Question

The equation sin ( (PI/3) - x ) is equal to ____.

A. 1/2( sqrt(3) cosx + sinx)
B. 1/2(sqrt(3) cosx - sinx)
C. 1/2(cosx- sqrt(3) sinx)
D. 1/2 (cosx + sqrt(3) sinx)

Answer 1

The Value of

`Sin ((\pi)/(3)-x)=Sin((\pi)/(3))*cos x - Cos((\pi)/(3))* Sin x\\\\=(√(3))/(2)*Cos x -(Sin x)/(2)\\\\=(1)/(2)[√(3)Cos x - Sin x]`

Using the Sine Law

Sin (A-B)=Sin A *Cos B - Cos A * Sin B

as well as

`Sin ((\pi)/(3))=(√(3))/(2), Cos ((\pi)/(3))=(1)/(2)`

Option B:
`(1)/(2)[√(3)Cos x - Sin x]`

Answer 2

sin ( π / 3 - x ) =
= sin π/3 cos x - cos π/3 sin x = ( additional formulas )
= √3 / 2 cos x - 1/2 sin x =
= 1/2 ( √3 cos x - sin x )
Answer: B )