Animated Solution for Mathematics - Complex Numbers: The sum of all possible values of $\theta \in [-\pi ,2\pi ]$, for which $\frac{1+i\cos \theta }{1-2i\cos \theta }$ is purely imaginary, is equal

Imagine you are standing on the complex plane, looking at the number $Z=\frac{1+i\cos \theta }{1-2i\cos \theta }$. The problem asks us to find the values of $\theta$ in the range $[-\pi ,2\pi ]$ that make this number purely imaginary.

For a complex number to be purely imaginary, it must have no real component—it must sit perfectly on the vertical imaginary axis. Algebraically, this is our master key: we must force the real part of $Z$ to be exactly zero.

To isolate the real part, we must rationalize the denominator. We multiply the numerator and the denominator by the complex conjugate of the denominator, which is $1+2i\cos \theta$.

This transforms our expression into:

Finding the Real Soul

Now, let's expand the numerator carefully: $(1+i\cos \theta )(1+2i\cos \theta )$. Multiplying these terms, we get $1+2i\cos \theta +i\cos \theta +2i^2{\cos }^2\theta$.

Since $i^2=-1$, this simplifies beautifully to:

Next, look at the denominator. It follows the form $(a-b)(a+b)=a^2-b^2$. Here, $a=1$ and $b=2i\cos \theta$.

The denominator becomes $1^2-(2i\cos \theta {)}^2$, which is $1-4i^2{\cos }^2\theta$. Since $i^2=-1$, the denominator simplifies to the purely real value $1+4{\cos }^2\theta$.

Our complex number $Z$ is now revealed as:

Setting the Real Part to Zero

We have successfully separated the real and imaginary parts. The real part of $Z$ is:

For $Z$ to be purely imaginary, this real part must be zero. A fraction is zero only when its numerator is zero.

Thus, we arrive at the elegant equation: $1-2{\cos }^2\theta =0$. This simplifies to ${\cos }^2\theta =\frac{1}{2}$, which means $\cos \theta =\pm \frac{1}{\sqrt{2}}$.

Mapping the Solutions

Now, we must find all $\theta$ in the interval $[-\pi ,2\pi ]$ that satisfy $\cos \theta =\pm \frac{1}{\sqrt{2}}$.

Tracing the unit circle: In the negative cycle $[-\pi ,0]$, we find $\theta =-\frac{3\pi }{4}$ and $\theta =-\frac{\pi }{4}$. Moving into the first positive cycle $[0,\pi ]$, we find $\theta =\frac{\pi }{4}$ and $\theta =\frac{3\pi }{4}$. * Finally, in the second positive cycle $[\pi ,2\pi ]$, we find $\theta =\frac{5\pi }{4}$ and $\theta =\frac{7\pi }{4}$.

The Beauty of Symmetry

We have six distinct values for $\theta$: $-\frac{3\pi }{4},-\frac{\pi }{4},\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\text{and }\frac{7\pi }{4}$.

Notice the beautiful symmetry: $-\frac{3\pi }{4}$ cancels with $\frac{3\pi }{4}$, and $-\frac{\pi }{4}$ cancels with $\frac{\pi }{4}$. We are left with:

The journey is complete, and the final answer is $3\pi$.