# Inverse of Trig Functions

#### xyz_1965

##### Member
Take any trig function, say, arcsin (x). Why is the answer x when taking the inverse of sin (x)?

Why does arcsin (sin x) = x?

Can it be that trig functions and their inverse undo each other?

#### MarkFL

Staff member
You have to be mindful of the one-to-one interval over which the inverse function is defined. For example:

$$\displaystyle \arcsin\left(\sin\left(\frac{5\pi}{6}\right)\right)=\frac{\pi}{6}$$

#### xyz_1965

##### Member
You have to be mindful of the one-to-one interval over which the inverse function is defined. For example:

$$\displaystyle \arcsin\left(\sin\left(\frac{5\pi}{6}\right)\right)=\frac{\pi}{6}$$
What? Can you explain further? Why is the answer pi/6?

#### MarkFL

Staff member
What? Can you explain further? Why is the answer pi/6?
What domain do we use for the sine function such that we can define an inverse?

#### xyz_1965

##### Member
What domain do we use for the sine function such that we can define an inverse?
Domain: [-1, 1].

#### MarkFL

Staff member
No, that's the range.

#### MarkFL

Staff member
[-pi/2, pi/2]
Yes...is $$\displaystyle \frac{5\pi}{6}$$ in that domain?

#### xyz_1965

##### Member
Yes...is $$\displaystyle \frac{5\pi}{6}$$ in that domain?
Yes, it is.

#### MarkFL

Staff member
Yes, it is.
No, it is outside that since:

5/6 > 1/2

What is $$\displaystyle \sin\left(\frac{5\pi}{6}\right)$$ ?

#### xyz_1965

##### Member
No, it is outside that since:

5/6 > 1/2

What is $$\displaystyle \sin\left(\frac{5\pi}{6}\right)$$ ?
I just got home. Let me see: sin(5pi/6) = 1/2.

#### MarkFL

Staff member
I just got home. Let me see: sin(5pi/6) = 1/2.
Yes. Now what angle within the restricted domain returns that same value from the sine function?

#### xyz_1965

##### Member
Yes. Now what angle within the restricted domain returns that same value from the sine function?
Using the unit circle, I found the angle to be pi/6.

#### MarkFL

Staff member
Using the unit circle, I found the angle to be pi/6.
Good, the puzzle is thus completed.

#### xyz_1965

##### Member
Good, the puzzle is thus completed.
Wasted too much time solving this puzzle. If I do this for every problem, I'll never get to calculus 1.