If α ,β ,γ ∈ [ 0,(π /2) ], then the value of ( sin (α +β +γ )/sin text α +sin β +sin text γ ) is

Question

If α ,β ,&gamma; ∈ [ 0,(π /2) ], then the value of ( sin (α +β +&gamma; )/sin text α +sin β +sin text &gamma; ) is (A)  <1  (B)  =-1  (C)  <0  (D) None of these

Tardigrade Admin · Accepted Answer

We have, sin text α +sin β +sin text &gamma; -sin text (α +β +&gamma; ) =sin text α +sin text β +sin text &gamma; -sinα text cosβ text cos&gamma; -cos text α text sin text β text cos text &gamma; -cos text α text cos text β text sin text &gamma; + text sin text α text sin text β text sin text &gamma; = sin α (1- cos β cos &gamma; )+ sin β (1- cos α cos &gamma; ) + sin &gamma; (1- cos α cos β )+ sin α sin β sin &gamma; ∴ sin α + sin β + sin &gamma; > sin (α +β +&gamma; ) ⇒ ( sin (α +β +&gamma; )/ sin α + sin β + sin &gamma; )<1

Q. If $α, β, γ \in [0, \frac{π}{2}],$ then the value of $\frac{s i n ( α + β + γ )}{s in α + s in β + s in γ}$ is

$< 1$

$= - 1$

$< 0$

None of these

Q. If α,β,γ∈[0,2π​], then the value of sinα+sinβ+sinγsin(α+β+γ)​ is

<1

=−1

<0

None of these

We have, sinα+sinβ+sinγ−sin(α+β+γ) =sinα+sinβ+sinγ−sinαcosβcosγ −cosαsinβcosγ−cosαcosβsinγ +sinαsinβsinγ =sinα(1−cosβcosγ)+sinβ(1−cosαcosγ) +sinγ(1−cosαcosβ)+sinαsinβsinγ ∴ sinα+sinβ+sinγ>sin(α+β+γ) ⇒ sinα+sinβ+sinγsin(α+β+γ)​<1

Q. If $α, β, γ \in [0, \frac{π}{2}],$ then the value of $\frac{s i n ( α + β + γ )}{s in α + s in β + s in γ}$ is

$< 1$

$= - 1$

$< 0$