Key Concepts and Formulas

• A complex number $Z$ is purely imaginary if its real part is zero, i.e., $Re(Z) = 0$.
• To simplify a complex fraction, multiply the numerator and denominator by the conjugate of the denominator.
• The conjugate of $a + bi$ is $a - bi$, and $(a+bi)(a-bi) = a^2 + b^2$.
• The solutions to $\cos \theta = \pm \frac{1}{\sqrt{2}}$ are $\theta = n\pi \pm \frac{\pi}{4}$, where $n$ is an integer.

Step-by-Step Solution

Step 1: Simplify the Complex Number to Standard Form ($x+iy$)

We are given the complex number $Z = \frac{1+i \cos \theta}{1-2 i \cos \theta}$. To express this in the form $x+iy$, we multiply both the numerator and the denominator by the conjugate of the denominator, which is $1+2i \cos \theta$. This will eliminate the imaginary term from the denominator.

$Z = \frac{1+i \cos \theta}{1-2 i \cos \theta} \times \frac{1+2 i \cos \theta}{1+2 i \cos \theta}$

Numerator: \begin{align*} (1+i \cos \theta)(1+2 i \cos \theta) &= 1 + 2i \cos \theta + i \cos \theta + 2i^2 \cos^2 \theta \ &= 1 + 3i \cos \theta - 2 \cos^2 \theta \ &= (1 - 2 \cos^2 \theta) + i(3 \cos \theta)\end{align*}

Denominator: \begin{align*} (1-2i \cos \theta)(1+2i \cos \theta) &= 1^2 - (2i \cos \theta)^2 \ &= 1 - 4i^2 \cos^2 \theta \ &= 1 + 4 \cos^2 \theta \end{align*}

Now, substitute these back into the expression for $Z$: $Z = \frac{(1 - 2 \cos^2 \theta) + i(3 \cos \theta)}{1 + 4 \cos^2 \theta}$

Step 2: Separate the Real and Imaginary Parts

We can now clearly identify the real and imaginary components of $Z$: $Z = \frac{1 - 2 \cos^2 \theta}{1 + 4 \cos^2 \theta} + i \left( \frac{3 \cos \theta}{1 + 4 \cos^2 \theta} \right)$

Here, $Re(Z) = \frac{1 - 2 \cos^2 \theta}{1 + 4 \cos^2 \theta}$ and $Im(Z) = \frac{3 \cos \theta}{1 + 4 \cos^2 \theta}$.

Step 3: Apply the Condition for a Purely Imaginary Number

For $Z$ to be purely imaginary, its real part must be zero: $Re(Z) = 0$ $\frac{1 - 2 \cos^2 \theta}{1 + 4 \cos^2 \theta} = 0$

Since the denominator $1 + 4 \cos^2 \theta$ can never be zero, the numerator must be zero: $1 - 2 \cos^2 \theta = 0$

Step 4: Solve the Trigonometric Equation for $\cos \theta$

From the equation above, we solve for $\cos^2 \theta$: $2 \cos^2 \theta = 1$ $\cos^2 \theta = \frac{1}{2}$ Taking the square root of both sides gives us the possible values for $\cos \theta$: $\cos \theta = \pm \sqrt{\frac{1}{2}}$ $\cos \theta = \pm \frac{1}{\sqrt{2}}$

Step 5: Identify all values of $\theta$ in the Interval $[-\pi, 2\pi]$

We need to find all $\theta$ in the interval $[-\pi, 2\pi]$ that satisfy $\cos \theta = \frac{1}{\sqrt{2}}$ or $\cos \theta = -\frac{1}{\sqrt{2}}$.

• If $\cos \theta = \frac{1}{\sqrt{2}}$, then $\theta = \frac{\pi}{4}, -\frac{\pi}{4}, \frac{7\pi}{4}, \frac{9\pi}{4}, \frac{3\pi}{4}, -\frac{5\pi}{4} ...$. In $[-\pi, 2\pi]$, we have $\theta = \frac{\pi}{4}, -\frac{\pi}{4}, \frac{7\pi}{4}, -\frac{7\pi}{4}$
• If $\cos \theta = -\frac{1}{\sqrt{2}}$, then $\theta = \frac{3\pi}{4}, -\frac{3\pi}{4}, \frac{5\pi}{4}, -\frac{11\pi}{4} ...$. In $[-\pi, 2\pi]$, we have $\theta = \frac{3\pi}{4}, -\frac{3\pi}{4}, \frac{5\pi}{4}$

Combining them, the set of all possible values for $\theta$ in the interval $[-\pi, 2\pi]$ is: $\left\{ -\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \right\}$

Step 6: Calculate the Sum of all Possible Values of $\theta$

Now, we sum the identified values of $\theta$: $S = \left(-\frac{3\pi}{4}\right) + \left(-\frac{\pi}{4}\right) + \left(\frac{\pi}{4}\right) + \left(\frac{3\pi}{4}\right) + \left(\frac{5\pi}{4}\right) + \left(\frac{7\pi}{4}\right)$ $S = \frac{-3\pi - \pi + \pi + 3\pi + 5\pi + 7\pi}{4}$ $S = \frac{12\pi}{4}$ $S = 3\pi$

Common Mistakes & Tips

• Remember to consider both positive and negative roots when solving $\cos^2 \theta = \frac{1}{2}$.
• Double-check that all solutions for $\theta$ lie within the specified interval $[-\pi, 2\pi]$.
• Be careful with the signs when summing the values of $\theta$.

Summary

To find the sum of all possible values of $\theta$ for which the given complex number is purely imaginary, we first simplified the complex number to the form $x+iy$. Then, we set the real part equal to zero and solved for $\cos \theta$. Finally, we identified all values of $\theta$ within the interval $[-\pi, 2\pi]$ and summed them up, resulting in $3\pi$.

Final Answer

The final answer is \boxed{3\pi}, which corresponds to option (B).