Trig SUm and Difference formula homework?

Question

Simplify to a trig function of a single angle

sin (pi/6 + x) + sin (pi/6 - x)

x = thadah

Answer 1

sin (π/6 + θ ) = sin π/6 cos θ + cos π/6 sin θ
sin (π/6 - θ ) = sin π/6 cos θ - cos π/6 sin θ
Sum = 2 (sin π/6) (cos θ)
Sum = (2) (1/2) (cos θ)
Sum = cos θ

Answer 2

sin (pi/6 + x) + sin (pi/6 - x)
= (sin pi/6 cos x + sin x cos pi/6)
+ (sin pi/6 cos (-x) - sin (-x) cos pi/6)
= (1/2) cos x + ((sqrt 3)/2) sin x + (1/2) cos x + ((sqrt 3)/2) sin x
= (sqrt 3) sin x + cos x.

Answer 3

fidel got it close but not quite right...

The formula you need is:

sin (angle 1 + angle 2) = sin(angle1)*cos(angle2) + sin(angle2)*cos(angle1)

So, sin (pi/6 + x) + sin (pi/6 - x)
= sin(pi/6)*cosx + sinx*cos(pi/6) + sin(pi/6)*cos(-x) + sin(-x)*cos(pi/6)
= 0.5*cosx + 0.5*sqrt3*sinx + 0.5*cosx - 0.5*sqrt3*sinx

[Since cos(-x) = cosx, and sin (-x) = -sin(x)]

= cos x