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)

r=√(,9+9)

r=√(,18)

r=3√(,2)

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

tan(θ)=3/(−3)

tan(θ)=−1

4. Locate the quadrant of the complex number. Since a<0 and b>0 the point (−3,3) lies in Quadrant II.

θ=(3*π)/4

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

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

Final Answer

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

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