Overcoming Lying Inverse Trigonometric Functions.

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.


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