Expand Using De Moivre's Theorem sin(2x)

Problem

sin(2*x)

Solution

1. State Euler's Formula and De Moivre's Theorem, which relates complex exponentials to trigonometric functions.

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

2. Set the exponent to n=2 to match the argument of the target expression sin(2*x)

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

3. Expand the left side of the equation using the binomial expansion (a+b)2=a2+2*a*b+b2

cos2(x)+2*i*sin(x)*cos(x)+i2*sin2(x)=cos(2*x)+i*sin(2*x)

4. Simplify the imaginary unit using the property i2=−1

cos2(x)+2*i*sin(x)*cos(x)−sin2(x)=cos(2*x)+i*sin(2*x)

5. Group the real and imaginary parts on the left side to prepare for comparison.

(cos2(x)−sin2(x))+i*(2*sin(x)*cos(x))=cos(2*x)+i*sin(2*x)

6. Equate the imaginary parts from both sides of the equation to find the expansion for sin(2*x)

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

Final Answer

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

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