Recently I was reminded of the trickiness of inverse trigonometric functions (e.g. arcsine and arccosine). In some calculations I was doing* I had prior knowledge of the values of sin(x) and cos(x) without directly knowing what x was, and I had to calculate sin(2x). To solve this problem, I did what every beginning student of trigonometry would do:
The problem with this simple solution can be summarized by plotting the sin(2*arcsin(sin(x))) vs x:
Obviously that ain’t a sinusoidal graph. The problem lies in the fact that there aren’t any true inverse trigonometric functions. Instead, the following is true
only if t is between -pi/2 and pi/2.
The solution to my problem is extremely simple, trigonometric identities:
With the above identity my problem was solved. I already know what sin(x) and cos(x) are, so I can use the above formula to find sin(2x).
I also needed to calculate
where again I only know sin(x) , cos(x), and this time I know the value of a. To do this without running into the same problem illustrated in the above figure, I used the following identity:
To calculate cos(2x) I use a similar identity as I used for sin(2x):
which now enables me to calculate:
Moral of the story: don’t forget the wonders of trigonometric identities.
* The calculations concern the Stokes parameters of polarized radiation.