Question

how do you find the arcsin of -sqrt(3)/2

Answer 1

think of arcsin as an inverse function

f^(-1) (x) = - sqrt(3)/2

(I would use the math editor but it doesn't display the - sign anymore)

what do we know about the relationship between functions and their inverses?

if f is a function and f^(-1) is the inverse then

f(f^(-1))(x) = x

and we know that arcsin is the inverse of sin

so the question can be restated as

f(f^(-1))(x) = x

or

f(-sqrt(3)/2) = x

or, since f(x) = sin(x) in our case

sin(-sqrt(3)/2) = x

Answer 2

so how do you find the answer?

Answer 3

in other words

arcsin (-sqrt(3)/2) = x is the same thing as saying sin(x) = -sqrt(3)/2

Answer 4

just use knowledge of trigonometry

sin(x) = -sqrt(3)/2 (-60 degrees)

using the unit circle as our guide this corresponds to a 30-60-90 triangle

the x component is positive, the y component is negative

http://lscc.edu/faculty/dannyt/SiteAssets/SitePages/Trigonometry%20-%20Summer%20B/UnitCircle.gif

this translates to 5 pi / 3 radians or -pi/3 radian