If cosx=-12/13 and is in the interval (π,3π/2)

Question

find cosx/2 and tanx/2?

Answer 1

cos x = - 12/13 in the interval (π, 3π/2).

Half angle formulas : cos(x) = 2cos² (x/2) - 1 (or)

                                 cos(x) = 1 - 2sin² (x/2).

• cos(x) = 2cos² (x/2) - 1

⇒ cos (x/2) = √[(1 + cos x)/2]

= √[(1 - 12/13)/2]

= √[(13 - 12)/26]

= √[1/26]

= ± 1/√26.

cos (x/2) = - 1/√26.   (Since, the interval is (π, 3π/2))

• cos(x) = 1 - 2sin² (x/2).

⇒ sin (x/2) = √[(1 - cos x)/2]

= √[(1 - (- 12/13))/2]

= √[(1 + 12/13)/2]

= √[(13 + 12)/26]

= √[25/26]

= ± 5/√26.

sin (x/2) = - 5/√26. (Since, the interval is (π, 3π/2))

• tan(x/2) = sin(x/2)/cos(x/2)

= [- 5/√26]/[- 1/√26]

tan(x/2) = 5.        (Since, the interval is (π, 3π/2))

Therefore, cos (x/2) = - 1/√26 and tan (x/2) = 5.