Convert to Trigonometric Form 4-4i

Problem

4−4*i

Solution

1. Identify the real part a=4 and the imaginary part b=−4 from the complex number z=a+b*i

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

r=√(,4+(−4)2)

r=√(,16+16)

r=√(,32)

r=4√(,2)

3. Determine the argument θ using the formula tan(θ)=b/a

tan(θ)=(−4)/4

tan(θ)=−1

4. Find the specific angle θ by noting that the point (4,−4) lies in the fourth quadrant.

θ=360−45

θ=315

θ=(7*π)/4

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

z=4√(,2)*(cos(315)+i*sin(315))

Final Answer

4−4*i=4√(,2)*(cos(315)+i*sin(315))

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