If cos^3 θ+cos^3 ((2 π)/3+θ )+cos^3 ((4 π)/3+θ ) =α cos 3 θ , then α is equal to

Question

MARKS Content Team · Accepted Answer

Correct answer: (B) 3/4. We have, cos^3 θ+cos^3 ((2 π)/3+θ )+cos^3 ((4 π)/3+θ )=a cos 3 θ ⇒ (cos 3 θ+3 cos θ)/4+(cos 3 ((2 π)/3+θ )+3 cos ((2 π)/3+θ ))/4 +(cos 3 ((4 π)/3+θ )+3 cos ((4 π)/3+θ ))/4=a cos 3 θ ⇒ cos 3 θ+3 cos θ+cos (2 π+3 θ)+3 cos ((2 π)/3+θ ) +cos (4 π+3 θ)+3 cos ((4 π)/3+θ )=4 a cos 3 θ ⇒ cos 3 θ+3 cos θ+cos 3 θ+cos 3 θ+3 [cos ((2 π)/3+θ ) . ⇒ 3 cos 3 θ+3 cos θ+3 [2 cos (π+θ) cos (π)/3 ] =4 a cos 3 θ ⇒ 3 cos 3 θ+3 cos θ+3 × 2 ×(-cos θ) × 1/2 =4 a cos 3 θ ⇒ 3 cos 3 θ+3 cos θ-3 cos θ=4 a cos 3 θ ⇒ 3 cos 3 θ=4 a cos 3 θ ⇒ a=3/4

If $\cos ^3 \theta+\cos ^3\left(\frac{2 \pi}{3}+\theta\right)+\cos ^3\left(\frac{4 \pi}{3}+\theta\right)$ $=\alpha \cos 3 \theta$, then $\alpha$ is equal to

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