Expand Using De Moivre's Theorem cos(x)

Problem

cos(x)

Solution

1. Identify the objective, which is to express cos(x) in terms of complex exponentials using De Moivre's Theorem and Euler's formula.

2. State Euler's formula, which relates complex exponentials to trigonometric functions.

cos(x)+i*sin(x)=e(i*x)

3. State the formula for the conjugate, replacing x with −x

cos(−x)+i*sin(−x)=e(−i*x)

4. Simplify using the even-odd identities cos(−x)=cos(x) and sin(−x)=−sin(x)

cos(x)−i*sin(x)=e(−i*x)

5. Add the two equations to isolate the cosine term.

(cos(x)+i*sin(x))+(cos(x)−i*sin(x))=e(i*x)+e(−i*x)

6. Combine like terms on the left side.

2*cos(x)=e(i*x)+e(−i*x)

7. Divide by 2 to solve for cos(x)

cos(x)=(e(i*x)+e(−i*x))/2

Final Answer

cos(x)=(e(i*x)+e(−i*x))/2

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