## 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.

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* The calculations concern the Stokes parameters of polarized radiation.

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