Convert to Trigonometric Form -4-4i

Problem

−4−4*i

Solution

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

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

r=√(,(−4)2+(−4)2)

r=√(,16+16)

r=√(,32)

r=4√(,2)

3. Determine the quadrant of the complex number. Since both a<0 and b<0 the point (−4,−4) lies in Quadrant III.

4. Find the argument θ using the reference angle α=arctan(b/a)

α=arctan((−4)/(−4))

α=arctan(1)

α=π/4

5. Adjust the angle for Quadrant III using θ=π+α

θ=π+π/4

θ=(5*π)/4

6. Write in trigonometric form z=r*(cos(θ)+i*sin(θ))

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

Final Answer

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

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