Convert to Trigonometric Form -3-3i

Problem

−3−3*i

Solution

1. Identify the real part a=−3 and the imaginary part b=−3 of the complex number z=a+b*i

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

r=√(,(−3)2+(−3)2)

r=√(,9+9)

r=√(,18)

r=3√(,2)

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

tan(θ)=(−3)/(−3)

tan(θ)=1

4. Locate the quadrant of the complex number. Since both a and b are negative, the point (−3,−3) lies in Quadrant III.

5. Find the angle θ in Quadrant III where tan(θ)=1

θ=180+45

θ=225

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

Final Answer

−3−3*i=3√(,2)*(cos(225)+i*sin(225))

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