Question

How do you solve: arcsin(sin 5pi/4)

Answer 1

That is not Calculus.

sin ( 5 π / 4 ) = sin ( 4 π / 4 + π / 4 ) = sin ( π + π / 4 )

Since:

sin ( π + θ ) = - sin θ

sin ( 5 π / 4 ) = sin ( π + π / 4 ) = - sin ( π / 4 ) = - 1 / √ 2

The range of sin x :

−1 ≤ sin x ≤ 1

sin ( - π / 2 ) = - 1 , sin ( π / 2 ) = 1

So x must be in interval:

x ∈ [ - π / 2, π / 2 ]

The only angle in this interval whose sine is - 1 / √ 2 is x = - π / 4

because - π / 2 < - π / 4 < π / 2

So the solution is:

x = - π / 4

Answer 2

arcsin [ sin ( 5 π / 4 ) ] = - π / 4