Find the exact value by using a half-angle identity. sin (5pi/12)

Find tan (5π 12) and sin ((5pi) / 12)

Answer: ± (2± √3) and± √ 2 + √3 2

Explanation:

Call tan ((5pi/12) = t.

Use trig identity: tan2 a = 2 tana 1 - tan2 a

tan (10π 12) = tan (5π 6) = - 1 √3 = 2t 1 - t2

t2 - 2 √3 t-1=0

D = d2 = b2 - 4ac=12+4=16 - - > d=±4

t = tan (5π 12) = 2 √3 2 ± 42 = 2± √3

Call sin (5π 12) = siny

Use trig identity: cos2 a=1-2 sin2 a

cos (10π 12) = cos (5π 6) = - √3 2 = 1-2 sin2 y

sin2 y = 2 + √3 4

siny = sin (5π 12) = ± √ 2 + √3 2