Convert to Trigonometric Form -3+3 square root of 3i

Problem

−3+3√(,3)*i

Solution

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

a=−3

b=3√(,3)

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

r=√(,(−3)2+(3√(,3))2)

r=√(,9+27)

r=√(,36)

r=6

3. Determine the reference angle θ′ using the formula tan(θ′)=|b/a|

tan(θ′)=|(3√(,3))/(−3)|

tan(θ′)=√(,3)

θ′=π/3

4. Find the argument θ based on the quadrant. Since a<0 and b>0 the complex number lies in the second quadrant.

θ=π−θ′

θ=π−π/3

θ=(2*π)/3

5. Write the complex number in trigonometric form z=r*(cos(θ)+i*sin(θ))

z=6*(cos((2*π)/3)+i*sin((2*π)/3))

Final Answer

−3+3√(,3)*i=6*(cos((2*π)/3)+i*sin((2*π)/3))

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