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The Inverse Tangent Function - Problem 2
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
We’re talking about the inverse tangent function. Now inverse tangent is the inverse of the restricted tangent function that’s something that you should always remember, it's not the inverse of the inverse of the entire tangent function.
So here I’ve written down f of x equals tangent x and f inverse of x equals inverse tangent x because these two functions are inverses there are certain identities that we can derive. For example f of f inverse of x equals x. This identity for these two functions becomes tangent of inverse tangent of x equals x. The other one, f inverse of f of x equals x becomes inverse tangent of tangent x equals x.
Now these two identities are not always true. They’re only true as long as x is in the domain of the inside function. Here the inside function is inverse tangent and luckily inverse tangent is defined for all real numbers, so this one is actually true for all real numbers; for all real x. This one however, x has to be in the domain of, not tangent, but the restricted tangent function. This is only if x is between negative pi over 2 and pi over 2. So that’s really important. Let’s use these identities in a problem.
First one says evaluate the inverse tangent of tangent of pi over 4. Well pi over 4 is in the domain of the restricted tangent function, so this identity should be true. Inverse tangent of tangent of pi over 4 is pi over 4. What about inverse tangent of 2 pi over 3. It’s not actually in the domain of the restricted tangent function it’s a little bigger than pi over 2, so we have to do something here. One thing we could do is we could actually evaluate the tangent of 2 pi over 3 but you might not remember what it is off hand. If you remember the tangent is a periodic function with period pi what you could do is move this point to the left, as many multiples of pi as you need until you get into the interval from negative pi over 2 to pi over 2. For example I could subtract pi. If I subtract pi over 2 from 2 I over 3 I get negative pi over 3. Now these two tangent values should be exactly the same because tangent’s periodic with periodic pi and this value negative pi over 3 is in the interval between negative pi over 2 and pi over 2. So the identity does hold and we get pi over 3.
What about this one? Tangent of inverse tangent of 2. I don’t know what the inverse tangent of 2 is but it doesn’t matter. 2 is in the domain of inverse tangent. Every number is in the domain of inverse tangent. So the tangent of inverse tangent of 2 is just 2. Tangent of inverse tangent of anything is that number. These two identities, these are the inverse identities for tangent and inverse tangent.
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