Problem 36 - Advanced Engineering Mathematics

Question

$$\frac{\left[8\left(\cos \frac{3 \pi}{8}+i \sin \frac{3 \pi}{8}\right)\right]^{3}}{\left[2\left(\cos \frac{\pi}{16}+i \sin \frac{\pi}{16}\right)\right]^{10}}=\frac{2^{9}}{2^{10}}\left[\cos \left(\frac{9 \pi}{8}-\frac{10 \pi}{16}\right)+i\left(\frac{9 \pi}{8}-\frac{10 \pi}{16}\right)\right]=\frac{1}{2}\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)=\frac{1}{2} i$$

Step-by-Step Solution

Answer

The final result is \(\frac{i}{2}\).

Step 1: Simplify the Numerator

The numerator is given as \([8(\cos \frac{3\pi}{8}+i \sin \frac{3\pi}{8})]^3\). We will use the formula for multiplying complex numbers in polar form \(r^n [\cos(n\theta) + i \sin(n\theta)]\). Here, \(r = 8\) and \(\theta = \frac{3\pi}{8}\). Compute: \[8^3 = 512\]\[(\cos \frac{3\pi}{8} + i \sin \frac{3\pi}{8})^3 = \cos \left(3 \times \frac{3\pi}{8}\right) + i \sin \left(3 \times \frac{3\pi}{8}\right) = \cos \frac{9\pi}{8} + i \sin \frac{9\pi}{8}\]So, the numerator becomes:\[512(\cos \frac{9\pi}{8} + i \sin \frac{9\pi}{8})\]

Step 2: Simplify the Denominator

The denominator is \([2(\cos \frac{\pi}{16}+i \sin \frac{\pi}{16})]^{10}\). Similarly, use the formula for complex powers:\[2^{10} = 1024\]\[(\cos \frac{\pi}{16} + i \sin \frac{\pi}{16})^{10} = \cos \left(10 \times \frac{\pi}{16}\right) + i \sin \left(10 \times \frac{\pi}{16}\right) = \cos \frac{10\pi}{16} + i \sin \frac{10\pi}{16}\]Thus, the denominator is:\[1024(\cos \frac{10\pi}{16} + i \sin \frac{10\pi}{16})\]

Step 3: Perform Division of Complex Numbers

Now we divide the numerator by the denominator:\[\frac{512(\cos \frac{9\pi}{8} + i \sin \frac{9\pi}{8})}{1024(\cos \frac{10\pi}{16} + i \sin \frac{10\pi}{16})}\]This simplifies to:\[\frac{512}{1024} \cdot \frac{\cos \frac{9\pi}{8} + i \sin \frac{9\pi}{8}}{\cos \frac{10\pi}{16} + i \sin \frac{10\pi}{16}}\]The coefficient \(\frac{512}{1024}\) is \(\frac{1}{2}\). Calculate the angle division:\[\left(\cos \frac{9\pi}{8} + i \sin \frac{9\pi}{8}\right) \left(\cos \left(-\frac{10\pi}{16}\right) + i \sin \left(-\frac{10\pi}{16}\right)\right)\]using the angle subtraction formula gives:\[\cos \left(\frac{9\pi}{8} - \frac{10\pi}{16}\right) + i \sin \left(\frac{9\pi}{8} - \frac{10\pi}{16}\right)\]

Step 4: Final Simplification

Calculate the angle:\[\frac{9\pi}{8} - \frac{10\pi}{16} = \frac{9\pi}{8} - \frac{5\pi}{8} = \frac{4\pi}{8} = \frac{\pi}{2}\]Therefore, the expression inside the cosine and sine becomes:\[\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}\]which simplifies to:\[i\]So, our expression now is:\[\frac{1}{2}(i) = \frac{i}{2}\]

Step 5: Result Verification

The terms have reduced consistently across all calculations. Therefore, the mathematical manipulations and the simplification are verified as correct.

Key Concepts

Polar Form of Complex Numbers

Complex numbers can be expressed in polar form which is often very useful for multiplication and division. The polar form of a complex number is given by:\[ z = r(\cos \theta + i \sin \theta) \]Here:

• \( r \) is the magnitude (or modulus) of the complex number.
• \( \theta \) is the argument (or angle) with respect to the positive x-axis.

To convert a complex number from rectangular (Cartesian) form to polar form, you can use:

• The magnitude \( r = \sqrt{x^2 + y^2} \) where \( z = x + iy \).
• The angle \( \theta = \text{atan2}(y, x) \).

The polar form is particularly handy when performing operations like raising numbers to powers or roots. For example, to raise a complex number \( r(\cos \theta + i \sin \theta) \) to the power of 3, you would compute:\[ r^3(\cos(3\theta) + i \sin(3\theta))\] This transformation simplifies complex calculations significantly.

Trigonometric Functions in Complex Numbers

Trigonometric functions play a crucial role when dealing with complex numbers in polar form. The expressions \( \cos \theta \) and \( \sin \theta \) define the horizontal and vertical coordinates of the point on the unit circle representing the complex number's angle.
When you see a complex number expressed as \( r(\cos \theta + i \sin \theta) \), it fundamentally represents the combination of these trigonometric coordinates scaled by \( r \). This makes use of Euler's formula, which contains the relationships:

• \( e^{i\theta} = \cos \theta + i \sin \theta \)

The change of an angle involved in operations such as multiplication of complex numbers can be understood through these functions. Adjusting the angle \( \theta \) results in rotating the complex number around the origin of the complex plane.Understanding trigonometric functions allows for easy transition between polar and rectangular forms. It also enables smooth division and multiplication, owing to properties of trigonometric identities and operations.

Multiplication and Division of Complex Numbers

Multiplication and division of complex numbers in their polar form is made simpler through De Moivre's Theorem. This theorem states:

• To multiply two complex numbers, multiply their magnitudes and add their angles.
• To divide two complex numbers, divide their magnitudes and subtract their angles.

For example, when multiplying:\[ z_1 = r_1(\cos \theta_1 + i\sin \theta_1)\space \text{and}\space z_2 = r_2(\cos \theta_2 + i\sin \theta_2),\] the product is given by:\[ r_1r_2(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)).\] Similarly, when dividing:\[ \frac{z_1}{z_2} = \frac{r_1}{r_2}(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)).\]These operations illustrate the elegant nature of complex numbers in polar form and provide an intuitive understanding of how multiplication corresponds to scaling and rotating in the complex plane, while division represents undoing these rotations and scalings.