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Question
If θ lies in the first quadrant and cosθ = `8/17`, then find the value of cos(30° + θ) + cos(45° – θ) + cos(120° – θ).
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Solution
cosθ = `8/17`
sinθ = `+- sqrt(1 – cos^2theta)`
Since θ lies in the first quadrant, only a positive sign can be considered.
⇒ sinθ = `sqrt(1 – 64/289)`
= `15/17`
Let, y = cos(30° + θ) + cos(45° – θ) + cos(120° – θ)
We know that,
cos(x + y) = cosx cosy – sinx siny
Therefore,
y = cos30° cosθ – sin30° sinθ + cos45° cosθ + sin45°sinθ + cos120° cosθ + sin120° sinθ
= `sqrt3/2(costheta + sintheta) - 1/2(costheta + sintheta) + 1/sqrt2(costheta + sintheta)`
= `(sqrt3/2 - 1/2 + 1/sqrt2)(costheta + sintheta)`
= `(sqrt3/2 - 1/2 + 1/sqrt2)(8/17 + 15/17)`
on solving,
= `((sqrt3 - 1)/2 + 1/sqrt2)(23/17)`
= `(23/17)((sqrt3 - 1)/2 + 1/sqrt2)`
= `23/17((sqrt3 - 1)/2 + 1/sqrt2)`
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