Convert to Trigonometric Form -1+i square root of 3

Problem

−1+i√(,3)

Solution

1. Identify the real part a=−1 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=√(,(−1)2+(√(,3))2)

r=√(,1+3)

r=2

3. Determine the argument θ by finding the angle such that cos(θ)=a/r and sin(θ)=b/r

cos(θ)=−1/2

sin(θ)=√(,3)/2

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

θ=(2*π)/3

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

Final Answer

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

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