Convert to Trigonometric Form 1- square root of 3i

Problem

1−√(,3)*i

Solution

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

a=1

b=−√(,3)

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

r=√(,1+(−√(,3))2)

r=√(,1+3)

r=2

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

tan(θ)=(−√(,3))/1

tan(θ)=−√(,3)

4. Locate the quadrant of the complex number. Since a>0 and b<0 the point (1,−√(,3)) is in Quadrant IV.

θ=300

θ=(5*π)/3

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

z=2*(cos((5*π)/3)+i*sin((5*π)/3))

Final Answer

1−√(,3)*i=2*(cos((5*π)/3)+i*sin((5*π)/3))

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