6 pts Simplify the expression below using a sum and differen

(6 pts) Simplify the expression below using a sum and difference formula. sina sin(x + 6 3 sin (x--)- cos(x + )- Remark: Your answers cannot contain trigonometric functions. It must be an arithmetic expression. 3

Solution

sin(x-pi/6) = sin x cos(pi/6)- cos x sin(pi/6)

             = sqrt(3)/2 sin x -cos x/2 = (1/2)(sqrt3 sin x -cos x)

sin(x + 7pi/3) = sin x cos (7pi/3) + cos x sin(7pi/3) =sin x (1/2) +cos x(sqrt3/2) =(1/2)(sin x +sqrt(3)cos x)

sin (x-3pi/4) = sin x cos(3pi/4)-cos x sin(3pi/4) =(-1/sqrt2)sin x -(1/sqrt2)cos x=(-1/sqrt2)(sin x +cos x)

cos(x+2pi/3) = cos x cos(2pi/3)-sinx sin(2pi/3) = cos x(-1/2) -sinx(sqrt3/2) =(-1/2)(cos x + sqrt(3) sin x)