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Inverse of Trig Functions

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xyz_1965

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Jul 26, 2020
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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

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Feb 24, 2012
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You have to be mindful of the one-to-one interval over which the inverse function is defined. For example:

\(\displaystyle \arcsin\left(\sin\left(\frac{5\pi}{6}\right)\right)=\frac{\pi}{6}\)
 
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xyz_1965

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Jul 26, 2020
81
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

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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?
 
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xyz_1965

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MarkFL

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No, that's the range.
 
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xyz_1965

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MarkFL

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xyz_1965

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Jul 26, 2020
81

MarkFL

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xyz_1965

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Jul 26, 2020
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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

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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

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Jul 26, 2020
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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

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xyz_1965

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Jul 26, 2020
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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

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

xyz_1965

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Jul 26, 2020
81
If you now understand how this works I'd say it was time well spent.
I gotta speed up this precalculus trek. It is on hold as I wait for my Michael Sullivan 5th Edition Precalculus textbook to arrive.