Convert to Trigonometric Form -1-i

Problem

−1−i

Solution

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

a=−1

b=−1

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

r=√(,(−1)2+(−1)2)

r=√(,1+1)

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)=−√(,2)/2

sin(θ)=(−1)/√(,2)=−√(,2)/2

4. Locate the quadrant of the complex number. Since both a and b are negative, the point (−1,−1) lies in Quadrant III.

θ=(5*π)/4

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

−1−i=√(,2)*(cos((5*π)/4)+i*sin((5*π)/4))

Final Answer

−1−i=√(,2)*(cos((5*π)/4)+i*sin((5*π)/4))

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