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!
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.
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.
Very true, thanks Steve.
To find the exact value of the given expression, we need to use some trigonometric identities and properties. Let's break down the steps:
Step 1: Convert the given angle to radians
The given angle is -120651/47π. To convert it to radians, we can multiply by π/180 since there are π radians in 180 degrees:
=> (-120651/47π) * (π/180) = -120651/47 * (1/180) = -120651/(47 * 180)
Step 2: Simplify the expression
Simplify the fraction -120651/(47 * 180) as much as possible. If there are common factors in the numerator and denominator, you can cancel them out.
Step 3: Use the tan(tan^-1(x)) identity
The identity tan(tan^-1(x)) = x helps us find the exact value. In this case, we have:
=> tan(tan^-1(tan(-120651/47π)))
Step 4: Apply the identity
Using the identity in step 3, we can simplify the expression:
=> tan(-120651/47π) = -120651/(47 * 180)
Step 5: Calculate the exact value
Now, you can calculate the exact value of the expression. The value will be a number, not an angle measure.
Remember, this process helps you find the exact value, which is different from an approximate value or decimal approximation.
Note: Some calculators have the ability to calculate inverse trigonometric functions and trigonometric functions of angles in degrees or radians. If you're looking for an approximate value, you can use a calculator accordingly.