Solving Fiber Optics Glass Characteristic Equations with Maple

[windsupernova]

Homework Statement

They ask me to solve the characteristic equation of a fiber optics glass. The equations I need to solve (separately) are:

Homework Equations

solve(2000000*Pi*sin(x)-2*arctan(.6666666667*(2.25*cos(x)^2-2.1904)^.5/sin(x)) = Pi, x):

solve(2000000*Pi*sin(x)-2*arctan(.6666666667*(2.25*cos(x)^2-2.1904)^.5/sin(x)) = 0, x)

That's how I tried to solve them in MAPLE.

The Attempt at a Solution

Maple is not giving me answers, I used my TI and it gave me some results but they were extremely similar and I don't think they are the real ones. How can I solve for the propagation angle x?

 

[Galileo]

The expressions on the left look the same. How can the same expression be both Pi and 0 at the same time?

And please try using tex or at least 2*10^6 and 2/3 instead of 2000000 and .6666666667
More people may decide to help if you put effort in presenting your problem neatly.

 

[windsupernova]

Well I'm Maple I used fractions, when I copy pasted them here they appeared as decimals. The equations are supposed to be different modes or propagation, so I need to solve for each angle individually.

The equation (general) is:

m*PI=2*PI*d/lambda -2 arctan(SQRT((n12Cos(x))^2 -n2^2)/n1*sin(x))

Where N1=1.5
n2=1.48
lambda=1(10^-6)
d=(3.192(10^-6)
And m=0,1,2,3... Where I only need the first (0 and 1)

 

[Galileo]

If you define
$$\alpha^2 =1-(n_2/n_1)^2$$
and let
$$y=sin(x)/\alpha$$, then I think you can write the arctan as:
$$arctan\left(\frac{\sqrt{1-y^2}}{y}\right)$$

Try some trig identities from there.