Expand the Trigonometric Expression sin(x)^3

Problem

sin3(x)

Solution

1. Identify the expression as a power of a sine function that can be rewritten using the product of a squared term and a linear term.

sin3(x)=sin2(x)*sin(x)

2. Apply the Pythagorean identity to replace the squared sine term with an expression involving cosine.

sin2(x)=1−cos2(x)

3. Substitute the identity back into the expression.

sin3(x)=(1−cos2(x))*sin(x)

4. Distribute the sine term to expand the expression.

sin3(x)=sin(x)−cos2(x)*sin(x)

5. Apply the power reduction formula if the goal is to express the result in terms of multiple angles. Use the identity sin3(x)=(3*sin(x)−sin(3*x))/4

sin3(x)=3/4*sin(x)−1/4*sin(3*x)

Final Answer

sin3(x)=(3*sin(x)−sin(3*x))/4

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