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

1 Answer
George C.
May 10, 2015

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).

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