Question

If 4sinθ = `sqrt(13)`, find the value of 4sin³θ - 3sinθ

Answer 1

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)`.