Step 4: Substitute into the Expression

Substitute the computed trigonometric values into the expression: \[ z = \sqrt{6} \left(-\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right). \]

Step 5: Simplify the Expression

Multiply \( \sqrt{6} \) with both real and imaginary components:\[ z = -\sqrt{6} \cdot \frac{\sqrt{2}}{2} + i \sqrt{6} \cdot \frac{\sqrt{2}}{2}. \]Simplifying further, we get \[ z = -\frac{\sqrt{12}}{2} + i \frac{\sqrt{12}}{2}. \]

Step 6: Finalize the Simplification

Recognize \( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \), so we have:\[ z = -\frac{2\sqrt{3}}{2} + i \frac{2\sqrt{3}}{2}. \]This simplifies to:\[ z = -\sqrt{3} + i \sqrt{3}. \]

Step 7: Solution Summary

The rectangular form of the complex number \( z \) is \( z = -\sqrt{3} + i \sqrt{3} \).

Key Concepts

Polar Form

Polar form is a way of expressing complex numbers using a magnitude and an angle. A complex number in polar form is represented as \( z = r \operatorname{cis}(\theta) = r(\cos \theta + i \sin \theta) \). This format brings a geometrical perspective that simplifies many operations, such as multiplication and division of complex numbers. In polar form:

• \( r \) is the magnitude, or distance, of the complex number from the origin in the complex plane.
• \( \theta \) is the angle formed with the positive real axis, also called the argument of the complex number.

Transforming a complex number into polar form involves calculating \( r = \sqrt{x^2 + y^2} \) where \( x \) and \( y \) are its rectangular components, and the angle \( \theta = \tan^{-1}(\frac{y}{x}) \). This representation is particularly helpful for visualizing complex numbers as it leverages trigonometric functions.

Rectangular Form

Rectangular form of complex numbers expresses the number in terms of its real and imaginary parts as \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. Converting from polar to rectangular form involves using trigonometric identities:

• \( a = r \cos \theta \)
• \( b = r \sin \theta \)

In this transformation:- The real part \( a \) involves the cosine of the angle, linked with the horizontal axis of the complex plane.- The imaginary part \( b \) involves the sine of the angle, relating to the vertical axis.By multiplying the magnitude \( r \) by these trigonometric values, the complex number is smoothly translated into its rectangular components. This form is most useful for addition and subtraction of complex numbers.

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that relate the angles and ratios of right-angled triangles. In the context of complex numbers, these identities help bridge polar and rectangular forms. The two most useful identities here are:

• \( \cos^2 \theta + \sin^2 \theta = 1 \)
• Euler's identity: \( e^{i\theta} = \cos \theta + i \sin \theta \)

When converting from polar to rectangular form, \( \cos(\theta) \) and \( \sin(\theta) \) are derived from these identities for trigonometric calculations. They allow us to express the complex number's components in a meaningful way, determining precise values for both real and imaginary parts through known angles like \( \theta = \frac{3\pi}{4} \). Understanding these identities deepens comprehension of how angles and magnitudes translate into the real-imaginary framework of complex numbers.

Complex Plane

The complex plane is a two-dimensional plane that visualizes complex numbers. This system maps each complex number \( z = a + bi \) to a coordinate point \((a, b)\) where:

• The horizontal axis (real axis) represents the real part \( a \).
• The vertical axis (imaginary axis) represents the imaginary part \( b \).

Each point can also be expressed in polar coordinates as \( (r, \theta) \), providing an angle and a distance from the origin. The complex plane serves as an essential tool for understanding the geometry of complex numbers and aids in operations such as addition, multiplication, and the determination of magnitude and angle.By visualizing complex numbers in this plane, one achieves a clearer picture of their structure and interactions, particularly through geometric transformations associated with multiplication and rotation.