Problem 6 - Advanced Engineering Mathematics

Question

$$5 \sqrt{2}\left(\cos \frac{7 \pi}{4}+i \sin \frac{7 \pi}{4}\right)$$

Step-by-Step Solution

Answer

The expression simplifies to \(5 - 5i\).

Step 1: Simplify the Angle

The angle \(\frac{7\pi}{4}\) needs to be simplified. It's equivalent to \(-\frac{\pi}{4}\) because \(\frac{7\pi}{4} = 2\pi - \frac{\pi}{4}\). This places the angle in the fourth quadrant.

Step 2: Determine the Trigonometric Values

The trigonometric values for \(\cos \frac{\pi}{4}\) and \(\sin \frac{\pi}{4}\) are \(\frac{\sqrt{2}}{2}\). Because the angle is in the fourth quadrant, \(\cos (-\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\) and \(\sin (-\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}\).

Step 3: Substitute Trigonometric Values into the Expression

Substitute the values of the trigonometric functions into the equation: \[5 \sqrt{2} \left(\cos \left(-\frac{\pi}{4}\right) + i \sin \left(-\frac{\pi}{4}\right)\right) = 5 \sqrt{2} \left(\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\right)\]

Step 4: Simplify the Expression

Multiply \(5\sqrt{2}\) with each term inside the parentheses:\[= 5 \sqrt{2} \cdot \frac{\sqrt{2}}{2} - 5 \sqrt{2} \cdot i \frac{\sqrt{2}}{2}\]Calculate each part separately:\[= 5 \cdot 1 - 5 \cdot i\]\[= 5 - 5i\]

Key Concepts

Trigonometric Form

The trigonometric form of a complex number is a powerful way to express complex numbers using angles and magnitudes. A complex number, generally written as \(a + bi\), can also be represented in the trigonometric form as \(r(\cos\theta + i\sin\theta)\). In this representation:

• \(r\) is the modulus, or magnitude, of the complex number.
• \(\theta\) is the angle, or argument, which the complex number makes with the positive x-axis on the complex plane.

This representation is especially useful for multiplying or dividing complex numbers. In these operations, it simplifies calculations involving the magnitude and angles. The given exercise is expressed in this trigonometric form, which includes converting the complex number into specific cosine and sine values.

Polar Coordinates

Polar coordinates provide a two-dimensional coordinate system that is perfect for working with rotation and angles. In this system, each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

• The distance, often indicated as \(r\), is the length of a line segment from the origin to the point.
• The angle, or \(\theta\), can represent the direction of the point from the positive x-axis.

In the context of the exercise solution, polar coordinates help simplify the complex number \(5 \sqrt{2} (\cos \frac{7 \pi}{4} + i\sin \frac{7 \pi}{4})\). This system allows you to visualize complex numbers as rotations and scalings on the complex plane, making it easier to see how they behave.

Complex Plane

The complex plane is a valuable tool for visualizing complex numbers. It combines real and imaginary parts to form a coordinate system.

• The horizontal axis represents the real component of complex numbers.
• The vertical axis denotes the imaginary component.

Using the complex plane, each complex number is a point with its coordinates given in the form \(a + bi\). On this plane, transformations of complex numbers like additions, subtractions, and multiplications can all be represented as geometric transformations, such as translations and rotations.
In this exercise, the complex plane helps in graphically interpreting the polar form \(r(\cos \theta + i \sin \theta)\). By transforming to the trigonometric form, you can understand how the angle determines the rotation and the modulus determines the size.