If: tanx = 3/4, π < x < 3π/2, find the value of: sin(x/2), cos(x/2), tan(x/2)

Since π < x < 3π/2, cosx is negative.

⇒ π/2 < x/2 < 3π/4

Therefore, sin(x/2) is positive and cos(x/2) is negative.

Now, sec²(x) = 1 + tan²(x) = 1 + 9/16 = 25/16

Therefore, cos²(x) = 16/25 or cos(x) = −4/5 (since x lies in 3rd quadrant)

sin(x/2) = √[(1 − cos(x))/2] = √[(1 − (−4/5))/2] = 3/√10

cos(x/2) = √[(1 + cos(x))/2] = √[(1 + (−4/5))/2] = 1/√10

Hence, tan(x/2) = (sin(x/2))/(cos(x/2)) = (3/√10)/(1/√10) = −3