Question

How do you evaluate `sin^-1(1/sqrt2)` without a calculator?

Answer 1

`sin^-1(1/sqrt2)= 45^0`

Explanation:

Let `sin^-1(1/sqrt2)=theta :. sin theta= 1/sqrt2`

In the right triangle `sin theta = P/H :. P=1; H= sqrt2; P` is

perpendiculur and `H` is hypotenuse. In right triangle

`H^2=P^2+B^2 :. B^2=H^2-P^2=2-1 :. B=1` . Since

`P=1 and B=1`, it is isosceles triangle `:. theta =45^0`

`:. sin^-1(1/sqrt2)= 45^0` [Ans]

Answer 2

`pi/4 + 2kpi`
`(5pi)/4 + 2kpi`

Explanation:

`sin x = - 1/sqrt2 = - sqrt2/2`. Find arcsin x
Trig Table and unit circle give 2 solutions -->
arc `x = - pi/4` , and
`x = pi - (-pi/4) = pi + pi/4 = (5pi)/4`
Note. `x = (7pi)/4` is co-terminal to `x = (-pi/4)`
General answers:
`x = pi/4 + 2kpi`
`x = (5pi)/4 + 2kpi`