MHB How do trigonometric functions and their inverses relate to each other?

[xyz_1965]

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]

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

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

 

[xyz_1965]

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

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

What? Can you explain further? Why is the answer pi/6?

 

[MarkFL]

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]

What domain do we use for the sine function such that we can define an inverse?

Domain: [-1, 1].

 

[MarkFL]

No, that's the range.

 

[xyz_1965]

No, that's the range.

[-pi/2, pi/2]

 

[MarkFL]

[-pi/2, pi/2]

Yes...is $$\frac{5\pi}{6}$$ in that domain?

 

[xyz_1965]

Yes...is $$\frac{5\pi}{6}$$ in that domain?

Yes, it is.

 

[MarkFL]

Yes, it is.

No, it is outside that since:

5/6 > 1/2

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

 

[xyz_1965]

No, it is outside that since:

5/6 > 1/2

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

I just got home. Let me see: sin(5pi/6) = 1/2.

 

[MarkFL]

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]

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]

Using the unit circle, I found the angle to be pi/6.

Good, the puzzle is thus completed.

 

[xyz_1965]

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.

 

[MarkFL]

If you now understand how this works I'd say it was time well spent.

 

[xyz_1965]

If you now understand how this works I'd say it was time well spent.

I got to speed up this precalculus trek. It is on hold as I wait for my Michael Sullivan 5th Edition Precalculus textbook to arrive.