Problem 10 - Advanced Engineering Mathematics

Question

$$6\left[\cos \left(-\frac{\pi}{6}\right)+i \sin \left(-\frac{\pi}{6}\right)\right]$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Formula

We can recognize the expression inside the brackets as a polar form of a complex number. In polar form, a complex number is expressed as \( r(\cos \theta + i \sin \theta) \), which is also known as Euler's formula. Here, \( r = 6 \) and \( \theta = -\frac{\pi}{6} \).

Step 2: Evaluate \( \cos \left(-\frac{\pi}{6}\right) \)

Recall that cosine is an even function, meaning \( \cos(-x) = \cos(x) \). Thus, \( \cos \left(-\frac{\pi}{6}\right) = \cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \).

Step 3: Evaluate \( \sin \left(-\frac{\pi}{6}\right) \)

Sine is an odd function, which means \( \sin(-x) = -\sin(x) \). So, \( \sin \left(-\frac{\pi}{6}\right) = -\sin \left(\frac{\pi}{6}\right) = -\frac{1}{2} \).

Step 4: Substitute and Simplify

Substitute the values from Steps 2 and 3 into the original expression:\[6 \left(\frac{\sqrt{3}}{2} + i \left(-\frac{1}{2}\right)\right)= 6 \cdot \frac{\sqrt{3}}{2} + 6 \cdot \left(-\frac{1}{2}\right)i\]Simplifying each term, we get:\[3\sqrt{3} - 3i\]

Step 5: Solution of the Expression

The final expression, \( 3\sqrt{3} - 3i \), represents the Cartesian form of the given complex number in polar coordinates.

Key Concepts

Polar Form

Complex numbers can be expressed in different forms. One common way is the polar form. This is where we express a complex number using a magnitude and an angle. It's written as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude, and \( \theta \) is the angle in radians from the positive x-axis.
In the expression from the exercise:

• \( r = 6 \) is the magnitude of the complex number.
• \( \theta = -\frac{\pi}{6} \) represents the angle.

Polar form is especially useful in scenarios involving multiplication and division of complex numbers, as it simplifies computing powers and roots.

Euler's Formula

Euler's formula is a key to connecting polar and Cartesian forms of complex numbers. It states that for any real number \( \theta \), \( e^{i\theta} = \cos \theta + i \sin \theta \). This elegant formula allows us to easily switch between forms and simplifies calculations involving complex numbers.
By using Euler’s formula, the polar form \( r(\cos \theta + i \sin \theta) \) can be rewritten as \( re^{i\theta} \). This form significantly streamlines mathematical operations with complex numbers.
In the given exercise, Euler’s formula highlights the intricate relationship between these forms, providing a clear path from the polar description to any Cartesian representation.

Cartesian Form

The Cartesian form of a complex number is also known as the standard form. It represents the complex number as a combination of a real part and an imaginary part: \( a + bi \).

• Here, \( a = 3\sqrt{3} \) is the real part.
• \( b = -3 \) is the imaginary part.

To convert from polar to Cartesian form, you evaluate the trigonometric functions and multiply them by the magnitude \( r \). This transformation involves computing \( r\cos\theta \) and \( r\sin\theta \), followed by a simple addition of real numbers to form the complex result.
In our solved exercise, the conversion results in the Cartesian form \( 3\sqrt{3} - 3i \), showing the equivalence of different expressions of the same number.