Problem 7 - Advanced Engineering Mathematics

Question

$$2\left(\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}\right)$$

Step-by-Step Solution

Answer

The expression simplifies to \(-\sqrt{3} + i\).

Step 1: Identify the Trigonometric Expressions

The given expression can be recognized as a complex number in polar form: \(2\left(\cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6}\right)\). Here, \(r = 2\), and the angle \(\theta = \frac{5\pi}{6}\).

Step 2: Understand the Polar Form

In polar form, a complex number is represented as \(r(\cos \theta + i\sin \theta)\), which can be converted into Cartesian form using \(a + bi = r\cos \theta + r(i\sin \theta)\).

Step 3: Calculate \(\cos \frac{5\pi}{6}\) and \(\sin \frac{5\pi}{6}\)

Evaluate the trigonometric functions for the angle: \(\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}\) and \(\sin \frac{5\pi}{6} = \frac{1}{2}\).

Step 4: Apply the Values to the Polar Form

Substitute the values from Step 3 into the polar form expression: \[2(\cos \frac{5\pi}{6} + i\sin \frac{5\pi}{6}) = 2\left(-\frac{\sqrt{3}}{2} + i \times \frac{1}{2}\right)\].

Step 5: Simplify the Expression

Distribute the 2 through the terms inside the parentheses: \[-\sqrt{3} + i\]. The corresponding complex number in standard form is \(-\sqrt{3} + i\).

Step 6: Confirm the Cartesian Form

The simplified standard form of the given expression is \(-\sqrt{3} + i\). You can double-check each step and substitutions to ensure accuracy.

Key Concepts

Polar Form

The polar form is an alternative way of expressing complex numbers. This form is particularly useful in various applications, such as in physics and engineering, due to its emphasis on magnitude and angle, which can simplify multiplication and division of complex numbers.
A complex number in polar form is written as:

• \[ r(\cos \theta + i\sin \theta) \]

Here:

• \( r \) is the modulus or absolute value of the complex number, representing the distance from the origin to the point in the complex plane.
• \( \theta \) is the argument, measuring the angle from the positive x-axis to the line representing the complex number.

Using this form, you can easily visualize the complex number's position by plotting \( r \) as the radius and representing \( \theta \) as the angle.Another significant advantage of the polar form is when converting between forms or solving equations, especially when the equations or roots involve powers of complex numbers.

Trigonometric Functions

Trigonometric functions are key components in expressing the polar form of complex numbers. They relate angles to the ratios of the sides of right triangles.
In the context of complex numbers, they help reflect how the number is positioned in the complex plane.
The two primary trigonometric functions used here are:

• Cosine (\( \cos \theta \)) - It represents the x-coordinate of a point on a unit circle at an angle \( \theta \) from the x-axis.
• Sine (\( \sin \theta \)) - It represents the y-coordinate of that same point.

In our example, we needed to evaluate:

• \( \cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2} \)
• \( \sin \frac{5\pi}{6} = \frac{1}{2} \)

These values are crucial for transitioning from polar to Cartesian forms.
Trigonometric identities and functions like these enable mathematicians to analyze periodic phenomena and simplify complex calculations.

Cartesian Form

The Cartesian form of a complex number utilizes the real number line and the imaginary number line for representation. This form is very intuitive for plotting numbers as it directly uses two perpendicular axes.
In Cartesian form, a complex number is written as:

• \( a + bi \)

Here:

• \( a \) is the real part.
• \( b \) is the imaginary part (multiplied by \( i \), the imaginary unit where \( i^2 = -1 \)).

For example, the polar expression \( 2(\cos \frac{5\pi}{6} + i\sin \frac{5\pi}{6}) \) becomes \( -\sqrt{3} + i \) in Cartesian form after calculations.
To convert from polar to Cartesian form, follow these steps:

• Multiply \( r \) by \( \cos \theta \) to find the real part.
• Multiply \( r \) by \( \sin \theta \) to find the imaginary part.

Understanding how to switch between these representations is valuable for calculations involving geometry in the complex plane.