Question

How do you evaluate `arcsin(sqrt 3/2)`?

Answer 1

`arcsin(sqrt(3)/2)=60°=pi/3`

Explanation:

`sqrt(3)/2` is a known value, and the main angle `alpha` that has `sin(alpha)=sqrt(3)/2` is `alpha=60°=pi/3`.
Because arcsin is a function `RR->[-1;1]`, we take only the value `alpha=pi/3`, without the periodic values.
So `arcsin(sqrt(3)/2)=60°=pi/3`.

Answer 2

Make a right triangle with one side = sqrt 3 and the hypotenuse = 2 and use Pythagoras to find the other leg = 1

Explanation:

If you know that the sin 30 deg = 1/2 .............
Make a right triangle with one side = sqrt 3 and the hypotenuse = 2 and use Pythagoras to find the other leg = 1
That makes the sign of the complementary angle = 1/2 which implies the angle = 30 deg, `pi`/6, so
the angle in question = 90 - 30 = 60 degrees or `pi` / 3
OR
Just calculate (sqrt 3) / 2 and find the arcsin with a calculator

Answer 3

`pi/3, (2pi)/3`

Explanation:

`sin x = sqrt3/2`
Trig Table gives as solution:
`x = pi/3` , or `x = 60^@`
The unit circle gives another x that has the same sin value (sqrt3/2)
`x = pi - pi/3 = (2pi)/3`, or `x = 120^@`
Answers for `(0, 2pi)`:
`pi/3, (2pi)/3`
For general answer, add `2kpi`