Trigonometry Examples

Expand Using De Moivre's Theorem cos(4x)
Step 1
A good method to expand is by using De Moivre's theorem . When , .
Step 2
Expand the right hand side of using the binomial theorem.
Expand:
Step 3
Use the Binomial Theorem.
Step 4
Simplify terms.
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Step 4.1
Simplify each term.
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Step 4.1.1
Apply the product rule to .
Step 4.1.2
Rewrite using the commutative property of multiplication.
Step 4.1.3
Rewrite as .
Step 4.1.4
Multiply by .
Step 4.1.5
Apply the product rule to .
Step 4.1.6
Factor out .
Step 4.1.7
Rewrite as .
Step 4.1.8
Rewrite as .
Step 4.1.9
Multiply by .
Step 4.1.10
Apply the product rule to .
Step 4.1.11
Rewrite as .
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Step 4.1.11.1
Rewrite as .
Step 4.1.11.2
Rewrite as .
Step 4.1.11.3
Raise to the power of .
Step 4.1.12
Multiply by .
Step 4.2
Reorder factors in .
Step 5
Take out the expressions with the imaginary part, which are equal to . Remove the imaginary number .