Expand Using DeMoivre’s Theorem sin(7x)

Math
A good method to expand is by using De Moivre’s theorem . When , .
Expand the right hand side of using the binomial theorem.
Expand:
Use the Binomial Theorem.
Simplify terms.
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Simplify each term.
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Apply the product rule to .
Rewrite using the commutative property of multiplication.
Rewrite as .
Multiply by .
Apply the product rule to .
Rewrite using the commutative property of multiplication.
Factor out .
Rewrite as .
Rewrite as .
Multiply by .
Apply the product rule to .
Rewrite using the commutative property of multiplication.
Rewrite as .
Tap for more steps…
Rewrite as .
Rewrite as .
Raise to the power of .
Multiply by .
Apply the product rule to .
Rewrite using the commutative property of multiplication.
Factor out .
Rewrite as .
Tap for more steps…
Rewrite as .
Rewrite as .
Raise to the power of .
Multiply by .
Apply the product rule to .
Factor out .
Rewrite as .
Tap for more steps…
Rewrite as .
Rewrite as .
Raise to the power of .
Multiply by .
Rewrite as .
Rewrite as .
Multiply by .
Apply the product rule to .
Rewrite as .
Tap for more steps…
Factor out .
Factor out .
Rewrite as .
Tap for more steps…
Rewrite as .
Rewrite as .
Raise to the power of .
Multiply by .
Rewrite as .
Rewrite as .
Reorder factors in .
Take out the expressions with the imaginary part, which are equal to . Remove the imaginary number .
Expand Using DeMoivre’s Theorem sin(7x)

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