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प्रश्न
If 4sinθ = `sqrt(13)`, find the value of 4sin3θ - 3sinθ
उत्तर
Consider ΔABC, where ∠B = 90°
⇒ 4sinθ = `sqrt(13)`
⇒ 4sinθ = `sqrt(13)/(4)`
⇒ sinθ = `"Perpendicular"/"Hypotenuse" = "BC"/"AC" = sqrt(13)/(4)`
By Pythagoras theorem,
AC2 = AB2 + BC2
⇒ AB2
= AC2 - BC2
= `4^2 - (sqrt(13))^2`
= 16 - 13
= 3
⇒ AB = `sqrt(3)`
Now,
cosθ = `"Base"/"Hypotenuse" = "AB"/"AC" = sqrt(3)/(4)`
4sin3θ - 3sinθ
= `4(sqrt(13)/4)^3 - 3 xx sqrt(13)/(4)`
= `(13sqrt(13))/(16) - (3sqrt(13))/(4)`
= `(13sqrt(13) - 12sqrt(13))/(16)`
= `sqrt(13)/(16)`.
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