Question

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

Answer 1

`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`