Convert to Trigonometric Form (1+i)^5

Problem

(1+i)^5

Solution

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

2. Calculate the modulus r of z using the formula r=√(,a^2+b^2)

r=√(,1^2+1^2)

r=√(,2)

3. Determine 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 trigonometric form of z

z=√(,2)*(cos(π/4)+i*sin(π/4))

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

(1+i)^5=(√(,2))^5*(cos(5⋅π/4)+i*sin(5⋅π/4))

6. Simplify the modulus (√(,2))^5=2^2⋅√(,2)=4√(,2) and the argument (5*π)/4

(1+i)^5=4√(,2)*(cos((5*π)/4)+i*sin((5*π)/4))

Final Answer

(1+i)^5=4√(,2)*(cos((5*π)/4)+i*sin((5*π)/4))

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