How do you use the half angle formula to determine the exact values of the sine, cosine, and tangent (19pi)/12?

1 Answer
Feb 5, 2018

sin ((19pi)/12) = - sqrt(2 + sqrt3)/2
cos ((19pi)/12) = sqrt(2 - sqrt3)/2

Explanation:

First, call (t/2) = (19pi)/12 -->
t = (38pi)/12 = (2pi)/12 + 3pi = pi/6 + 3pi
Next, use the half angle formulas:
sin (t/2) = +- sqrt((1 - cos t)/2)
cos (t/2) = +- sqrt((1 + cos t)/2)
Note that
cos t = cos ((pi)/6 + 3pi) = - cos (pi/6) = - sqrt3/2.
Therefor, since (t/2) is in Quadrant 4 -->
sin (t/2) = - sqrt((1 + sqrt3/2)/2) = - sqrt(2 + sqrt3)/2
By the same process, with cos (t/2) positive -->
cos (t/2) = + sqrt((1 - sqrt3/2)/2) = sqrt(2 - sqrt3)/2
From there -->
tan (t/2) = sin/(cos) = - (2 + sqrt3)/(2 - sqrt3)