Convert to Trigonometric Form -2-2i

Problem

−2−2*i

Solution

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

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

r=√(,(−2)2+(−2)2)

r=√(,4+4)

r=√(,8)

r=2√(,2)

3. Determine the argument θ by finding the angle in the third quadrant (since both a and b are negative) where tan(θ)=b/a

tan(θ)=(−2)/(−2)

tan(θ)=1

θ=π+π/4

θ=(5*π)/4

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

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

Final Answer

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

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