Question

if, sinx+siny+sinz = 0 = cosx+cosy+cosz
then the expression
cos(theta-x) +cos(theta-y) +cos(theta-z) for theta belongs to R is,

1) independent of theta but dependent on x, y, z.
2) dependent on theta but independent of x, y, z.
3) dependent on x, y, z and theta.4) independent of x, y, z and theta.​

Answer 1

Given: sinx + siny + sinz = 0 = cosx + cosy + cosz

To find: cos(θ - x) + cos(θ - y) + cos(θ - z)

Solution:

• We must know that,
• cos(A - B) = cosA cosB + sinA sinB

Now, cos(θ - x) + cos(θ - y) + cos(θ - z)

= cosθ cosx + sinθ sinx + cosθ cosy + sinθ siny + cosθ cosz + sinθ sinz

= cosθ cosx + cosθ cosy + cosθ cosz + sinθ sinx + sinθ siny + sinθ sinz

= cosθ (cosx + cosy + cosz) + sinθ (sinx + siny + sinz)

= cosθ * 0 + sinθ * 0

= 0 + 0

= 0

Answer:

• Option 4 is correct.
• The given expression is independent of x, y, z and θ.