How do you find the value of sin -((5pi)/12) using the double or half angle formula?

1 Answer
Oct 25, 2016

sqrt(2 + sqrt3)/2

Explanation:

Unit circle and property of complementary arcs -->
sin ((-5pi)/12) = sin (pi/12 - (6pi)/12) = sin (pi/12 - pi/2) = cos (pi/12)
Evaluate cos (pi/12), knowing cos ((2pi)/12) = cos (pi/6) = sqrt3/2
Use trig identity
2cos^2 a = 1 + cos 2a
2cos^2 (pi/12) = 1 + cos (pi/6) = 1 + sqt3/2 = (2 + sqrt3)/2
cos^2 (pi/12) = (2 + sqrt3)/4
cos (pi/12) = +- sqrt(2 + sqrt3)/2
Since cos ((pi/12) is positive, then only the positive value is accepted.
Finally:
sin ((-5pi)/12) = cos (pi/12) = sqrt(2 + sqrt3)/2