Question

Find the exact value of sin^-1 (sin 2pi/3) . Answer in radians in terms of pi

Answer 1

So what is sin 2pi/3 = ... ?

Answer 2

sin 2pi/3 = (sqrt 3) / 2

Answer 3

is it pi/3 ?

Answer 4

yes

Answer 5

sin^-1 (x) as notation I will denote as arcsin(x)
sin(x) has a domain of a number and a range of a number. However, the domain is typically an angle measure and the range is the corresponding ratio of the side lengths in a unit circle.

arcsin(x) is the inverse function of sin(x) notably, that the domain of arcsin(x) is the range of sin(x) and the range of arcsinx is the domain of sinx. That means that you give arcsinx a number that is a ratio of sides of the a triangle and it will spit out the angle measure.

Intuitively, this is 2pi/3. But more generally, the composition of a function and its inverse just spits out whatever the original input was. arcsinx undoes what sinx did therefore, you get back 2pi/3. But as you noticed the range of arcsinx is restricted from -pi/2 to positive pi/2 so the equivalent angle is pi/3.