# Thread: Eigenvalues involving trig functions and imaginary/complex numbers

1. ## Eigenvalues involving trig functions and imaginary/complex numbers

Hi overlords.

I have a pretty grand request if someone has the time. In my revision for an upcoming exam there is a fairly advanced (for intro level LA) question regarding eigenvectors. Now, I'm very familiar with what eigenvectors are and how to obtain them, however because of my poor math background I am really unfamiliar with factorising trig functions and have literally no idea how to operate with imaginary/complex numbers. Could someone possibly give a 10 seconds lesson on how to approach this question after the determinant has been found? Everything above the blue line I understand, everything else I need some help.

I understand this is quite a task, but I don't have the time to fully learn all of the above, as well as revise for everything else in time for exams, so I figured I'd ask here as a last resort. No feelings hurt if no takers

2. ## Re: Eigenvalues involving trig functions and imaginary/complex numbers

The first line under you blue line is obtained by expanding the term $(\cos \theta-\lambda)^2=\cos^2 \theta -2\lambda \cos \theta+\lambda^2$ on the line above.

Then you need to use the trig-identity corresponding to Pythagoras' theorem $\cos^2 \theta +\sin^2 \theta =1$ to eliminate the squared trig functions:

$$\begin{array}{ll} (\cos \theta-\lambda)^2+\sin^2 \theta &=\cos^2 \theta -2\lambda \cos \theta+\lambda^2+\sin^2 \theta\\ &=1-2\lambda \cos \theta+\lambda^2 \end{array}$$

To find the roots of $1-2\lambda \cos \theta+\lambda^2$ you use the quadratic formula to get:
$$\begin{array}{ll} \lambda &=\frac{2\cos(\theta \pm \sqrt{4 \cos^2 \theta -4}}{2}\\ &=\frac{2\cos\theta \pm 2\sqrt{ \cos^2 \theta -1}}{2}\\ &=\cos\theta \pm \sqrt{ \cos^2 \theta -1} \end{array}$$

Now we use the same trig-identity we used before $\cos^2 \theta+\sin \theta=1$, so $\cos^2 \theta -1=-\sin \theta$, so we have:
$$\lambda=\cos\theta \pm \sqrt{-\sin^2 \theta}=\cos \theta \pm \rm{i} \sin \theta$$
Then that $\cos$ is an even function and $\sin$ and odd function means that:
$$\lambda=\cos \theta + \rm{i} \sin \pm \theta$$