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

Jun 10, 2015

arcsin(sqrt(3)/2)=60°=pi/3

#### Explanation:

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

Mar 6, 2018

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

Mar 6, 2018

$\frac{\pi}{3} , \frac{2 \pi}{3}$

#### Explanation:

$\sin x = \frac{\sqrt{3}}{2}$
Trig Table gives as solution:
$x = \frac{\pi}{3}$ , or $x = {60}^{\circ}$
The unit circle gives another x that has the same sin value (sqrt3/2)
$x = \pi - \frac{\pi}{3} = \frac{2 \pi}{3}$, or $x = {120}^{\circ}$
Answers for $\left(0 , 2 \pi\right)$:
$\frac{\pi}{3} , \frac{2 \pi}{3}$
For general answer, add $2 k \pi$