Math Problem!?

Question

What is the exact value of cos x - sin(x + 150) + cos(x + 120)?

step-by-step solution would be greatly appreciated

N.B. All numbers are in degrees

Sorry! I forgot to mention that I had expanded the sin and cos brackets.

When I convert everything into radians, and cancel out, I'm just left with cos x.

Is this right! If not, what is the best way to do it (aprat from Euler's theorem).

I thought if sin(x + 150) and cos(x+120) were seperate terms then:

sin(x + 150) = sinx*cos150 + sin150*cosx

and

cos(x + 120) = cos120cosx - sin120-sinx

Sorry if this has confused anyone

Answer 1

cos(a+b) = cos(a) cos(b) - sin(a) sin(b)
and
sin(a+b) = sin(a) cos(b) + sin (b) cos(a)

So first expand sin(x + 150°) into sin(x)cos(150°) +sin(150°)cos(x)
and expand cos(x + 120°) into cos(x)cos(120°) - sin(x)sin(120°).
now sin (150°) is 1/2 and cos(150°) is sqrt(3)/2, sin(120°) is sqrt(3)/2 and cos(120°) is 1/2. So you have cos x - sqrt(3)/2(sin x) - 1/2cos(x) + 1/2cos x - sqrt(3)/2 sin x = cos x -sqrt(3)sin x. I assume you followed the same steps and mixed up a minus sign. I did the same thing first time around.

Since you are not leaving a degree terms, it doesn't actually matter if you do this in degrees or radians.

Answer 2

Use the sin(a+b) and cos(a+b) equalities. 150° = 5pi/6, 120° = 2pi/3.

Answer 3

sin(x+150) = sinx*cos150 + cosx*sin150 = rt3/2*sinx + cosx/2

cos(x+120) = cosx*cos120 - sinx*sin120 = cosx/2 - rt3/2*sinx

Combining all terms,

cosx - rt3/2*sinx - cosx/2 + cosx/2 -rt3/2*sinx =

cosx - rt3*sinx

Answer 4

cosx - sinxcos150 - cosxsin150 + cosxcos150 - sinxsin150
Take it from here

Answer 5

you can use eulers therium to figure this out