SOLUTION: What is the exact value of this expression? arcsin(sin(-3pi/5)) How do I get the answer? Thanks!

Question 808932: What is the exact value of this expression? arcsin(sin(-3pi/5))
How do I get the answer? Thanks!
Found 2 solutions by KMST, DrBeeee:
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
THE SHORT ANSWER:
because
and
To find the answer you have to look for the angle that
has the same sine as
and is between and

THE LONG STORY:
is a function and its graph looks like this:

It is a periodic function with period ,
meaning that the same value of is guaranteed to repeat at intervals.
The function has a maximum where and it repeats that value later, at , where again.
For example, are three consecutive crests (maxima) in the wavy graph, at , and
and are two consecutive minima, at and .
In between those extremes, the function takes the same values more often, once on the way up, and again on the way down.
For example

With a domain of all the real numbers as possible values, the values of repeat, so that function does not have an inverse.
For that reason, we define as the inverse function of
define on the domain
In that restricted domain, the only that has is .
The same goes for any other value of within the domain defined as .
The other inverse trigonometric functions are defined similarly.
is the angle such that
and
Answer by DrBeeee(684) (Show Source): You can put this solution on YOUR website!
Let
(1) sin(x) = y
Then
(2) x = arcsin(y)
Put y of (1) into (2) and see that
(3) x = arcsin(sin(x))
In your problem
(4) x = -3pi/5
So the exact value of
(5) arcsin(sin(-3pi/5)) = -3pi/5
When the exact answer is asked for, use the symbol, pi in this case, for the irrational number not a decimal.
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