Question

if tanx=3/9 , x lies in third quadrant , find the value of sin(x/2) , cos(x/2) , tan(x/2)

Answer 1

tanx = 3/9 = 1/3 , where x lies in 3rd quadrant.
e.g., 180° < x < 270° so, 90° < (x/2) < 135°
hence, (x/2) is in 2nd quadrant. we know in 2nd quadrant , sin and cosec are positive.

tanx = 2tan(x/2)/{1 - tan² (x/2)} [ from formula]
1/3 = 2tan(x/2)/{1 - tan² (x/2)}
6tan(x/2) = 1 - tan²(x/2)
tan²(x/2) + 6tan(x/2) - 1 = 0
tan(x/2) = {-6 ± √(60)}/2 = -3 ± √15
tan(x/2) = - (3 + √15) is correct for (x/2) lies on 2nd quadrant.

now, sin(x/2) = (3 + √15)/√(1² + (-3-√15)²}
= (3 + √15)/√(1 + 9 + 15 + 6√15)
= (3 + √15)/√(25 + 6√15)

cos(x/2) = -1/√(25 + 6√15)

tan(x/2) = -(3 + √15)

Answer 2

tan x = 3/4, the equation will be. tan 2x = 2 tan x / 1 - tan 2 x.

Solving this equation comes to 3 tan 2 x/2 + 8 tan x/2 - 3 = 0. x becomes 1/3 or - 3.

Hence tan x / 2 is - 3.

Now using this we can find others. sin  x /2 is 3 / root 10, cost x / 2 is - 1 / root 10