Solving a Physics Problem Involving Inverse Trig Function

[mindcircus]

This is a physics problem, but I have some trouble finishing it up. I evaluated an integral, which gave me

inv tan of (1-E)tan(theta/2)/(sqrt (1-E^2)

evaluated from 0 to 2pi. I changed the limits to 0 to pi, and multipied by 2, because tan x doesn't exist at 2pi.
I know that if you take the inverse tangent of a tangent, you'll just get the angle. But because the tan has this ugly coefficient, I can't simply get the angle, right?
The answer is pi/2 so I'm inclined to just ignore it...but I know you can't do that...?

Thanks a lot!

 

[philosophking]

tan (x) is defined at 2pi. tan(x) = sin(x)/cos(x), and cos(2pi) = 1. Might want to rethink that. I'm just wondering, what's the E? that's where I'm having trouble! After all the dirty work (not really i guess) i get tan^-1(E-1)/sqrt(1-E^2). I can only guess that E is 0 or something , which would mean that you'd get tan^-1(-1) which is only defined in the second and fourth quadrants.

Maybe your integral is wrong? It sounds to me by your language that you used a computer ("it gave me") or other device to get this integral. You might want to go back and type it in correctly. That's all I have to say.

 

[M98Ranger]

First of all, great job on evaluating the integral and recognizing the need to change the limits and multiply by 2! It seems like you have a good understanding of the concept so far.

To address your question about the inverse tangent function, you are correct in saying that if you take the inverse tangent of a tangent, you will get the angle. However, in this case, the coefficient of the tangent function is not just any number, it is a function of the variable E. This means that the angle you get from taking the inverse tangent will also be a function of E.

So, in order to get a numerical value for the angle, you will need to plug in a specific value for E. This is where the limits of integration come into play. By changing the limits from 0 to 2pi to 0 to pi, you are essentially plugging in a value of E=1 into the function. This is why the answer you got is pi/2, because at E=1, the inverse tangent function simplifies to just the angle.

In general, when solving physics problems involving inverse trig functions, it is important to pay attention to the limits of integration and make sure they correspond to the specific values of the variables in the problem. And remember, just because the coefficient of the inverse tangent function may look complicated, it doesn't mean you can ignore it. It is still an important factor in finding the solution to the problem.

Keep up the good work and don't be afraid to ask for clarification if you come across any other confusing concepts in your physics problems. Good luck!