Convert to Trigonometric Form (1+i)^4

Problem

(1+i)4

Solution

1. Identify the complex number z=1+i inside the exponent.

2. Calculate the modulus r of z using the formula r=√(,a2+b2)

r=√(,1+1)

r=√(,2)

3. Find the argument θ using tan(θ)=b/a Since the point (1,1) is in the first quadrant, θ=π/4

θ=arctan(1/1)

θ=π/4

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

z=√(,2)*(cos(π/4)+i*sin(π/4))

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

z4=(√(,2))4*(cos(4⋅π/4)+i*sin(4⋅π/4))

6. Simplify the modulus and the argument.

(√(,2))4=4

4⋅π/4=π

Final Answer

(1+i)4=4*(cos(π)+i*sin(π))

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