Find sin(x/2), cos(x/2) and tan(x/2), if tan(x) = −4/3, x in quadrant II

Here, x is in quadrant II.

i.e., π/2 < x < π 

⇒ π/4 < x/2 < π/2

Therefore, sin(x/2), cos(x/2) and tan(x/2) are all positive.

It is given that tanx = −4/3.

sec²(x) = 1 + tan²(x) = 1 + (−4/3)² = 1 + 16/9 = 25/9

Therefore, cos²x = 9/25 ⇒ cos(x) = ±3/5

As x is in quadrant II, cos(x) is negative.

⇒ cos(x) = −3/5

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

cos(x/2) = √[(1 + cos(x))/2] = √[(1 + (−3/5))/2] = 1/√5 = √5/5

tan(x/2) = sin(x/2) / cos(x/2) = (2/√5) / (1/√5) = 2