Question

If θ lies in third quadrant and sin θ = $\frac{-4}{5}$, find the value of cosec($\frac{\theta}{2}$) and tan ($\frac{\theta}{2}$).

Answer 1

θ lies in 3rd quadrant .
it means , θ lies between 180° to 270° .
e.g., 180° ≤ θ ≤ 270° or, 90° ≤ θ/2 ≤ 135°
hence, (θ/2) lies in 2nd quadrant . and we know, sine and cosecant be positive in 2nd quadrant.

given , sinθ = -4/5

use formula, sin2x = 2tanx/(1 + tan²x)

then, sinθ = 2tanθ/2/ (1 + tan²θ/2) = -4/5

5tanθ/2 = -2(1 + tan²θ/2)

2tan²θ/2 + 5tanθ/2 + 2 = 0

(2tanθ/2 + 1)(tanθ/2 + 2) = 0

tanθ/2 = -1/2 , -2

sinθ = 1/√5 , 2/√5

so, cosecθ/2 = √5 , √5/2

Answer 2

HELLO DEAR,



given , sinθ = -4/5

[as, sin2x = 2tanx/(1 + tan²x) ]

then, sinθ = 2tanθ/2/ (1 + tan²θ/2) = -4/5

5tanθ/2 = -2(1 + tan²θ/2)

2tan²θ/2 + 5tanθ/2 + 2 = 0

(2tanθ/2 + 1)(tanθ/2 + 2) = 0

tanθ/2 = -1/2 , -2

sinθ = 1/√5 , 2/√5

so, cosecθ/2 = √5 , √5/2



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THANKS