If tan a= 12/15, 0 degrees < a < 90 degrees, then find the exact value of each of the following:

Question

A. Sin a/2
B. Cos a/2
C. Tan a/2

Answer 1

Make a sketch using tana = 12/15
then x = 15, y = 12
r^2 = 12^2 + 15^2
r = √369 = 3√41
sina = 12/(3√41) = 4/√41
cosa = 5/√41

Using cos 2A = 1 - 2sin^2 A
or cos a = 1 - 2sin^2 (a/2)
5/√41 = 1 - 2sin^2 (a/2)
2sin^2 (a/2) = 1 - 5/√41
2sin^2 (a/2) = (√41 - 5)/√41
sin^2 (a/2) = (√41-5)/(2√41)
sin (a/2) = √[(√41-5)/(2√41)]

Now use the property: sin^2 x + cos^2 x = 1
to find cos (a/2)
and once you have that, recall that tan(a/2) = sin(a/2) / cos(a/2)