# How do you evaluate #cos(sin^-1((sqrt3/2))# without a calculator?

##### 2 Answers

#### Explanation:

Consider an equilateral triangle with sides of length

Remembering that

#sin(pi/3) = sqrt(3)/2#

Since

#sin^(-1)(sqrt(3)/2) = pi/3#

From the same diagram, remembering

#cos(pi/3) = 1/2#

So:

#cos(sin^(-1)(sqrt(3)/2)) = cos(pi/3) = 1/2#

#### Explanation:

Starting from:

#cos^2 theta + sin^2 theta = 1#

Subtract

#cos^2 theta = 1 - sin^2 theta#

Take the square root to find:

#cos theta = +-sqrt(1-sin^2 theta)#

If

#cos theta = +-sqrt(1-(sqrt(3)/2)^2) = +-sqrt(1-3/4) = +-sqrt(1/4) = +-1/2#

Further note that

So:

#cos(sin^(-1)(sqrt(3)/2)) = 1/2#