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

In summary, the conversation discusses the relationship between trigonometric functions and their inverses. It is explained that the inverse function undoes the original function, but the domain must be carefully considered. An example is given using the arcsin function and it is determined that the answer is pi/6. The conversation also touches on the restricted domain of the sine function and how it can be used to find the correct angle. The conversation concludes with the realization that understanding this concept is important for success in calculus.
  • #1
xyz_1965
76
0
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?
 
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  • #2
You have to be mindful of the one-to-one interval over which the inverse function is defined. For example:

\(\displaystyle \arcsin\left(\sin\left(\frac{5\pi}{6}\right)\right)=\frac{\pi}{6}\)
 
  • #3
MarkFL said:
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?
 
  • #4
xyz_1965 said:
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?
 
  • #5
MarkFL said:
What domain do we use for the sine function such that we can define an inverse?

Domain: [-1, 1].
 
  • #6
No, that's the range.
 
  • #7
MarkFL said:
No, that's the range.

[-pi/2, pi/2]
 
  • #8
xyz_1965 said:
[-pi/2, pi/2]

Yes...is \(\displaystyle \frac{5\pi}{6}\) in that domain?
 
  • #9
MarkFL said:
Yes...is \(\displaystyle \frac{5\pi}{6}\) in that domain?

Yes, it is.
 
  • #10
xyz_1965 said:
Yes, it is.

No, it is outside that since:

5/6 > 1/2

What is \(\displaystyle \sin\left(\frac{5\pi}{6}\right)\) ?
 
  • #11
MarkFL said:
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.
 
  • #12
xyz_1965 said:
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?
 
  • #13
MarkFL said:
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.
 
  • #14
xyz_1965 said:
Using the unit circle, I found the angle to be pi/6.

Good, the puzzle is thus completed. 😁
 
  • #15
MarkFL said:
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.
 
  • #16
If you now understand how this works I'd say it was time well spent.
 
  • #17
MarkFL said:
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.
 

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

What is the inverse of a trigonometric function?

The inverse of a trigonometric function is a function that undoes the original trigonometric function. It is denoted by adding "arc" or "a" before the name of the original function. For example, the inverse of sine is denoted as "arcsin" or "asin".

What is the domain and range of inverse trigonometric functions?

The domain of inverse trigonometric functions is the range of the corresponding trigonometric function. The range of inverse trigonometric functions is the domain of the corresponding trigonometric function. For example, the domain of arcsin(x) is [-1,1] and the range is [-π/2,π/2].

What is the relationship between a trigonometric function and its inverse?

The inverse of a trigonometric function reverses the input and output values of the original function. This means that the input of the original function becomes the output of the inverse function and vice versa. For example, if sin(x) = 0.5, then arcsin(0.5) = x.

What are the key properties of inverse trigonometric functions?

The key properties of inverse trigonometric functions include: the range and domain are swapped, the inverse of a function is its reflection over the line y=x, the inverse of a function is always a one-to-one function, and the inverse of a function is not defined for certain values in the original function's range.

What are the common applications of inverse trigonometric functions?

Inverse trigonometric functions are commonly used in fields such as physics, engineering, and navigation. They are used to solve problems involving angles and sides of triangles, as well as to model periodic phenomena such as sound waves and electrical currents.

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