Calculating the Value of a Complex Expression

We are asked to find the value of the complex expression \({\left( {\frac{{\cos \theta + i\sin \theta }}{{i\cos \theta + \sin \theta }}} \right)^4}\). To do this, we first need to simplify the base of the expression, which is the fraction inside the parenthesis.

Simplifying the Base Complex Number

The base expression is \({\frac{{\cos \theta + i\sin \theta }}{{i\cos \theta + \sin \theta }}}\). Let's analyze the numerator and the denominator separately in terms of polar form or Euler's formula \(e^{ix} = \cos x + i\sin x\).

• The numerator is \( \cos \theta + i\sin \theta \). This is already in the standard polar form \(r(\cos \phi + i\sin \phi)\) with \(r=1\) and \( \phi = \theta \). In exponential form, this is \( e^{i\theta} \).
• The denominator is \( i\cos \theta + \sin \theta \), which can be written as \( \sin \theta + i\cos \theta \). To express this in polar form \(r(\cos \phi + i\sin \phi)\), we find its magnitude and argument.
  • Magnitude: \( | \sin \theta + i\cos \theta | = \sqrt{\sin^2 \theta + \cos^2 \theta} = \sqrt{1} = 1 \).
  • Argument: Let the argument be \( \phi \). Then \( \cos \phi = \sin \theta \) and \( \sin \phi = \cos \theta \). This implies \( \phi = \frac{\pi}{2} - \theta \).
  So the denominator is \( \cos(\frac{\pi}{2} - \theta) + i \sin(\frac{\pi}{2} - \theta) \). In exponential form, this is \( e^{i(\frac{\pi}{2} - \theta)} \).

Now we can write the base expression as the division of two complex numbers in polar or exponential form:

Base expression \( = \frac{\cos \theta + i \sin \theta}{\cos(\frac{\pi}{2} - \theta) + i \sin(\frac{\pi}{2} - \theta)} \)

Using the property for division of complex numbers in polar form, \( \frac{r_1 (\cos \theta_1 + i \sin \theta_1)}{r_2 (\cos \theta_2 + i \sin \theta_2)} = \frac{r_1}{r_2} (\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)) \). Here, \( r_1 = 1 \), \( \theta_1 = \theta \), \( r_2 = 1 \), and \( \theta_2 = \frac{\pi}{2} - \theta \).

So, the base expression is:

\( 1 \cdot (\cos(\theta - (\frac{\pi}{2} - \theta)) + i \sin(\theta - (\frac{\pi}{2} - \theta))) \)

\( = \cos(\theta - \frac{\pi}{2} + \theta) + i \sin(\theta - \frac{\pi}{2} + \theta) \)

\( = \cos(2\theta - \frac{\pi}{2}) + i \sin(2\theta - \frac{\pi}{2}) \)

This is the simplified form of the base complex number.

Applying De Moivre's Theorem

The question asks for the value of the base expression raised to the power of 4. We have the base in the form \( \cos \phi + i \sin \phi \), where \( \phi = 2\theta - \frac{\pi}{2} \). We need to calculate \( (\cos \phi + i \sin \phi)^4 \).

According to De Moivre's theorem, \( (\cos \phi + i \sin \phi)^n = \cos(n\phi) + i \sin(n\phi) \). Here \( n=4 \) and \( \phi = 2\theta - \frac{\pi}{2} \).

Applying De Moivre's theorem:

\( {\left( {\cos(2\theta - \frac{\pi}{2}) + i\sin(2\theta - \frac{\pi}{2})} \right)^4} = \cos(4(2\theta - \frac{\pi}{2})) + i \sin(4(2\theta - \frac{\pi}{2})) \)

\( = \cos(8\theta - 2\pi) + i \sin(8\theta - 2\pi) \)

Since the cosine and sine functions have a period of \( 2\pi \), \( \cos(x - 2\pi) = \cos x \) and \( \sin(x - 2\pi) = \sin x \).

So, \( \cos(8\theta - 2\pi) + i \sin(8\theta - 2\pi) = \cos(8\theta) + i \sin(8\theta) \).

The value of the given expression is \( \cos 8\theta + i \sin 8\theta \).

Final Value of the Expression

The computed value \( \cos 8\theta + i \sin 8\theta \) matches one of the given options.

Let's verify the steps involved in simplifying the complex number fraction and applying De Moivre's theorem to find the final value.

The steps were:

1. Identify the numerator and denominator of the base expression.
2. Convert both numerator and denominator into polar form \( r(\cos \phi + i \sin \phi) \).
3. Use the division rule for polar forms: \( \frac{r_1 (\cos \theta_1 + i \sin \theta_1)}{r_2 (\cos \theta_2 + i \sin \theta_2)} = \frac{r_1}{r_2} (\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)) \).
4. Simplify the resulting angle.
5. Apply De Moivre's theorem \( (\cos \phi + i \sin \phi)^n = \cos(n\phi) + i \sin(n\phi) \) to raise the simplified base to the power of 4.
6. Simplify the final angle using periodicity of sine and cosine.

Following these steps leads to the result \( \cos 8\theta + i \sin 8\theta \).