Convert to Trigonometric Form (1+i)^20

Problem

(1+i)20

Solution

1. Identify the complex number z=1+i and find its modulus r

r=√(,1+1)

r=√(,2)

2. Determine the argument θ for z=1+i Since the point (1,1) is in the first quadrant:

tan(θ)=1/1

θ=π/4

3. Write the complex number z in trigonometric form r*(cos(θ)+i*sin(θ))

z=√(,2)*(cos(π/4)+i*sin(π/4))

4. Apply De Moivre's Theorem, which states [r*(cos(θ)+i*sin(θ))]n=rn*(cos(n*θ)+i*sin(n*θ)) where n=20

z20=(√(,2))20*(cos(20⋅π/4)+i*sin(20⋅π/4))

5. Simplify the modulus and the argument.

(√(,2))20=(2(1/2))20=2=1024

20⋅π/4=5*π

6. Reduce the argument 5*π to its coterminal equivalent within the interval [0,2*π)

5*π−4*π=π

Final Answer

(1+i)20=1024*(cos(π)+i*sin(π))

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