Question #3504b

3 Answers
Apr 10, 2017

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

Explanation:

I'm assuming the function is this;

cos(1/2 * sin^-1(sqrt(3) /2))

Then we can simplify this as,

cos(1/2 * pi/3) since, sin(pi/3) = sqrt(3) /2

=> cos(pi/6)

=> sqrt(3)/2

Apr 10, 2017

sqrt3/2

Explanation:

cos(1/2(color(blue)(arcsin(sqrt3/2))))

arcsin(color(red)(sqrt3/2))=color(blue)60 degrees, because sincolor(blue)60=color(red)(sqrt3/2)

=cos(1/2(color(blue)60))

=cos(30)

=sqrt3/2

Apr 17, 2017

More generally:

cos(theta)=2cos^2(theta/2)-1

So:

cos(theta/2)=sqrt((1+cos(theta))/2)

Then:

cos(1/2arcsin(sqrt3/2))=sqrt((1+cos(arcsin(sqrt3/2)))/2)

Note that cos(arcsin(sqrt3/2))=1/2, since this refers to a 1-sqrt3-2 right triangle.

cos(1/2arcsin(sqrt3/2))=sqrt((1+1/2)/2)=sqrt(3/4)=sqrt3/2