Question

What is `\arcsin ( \sin ( \frac { 2\pi } { 3} ) )`?

Answer 1

`arcsin(sin((2pi)/3))=(2pi)/3`

Explanation:

By definition if `sintheta=x`, `arcsinx=theta` i.e. `arcsinx` is the angle whose sine ratio is `x`.

Here let `arcsin(sin((2pi)/3))=x`, then `sin((2pi)/3)=sinx`

Hence `arcsin(sin((2pi)/3))=(2pi)/3`.

Answer 2

`arcsin(sin((2pi)/3))=pi/3`

Explanation:

`arcsin(x)` has domain `-1<=x<=1` and range `-pi/2<=y<=pi/2`.

`sin((2pi)/3)=sqrt(3)/2` so the problem we're solving is really:

`arcsin(sqrt(3)/2)`. We're looking for the angle between `-pi/2` and `pi/2` that has a sine value of `sqrt(3)/2`. We know it has to be from QI because it's a positive value. The angle in QI with `sin(theta)=sqrt(3)/2` is `pi/3`, so `arcsin(sin((2pi)/3))=pi/3`