Asked by Dan
Find the exact value of the expression:
tan−1(tan(−120651/47π))
I just don't know how to find the exact value. Any explanation would be awesome!
tan−1(tan(−120651/47π))
I just don't know how to find the exact value. Any explanation would be awesome!
Answers
Answered by
MathMate
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<sup>-1</sup>f(x) = x, and
f(f<sup>-1</sup>)(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<sup>-1</sup>(tan(x)) = x
For example:
sin<sup>-1</sup>sin(3.5π) = 3.5π
The same principle applies to the other trigonometric functions, in fact, all other functions.
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<sup>-1</sup>f(x) = x, and
f(f<sup>-1</sup>)(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<sup>-1</sup>(tan(x)) = x
For example:
sin<sup>-1</sup>sin(3.5π) = 3.5π
The same principle applies to the other trigonometric functions, in fact, all other functions.
Answered by
Steve
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, <u>not</u> 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.
arctan(tan(3π/4)) = arctan(1) = π/4, <u>not</u> 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.
Answered by
MathMate
Very true, thanks Steve.