Expand the Trigonometric Expression sin(4x)

Problem

sin(4*x)

Solution

1. Apply the double angle formula for sine, which is sin(2*θ)=2*sin(θ)*cos(θ) by treating 4*x as 2*(2*x)

sin(4*x)=2*sin(2*x)*cos(2*x)

2. Expand the sine term using the double angle formula sin(2*x)=2*sin(x)*cos(x)

2*sin(2*x)*cos(2*x)=2*(2*sin(x)*cos(x))*cos(2*x)

3. Expand the cosine term using the double angle formula cos(2*x)=cos2(x)−sin2(x)

4*sin(x)*cos(x)*(cos2(x)−sin2(x))

4. Distribute the terms to find the expanded form.

4*sin(x)*cos3(x)−4*sin3(x)*cos(x)

Final Answer

sin(4*x)=4*sin(x)*cos3(x)−4*sin3(x)*cos(x)

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