Write 3theta - sin 4theta as a product.
I assume you meant (using x)
sin3x - sin4x
Use the sum-to-product formula:
sinA-sinB = 2 cos(A+B)/2 sin(A-B)/2
To write 3θ - sin(4θ) as a product, we can use the trigonometric identity:
sin(A) - sin(B) = 2sin((A - B)/2)cos((A + B)/2)
In this case, let A = 4θ and B = 0, then we have:
3θ - sin(4θ) = 3θ - 2sin(2θ)cos(2θ)
Now, we can use another trigonometric identity:
sin(2θ) = 2sin(θ)cos(θ)
Substituting this into the equation, we get:
3θ - 2sin(2θ)cos(2θ) = 3θ - 2(2sin(θ)cos(θ))cos(2θ)
Now, let's combine the terms and simplify further:
= 3θ - 4sin(θ)cos(θ)cos(2θ)
At this point, we can't simplify further or write it as a single product since we have a product of three terms. However, we have expressed 3θ - sin(4θ) as a sum of two terms.