Express sin(x y z) in terms of cosines and sines of x, y and z.
What does it even mean by sines of x, y and z? Sin(x), Sin(y), Sin(z)?
nvm solved it
Yes, when we say "sines of x, y, and z," we mean sin(x), sin(y), and sin(z) respectively. In order to express sin(x y z) in terms of cosines and sines of x, y, and z, we can use the trigonometric identity known as the product-to-sum formula.
The product-to-sum formula states that for any two angles θ and φ, the following identity holds:
sin(θ) * sin(φ) = (1/2) * [cos(θ - φ) - cos(θ + φ)]
Using this formula, we can apply it to the expression sin(x y z).
sin(x y z) = sin(x) * sin(y z)
Now, we can apply the product-to-sum formula to sin(y z) by considering θ = y and φ = z:
sin(y z) = sin(y) * sin(z) = (1/2) * [cos(y - z) - cos(y + z)]
Now, substituting this result back into the original expression, we get:
sin(x y z) = sin(x) * [sin(y) * sin(z)] = (1/2) * sin(x) * [cos(y - z) - cos(y + z)]