Sinusoidal Summations (12/2 – 12/15)
Dave, just after learning about trigonometry is giving his friend Buster a challenge. He notes for a certain value of \(\theta\), the summation
\[\sum_{n = 1}^\infty \frac{\cos(n \theta)}{2^{n-1}} = \cos(\theta) + \frac{\cos(2\theta)}{2} + \frac{\cos(3\theta)}{4} + \dots\]
is equal to \(\frac{1}{3}\). Knowing this, he wants Buster to find the value of the summation
\[\sum_{n = 1}^\infty \frac{\sin(n \theta)}{2^{n-1}} = \sin(\theta) + \frac{\sin(2\theta)}{2} + \frac{\sin(3\theta)}{4} + \dots\]
Help Buster find all possible values of this summation!
Solution:
Let \(x\) be the value of the sine summation and let \(z = e^{i \theta} = \cos(\theta) + i \sin(\theta))\).
Multiplying the sine summation by \(i\) and adding it to the cosine summation gives
\[\frac{1}{3} + xi = z + \frac{z^2}{2} + \frac{z^3}{4} + \dots = \frac{z}{1 – \frac{z}{2}}\]
Solving for \(z\), \[z = 2 – \frac{4}{\frac{7}{3} + xi}\]
Using \(|z| = 1\), we get
\begin{align}
|2 – \frac{12}{7 + 3xi}| &= 1 \\
|14 + 6xi – 12| &= |7 + 3xi| \\
|2 + 6xi| &= |7 + 3xi| \\
36x^2 + 4 &= 9x^2 + 49 \\
x^2 &= \frac{15}{3} \\
x &= \boxed{\pm \sqrt{\frac{15}{3}}}
\end{align}