Problem 8 - Advanced Engineering Mathematics

Question

$$4\left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right)$$

Step-by-Step Solution

Answer

The expression evaluates to \(-2 - 2i\sqrt{3}\).

Step 1: Identify the Trigonometric Form

The given expression is in the form of a complex number using trigonometric (or polar) coordinates. It can be written as: \[ 4 \left( \cos \theta + i \sin \theta \right) \] where \( \theta = \frac{4\pi}{3} \).

Step 2: Convert to Rectangular Form

To convert from trigonometric form to rectangular form, use the identities \( \cos \theta = x \) and \( \sin \theta = y \). This gives us: \[ 4(\cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3}) = 4\cos \frac{4\pi}{3} + 4i\sin \frac{4\pi}{3} \]

Step 3: Calculate \( \cos \frac{4\pi}{3} \) and \( \sin \frac{4\pi}{3} \)

The angle \( \frac{4\pi}{3} \) is in the third quadrant, where cosine is negative and sine is negative. Therefore, \( \cos \frac{4\pi}{3} = -\frac{1}{2} \) and \( \sin \frac{4\pi}{3} = -\frac{\sqrt{3}}{2} \).

Step 4: Substitute the Values

Substitute the values of cosine and sine into the equation from Step 2: \[ 4\left(-\frac{1}{2}\right) + 4i\left(-\frac{\sqrt{3}}{2}\right) \] Simplifying, we get: \[ -2 - 2i\sqrt{3} \]

Step 5: Write the Final Expression

The expression in rectangular form is: \(-2 - 2i\sqrt{3}\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing complex numbers in a plane using a distance and an angle. Instead of using a standard Cartesian coordinate system with an x and y axis, polar coordinates describe points by how far they are from the origin (radius) and the angle they make with the positive x-axis. This is particularly useful in complex numbers.

• The distance from the origin is called the modulus, represented as \( r \).
• The angle is known as the argument or angle \( \theta \).

A complex number in polar form uses these two components: \( r \text{cis} \theta \), where \( \text{cis} \theta = \cos \theta + i\sin \theta \). In the given exercise, the complex number is already expressed in this form, with a modulus of 4 and an angle \( \frac{4\pi}{3} \).

Rectangular Form

Rectangular form, also known as Cartesian form, expresses complex numbers like ordinary plane coordinates. Each complex number can be represented as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. This form clearly shows the position of the point in the plane by telling you how far to move along the x-axis (real) and y-axis (imaginary).
For the given expression, converting from the polar (trigonometric) form to rectangular form involves calculating the actual values of cos(\( \theta \)) and sin(\( \theta \)). You use these to find \( a \) and \( b \), enabling you to rewrite the complex number as \(-2 - 2i\sqrt{3}\). This form gives a clearer picture of the number's exact position on the complex plane.

Trigonometric Form

The trigonometric form of complex numbers makes understanding their magnitude and direction straightforward. This form is crucial when performing operations like multiplying and dividing complex numbers.

• Magnitude is represented by the modulus \( r \), which is the length of the vector from the origin to the point.
• The direction is given by the angle \( \theta \) from the positive x-axis.

In mathematical terms, a complex number \( z \) in trigonometric form appears as \( r(\cos \theta + i\sin \theta) \), often abbreviated as \( r \text{cis} \theta \). The provided example shows \( 4( \cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3} ) \), indicating a magnitude of 4 and an angle of \( \frac{4\pi}{3} \).

Conversion of Coordinates

Converting between different representations of complex numbers is a key skill. Such conversions often involve moving from polar to rectangular form or vice versa.

• From polar to rectangular: Use \( x = r\cos \theta \) and \( y = r\sin \theta \) to find the rectangular coordinates \( (a, b) \).
• From rectangular to polar: Calculate \( r = \sqrt{a^2 + b^2} \) and \( \theta = \tan^{-1}(\frac{b}{a}) \).

This transformation helps by emphasizing different aspects of the number. For operations like addition or subtraction, the rectangular form is often more convenient. For multiplication and division, polar (or trigonometric) form is preferred. In the given example, converting follows simple steps by determining cosine and sine values for the given angle, and subsequently computing to find \(-2 - 2i\sqrt{3}\) in rectangular form.