Convert to Trigonometric Form -2+2i

Problem

−2+2*i

Solution

1. Identify the real part a=−2 and the imaginary part b=2 from the complex number z=a+b*i

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

r=√(,(−2)2+2)

r=√(,4+4)

r=√(,8)

r=2√(,2)

3. Determine the argument θ by finding the angle such that cos(θ)=a/r and sin(θ)=b/r

cos(θ)=(−2)/(2√(,2))=−1/√(,2)=−√(,2)/2

sin(θ)=2/(2√(,2))=1/√(,2)=√(,2)/2

4. Find the angle θ in the second quadrant (since a<0 and b>0 that satisfies these values.

θ=(3*π)/4

5. Substitute r and θ into the trigonometric form z=r*(cos(θ)+i*sin(θ))

−2+2*i=2√(,2)*(cos((3*π)/4)+i*sin((3*π)/4))

Final Answer

−2+2*i=2√(,2)*(cos((3*π)/4)+i*sin((3*π)/4))

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