The minimum value of cos θ + sin θ+(2/ sin 2θ) for θ ∈ (0, π/2) is

Question

The minimum value of cos θ + sin θ+(2/ sin 2θ) for θ ∈ (0, π/2) is (A) 2+√2 (B) 2 (C) 1+√2 (D) 2√2

Tardigrade Admin · Accepted Answer

Here, cos θ+ sin θ+(2/ sin 2 θ), θ ∈(0, (π/2)) For minimum value, sin 2 θ must be maximum. ∴ 2 θ=(π/2) ⇒ θ=(π/4) Hence, cos (π/4)+ sin (π/4)+(2/ sin (π)2)=√2+2

Q. The minimum value of $cos θ + sin θ + \frac{2}{s i n 2 θ}$ for $θ \in (0, π /2)$ is

$2 + 2$

$2$

$1 + 2$

$22$

Here, $cos θ + sin θ + \frac{2}{s i n 2 θ}, θ \in (0, \frac{π}{2})$
For minimum value, $sin 2 θ$ must be maximum.
$∴ 2 θ = \frac{π}{2}$
$\Rightarrow θ = \frac{π}{4}$
Hence, $cos \frac{π}{4} + sin \frac{π}{4} + \frac{2}{s i n \frac{π}{2}} = 2 + 2$

Q. The minimum value of cosθ+sinθ+sin2θ2​ for θ∈(0,π/2) is

2+2​

2

1+2​

22​

Here, cosθ+sinθ+sin2θ2​,θ∈(0,2π​) For minimum value, sin2θ must be maximum. ∴2θ=2π​ ⇒θ=4π​ Hence, cos4π​+sin4π​+sin2π​2​=2​+2

Q. The minimum value of $cos θ + sin θ + \frac{2}{s i n 2 θ}$ for $θ \in (0, π /2)$ is

$2 + 2$

$2$

$1 + 2$

$22$

Here, $cos θ + sin θ + \frac{2}{s i n 2 θ}, θ \in (0, \frac{π}{2})$
For minimum value, $sin 2 θ$ must be maximum.
$∴ 2 θ = \frac{π}{2}$
$\Rightarrow θ = \frac{π}{4}$
Hence, $cos \frac{π}{4} + sin \frac{π}{4} + \frac{2}{s i n \frac{π}{2}} = 2 + 2$