Question

If 2 sin x = `sqrt3` , evaluate.
(i) 4 sin³ x - 3 sin x.
(ii) 3 cos x - 4 cos³ x.

Answer 1

2 sin x  = `sqrt3`

sin x = `sqrt3 /(2)`

i.e.`"perpendicular"/"base" = "BC"/"AC" = sqrt3/(2)`

Therefore if length of perpendicular = `sqrt3x` , length of = 2x

Since
AB² + BC² = AC²  ...[ Using Pythagoras Theorem]

(2x)² – (`sqrt3x`)² = AB²

AB² = x²

∴ AB = x

Now, cos x = `"AB"/"AC" = (1)/(2)`

(i) 4 sin³ x – 3sin x

= `4 (sqrt3/2)^3 –  3(sqrt3/2)`

= `(3sqrt3)/2 – (3sqrt3)/2`

= 0

(ii) 3  cos x – 4 cos³ x

= `3 * (1)/(2)  –  4 * (1/2)^3`

= `3 * 1/2 - 4 * 1/8`

= `3 * 1/2 - cancel4^1 * 1/cancel8_2`

= `3/2  –  1/2`

= 1