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 from 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)=6

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

tan(θ)=(−3√(,3))/3=−√(,3)

4. Find the quadrant of the complex number to select the correct angle. Since a>0 and b<0 the point (3,−3√(,3)) is in Quadrant IV.

θ=300

θ=(5*π)/3

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

z=6*(cos((5*π)/3)+i*sin((5*π)/3))

Final Answer

3−3√(,3)*i=6*(cos((5*π)/3)+i*sin((5*π)/3))

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