Express sin(x+y+z) in terms of cosines and sines of x, y and z
To express sin(x+y+z) in terms of cosines and sines of x, y, and z, we can use the trigonometric identity for the sine of the sum of three angles.
The identity is:
sin(A + B + C) = sin(A) * cos(B) * cos(C) + cos(A) * sin(B) * cos(C) + cos(A) * cos(B) * sin(C) - sin(A) * sin(B) * sin(C)
Applying this identity to sin(x+y+z), we have:
sin(x+y+z) = sin(x) * cos(y+z) * cos(y+z) + cos(x) * sin(y+z) * cos(y+z) + cos(x) * cos(y+z) * sin(y+z) - sin(x) * sin(y+z) * sin(y+z)
Now, we need to use the trigonometric identities for cosines and sines of the sum of two angles to simplify this expression further.
cos(y+z) = cos(y) * cos(z) - sin(y) * sin(z)
sin(y+z) = sin(y) * cos(z) + cos(y) * sin(z)
Substituting these identities into the expression, we get:
sin(x+y+z) = sin(x) * (cos(y) * cos(z) - sin(y) * sin(z)) * (cos(y) * cos(z) - sin(y) * sin(z)) + cos(x) * (sin(y) * cos(z) + cos(y) * sin(z)) * (cos(y) * cos(z) - sin(y) * sin(z)) + cos(x) * (cos(y) * cos(z) - sin(y) * sin(z)) * (sin(y) * cos(z) + cos(y) * sin(z)) - sin(x) * (sin(y) * cos(z) + cos(y) * sin(z)) * (sin(y) * cos(z) + cos(y) * sin(z))
Simplifying further by expanding and collecting like terms, we can arrive at the expression for sin(x+y+z) in terms of cosines and sines of x, y, and z.