Question

Find the Cartesian form of each complex number: \((a) 2\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)\) \((b) 4 e^{i \pi / 3}\) (c) \(\sqrt{2} e^{-i x / 4}\)

Short answer

(a) \( z = \sqrt{3} + i \) (b) \( z = 2 + 2\sqrt{3} i \) (c) \( z = \sqrt{2}(\cos \frac{x}{4} - i\sin \frac{x}{4}) \)

Step-by-step solution

Expressing in Rectangular Form for (a)

The complex number given is in trigonometric form: \[ z = 2 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right) \]First, find the values of \( \cos \frac{\pi}{6} \) and \( \sin \frac{\pi}{6} \). From trigonometric identities, we know:\[ \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \quad \text{and} \quad \sin \frac{\pi}{6} = \frac{1}{2} \]Substitute these into the original expression:\[ z = 2 \left( \frac{\sqrt{3}}{2} + i \frac{1}{2} \right) \]Simplify by multiplying through by 2:\[ z = \sqrt{3} + i \]

Applying Euler's Formula for (b)

The complex number given in exponential form is:\[ z = 4 e^{i \pi/3} \]Apply Euler's formula to express \( e^{i \pi/3} \) in terms of sine and cosine:\[ e^{i \pi/3} = \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \]Using trigonometric values, substitute:\[ \cos \frac{\pi}{3} = \frac{1}{2} \quad \text{and} \quad \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \]Substitute back:\[ z = 4 \left( \frac{1}{2} + i \frac{\sqrt{3}}{2} \right) \]Simplify by multiplying through by 4:\[ z = 2 + 2\sqrt{3} i \]

Finding Cartesian Form for (c)

The given complex number is:\[ z = \sqrt{2} e^{-i x/4} \]By Euler's formula, express it as:\[ e^{-i x/4} = \cos(-x/4) + i \sin(-x/4) \]Using properties of trigonometric functions, we recall that:\[ \cos(-\theta) = \cos(\theta) \quad \text{and} \quad \sin(-\theta) = -\sin(\theta) \]Thus, it becomes:\[ z = \sqrt{2} \left( \cos \frac{x}{4} - i \sin \frac{x}{4} \right) \]This is the Cartesian form.

Key concepts

Cartesian Form

The Cartesian form of a complex number, also known as the rectangular form, is a way to express a complex number using a combination of a real part and an imaginary part. It takes the form \( z = a + bi \), where \(a\) and \(b\) are real numbers. This representation is crucial because it allows us to visualize complex numbers in a two-dimensional space—the real part \(a\) on the x-axis and the imaginary part \(b\) on the y-axis.

• Real Part (\(a\)): This is the horizontal component. It represents the magnitude along the real axis.
• Imaginary Part (\(b\)): This is the vertical component. It shows the magnitude along the imaginary axis \(i\).

Understanding the Cartesian form helps in performing basic operations such as addition and subtraction easily by separating real and imaginary components. This form is used extensively in engineering and physics to simplify and solve problems involving complex numbers.

Trigonometry

In mathematics, trigonometry plays a pivotal role in understanding and converting complex numbers between different forms. It is instrumental when dealing with complex numbers in polar form, where a complex number is represented as \( z = r(cos \theta + isin \theta) \). Here, \( r \) is the modulus (distance from the origin) and \(\theta\) is the argument (the angle from the positive x-axis).

• Cosine and Sine Functions: These are used to find the horizontal and vertical components of a complex number based on a given angle.
• Values at Specific Angles: Easily recognizable values like \( \cos(\pi/6) = \frac{\sqrt{3}}{2} \) and \( \sin(\pi/6) = \frac{1}{2} \) are crucial for transformations between forms.

Trigonometry allows us to interpret the geometry of complex numbers and is essential for transitioning between the trigonometric and Cartesian forms. Calculators or trigonometric tables can often be used to find the exact cosine and sine values needed for specific angles.

Euler's Formula

Euler's Formula is a remarkable discovery in mathematics that links complex numbers to exponential functions through trigonometry. It states that \( e^{i\theta} = \cos \theta + i \sin \theta \). This formula forms the backbone for expressing complex numbers in exponential form and is particularly useful when multiplying or dividing complex numbers.

• Exponential Form: Euler's formula enables us to express complex numbers as \( z = re^{i\theta} \), where \( r \) is the modulus and \( \theta \) is the argument.
• Simplification of Calculations: By using the exponential form, many operations involving powers and roots of complex numbers become simpler.

Understanding Euler's formula is crucial for students dealing with complex number conversions and manipulations. It serves as a bridge between exponential functions and trigonometric representations, greatly enhancing the flexibility with which complex numbers can be worked upon.