If sin θ+ sin 2 θ+ sin 3 θ= sin α and cos θ+ cos 2 θ+ cos 3 θ= cos α, then θ is equal to:

Question

If sin θ+ sin 2 θ+ sin 3 θ= sin α and cos θ+ cos 2 θ+ cos 3 θ= cos α, then θ is equal to: (A) (α/2) (B) α (C) 2 α (D) (α/6)

Tardigrade Admin · Accepted Answer

sin θ+ sin 3 θ+ sin 2 θ= sin α ⇒ 2 sin 2 θ cos θ+ sin 2 θ= sin α ⇒ sin 2 θ(2 cos θ+1)= sin α ... (i) Now, cos θ+ cos 3 θ+ cos 2 θ= cos α cos 2 θ(2 cos θ+1)= cos α ... (ii) From (i) and (ii), tan 2 θ= tan α ⇒ 2 θ=α ⇒ θ=(α/2)

Q. If $sin θ + sin 2 θ + sin 3 θ = sin α$ and $cos θ + cos 2 θ + cos 3 θ = cos α$ , then $θ$ is equal to:

$\frac{α}{2}$

$α$

$2 α$

$\frac{α}{6}$

Q. If sinθ+sin2θ+sin3θ=sinα and cosθ+cos2θ+cos3θ=cosα, then θ is equal to:

2α​

α

2α

6α​

sinθ+sin3θ+sin2θ=sinα ⇒2sin2θcosθ+sin2θ=sinα ⇒sin2θ(2cosθ+1)=sinα ... (i) Now, cosθ+cos3θ+cos2θ=cosα cos2θ(2cosθ+1)=cosα... (ii) From (i) and (ii), tan2θ=tanα⇒2θ=α ⇒θ=2α​

Q. If $sin θ + sin 2 θ + sin 3 θ = sin α$ and $cos θ + cos 2 θ + cos 3 θ = cos α$ , then $θ$ is equal to:

$\frac{α}{2}$

$α$

$2 α$

$\frac{α}{6}$