If cos α + cos β =(3/2) and sin α + sin β =(1/2) and θ is the arithmetic mean of α and β then sin 2θ + cos 2θ is equal to :

Question

If cos α + cos β =(3/2) and sin α + sin β =(1/2) and θ is the arithmetic mean of α and β then sin 2θ + cos 2θ is equal to : (A) (3/5) (B) (4/5) (C) (7/5) (D) (8/5)

Tardigrade Admin · Accepted Answer

2 cos (α+β/2) cos (α-β/2)=(3/2) and 2 sin (α+β/2) cos (α-β/2)=(1/2) ⇒ tan ((α+β/2))=(1/3) ⇒ sin 2 θ+ cos 2 θ= sin (α+β)+ cos (α+β) =((2/3)/1+(1)9)+(1-(1/9)/1+(1)9)=(6/10)+(8/10) =(7/5)

Q. If $cos α + cos β = \frac{3}{2}$ and $sin α + sin β = \frac{1}{2}$ and $θ$ is the arithmetic mean of $α$ and $β$ then $sin 2 θ + cos 2 θ$ is equal to :

$\frac{3}{5}$

$\frac{4}{5}$

$\frac{7}{5}$

$\frac{8}{5}$

Q. If cosα+cosβ=23​ and sinα+sinβ=21​ and θ is the arithmetic mean of α and β then sin2θ+cos2θ is equal to :

53​

54​

57​

58​

2cos2α+β​cos2α−β​=23​ and 2sin2α+β​cos2α−β​=21​ ⇒tan(2α+β​)=31​ ⇒sin2θ+cos2θ=sin(α+β)+cos(α+β) =1+91​32​​+1+91​1−91​​=106​+108​ =57​

Q. If $cos α + cos β = \frac{3}{2}$ and $sin α + sin β = \frac{1}{2}$ and $θ$ is the arithmetic mean of $α$ and $β$ then $sin 2 θ + cos 2 θ$ is equal to :

$\frac{3}{5}$

$\frac{4}{5}$

$\frac{7}{5}$

$\frac{8}{5}$