Let θ, varphi ∈[0,2 π] be such that 2 cos θ(1- sin varphi)= sin 2 θ( tan (θ/2)+ cot (θ/2)) cos varphi-1, tan (2 π-θ)0 and -1

Question

Let θ, varphi ∈[0,2 π] be such that 2 cos θ(1- sin varphi)= sin 2 θ( tan (θ/2)+ cot (θ/2)) cos varphi-1, tan (2 π-θ)>0 and -1< sin θ<-(√3/2) . Then varphi cannot satisfy (A) 0< varphi<(π/2) (B) (π/2)< varphi<(4 π/3) (C) (4 π/3)< varphi<(3 π/2) (D) (3 π/2)< varphi<2 π

Tardigrade Admin · Accepted Answer

2 cos θ(1- sin varphi)=(2 sin 2 θ/ sin θ) cos varphi-1=2 sin θ cos varphi-1 2 cos θ-2 cos θ sin varphi=2 sin θ cos varphi-1 2 cos θ+1=2 sin (θ+ varphi) tan (2 π-θ)>0 ⇒ tan θ<0 text and -1< sin θ<-(√3/7) ⇒ θ ∈((3 π/2), (5 π/3)) (1/2)< sin (θ+ varphi)<1 ⇒ 2 π+(π/6)<θ+ varphi<(5 π/6)+2 π 2 π+(π/6)-θ max < varphi<2 π+(5 π/6)-θ min (π/2)< varphi<(4 π/3)

Q. Let $θ, φ \in [0, 2 π]$ be such that $2 cos θ (1 - sin φ) = sin^{2} θ (tan \frac{θ}{2} + cot \frac{θ}{2}) cos φ - 1, tan (2 π - θ) > 0$ and $- 1 < sin θ < - \frac{3}{2} .$ Then $φ$ cannot satisfy

$0 < φ < \frac{π}{2}$

$\frac{π}{2} < φ < \frac{4 π}{3}$

$\frac{4 π}{3} < φ < \frac{3 π}{2}$

$\frac{3 π}{2} < φ < 2 π$

Q. Let θ,φ∈[0,2π] be such that 2cosθ(1−sinφ)=sin2θ(tan2θ​+cot2θ​)cosφ−1,tan(2π−θ)>0 and −1<sinθ<−23​​. Then φ cannot satisfy

0<φ<2π​

2π​<φ<34π​

34π​<φ<23π​

23π​<φ<2π

Q. Let $θ, φ \in [0, 2 π]$ be such that $2 cos θ (1 - sin φ) = sin^{2} θ (tan \frac{θ}{2} + cot \frac{θ}{2}) cos φ - 1, tan (2 π - θ) > 0$ and $- 1 < sin θ < - \frac{3}{2} .$ Then $φ$ cannot satisfy

$0 < φ < \frac{π}{2}$

$\frac{π}{2} < φ < \frac{4 π}{3}$

$\frac{4 π}{3} < φ < \frac{3 π}{2}$

$\frac{3 π}{2} < φ < 2 π$