Find the exact value of the expression:

Question

tan−1(tan(−120651/47π))

I just don't know how to find the exact value. Any explanation would be awesome!

Answer 1

Many math problems can be solved by understanding the definition of the key term involved. In this case, the key term is the inverse of a (trigonometric) function.

From Wikipedia:
"In mathematics, an inverse function is a function that "reverses" another function. That is, if f is a function mapping x to y, then the inverse function of f maps y back to x."

One of the properties of the inverse of a function is:
f^-1f(x) = x, and
f(f^-1)(x) = x

The inverse of a function is like a mirror.
If we look into a mirror, the image is backwards. If we look at this image through another mirror (i.e. reflected twice), the final image looks exactly like the original.

so
Hint:
use the identity which is derived from the definition of an inverse of a function:
tan^-1(tan(x)) = x
For example:
sin^-1sin(3.5π) = 3.5π

The same principle applies to the other trigonometric functions, in fact, all other functions.

Answer 2

However, since inverse trig functions have principal values (because they are multi-valued), you have to be careful.

arctan(tan(3π/4)) = arctan(1) = π/4, not 3π/4.

So, you need to reduce your fraction to see how many multiples of π you can discard, and then make sure you end with an answer between -π/2 and π/2.

Answer 3

Very true, thanks Steve.