Question

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle.
67° 30'
7 sin(67° 30')=
cos(67° 30')=
4 tan(67° 30')=

Answer 1

1. 2 x 67.5 = 135
2. tan 135 = - tan 45 = -1
3. tan 2A = [2tan A]/[1-(tan A)^2]
4. Put A = 67.5 in (3) and get
tan135 = [2tan67.5]/[1-(tan67.5)^2]
-1 = [2tan 67.5]/[1-(tan 67.5)^2]
5. Let x = tan 67.5. Then,
-1 = 2x/[1 - x^2]
x^2 - 2x - 1 = 0
By quadratic formula,
x = 1 + sqrt(2) or 1 - sqrt(2)
reject 1 - sqrt(2)since tan 67.5 > 0
6. x = tan 67.5 = 1 + sqrt(2)
4 tan 67.5 = 4 + 4 sqrt(2)