Question

How do you evaluate `sin((-17pi)/3)`?

Answer 1

`sin(-17pi/3) = sin(pi/3-3(2pi)) = sin(pi/3) = sqrt(3)/2`

How do I know that `sin(pi/3) = sqrt(3)/2`?

Picture an equilateral triangle with sides of length 1. Now cut the triangle into two equal pieces with a line segment running from one vertex to the middle of the opposite side. This will split the triangle into two right-angled triangles with angles 30, 60 and 90 degrees i.e. `pi/6`, `pi/3` and `pi/2` radians. The shortest side of one of these right-angled triangles has length `1/2`. Using Pythagoras theorem, the other side adjoining the right angle must have length:

`sqrt(1^2 - (1/2)^2) = sqrt(1-1/4) = sqrt(3/4) = sqrt(3)/2`

Divide this by the length of the hypotenuse (1) to get the sine of the opposite angle (i.e. `sin (pi/3) = (sqrt(3)/2)/1 = sqrt(3)/2`).