Convert to Trigonometric Form -5+5i

Problem

−5+5*i

Solution

1. Identify the real part a=−5 and the imaginary part b=5 of the complex number z=a+b*i

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

r=√(,(−5)2+5)

r=√(,25+25)

r=√(,50)

r=5√(,2)

3. Determine the argument θ by finding the angle such that tan(θ)=b/a

tan(θ)=5/(−5)

tan(θ)=−1

4. Locate the quadrant of the complex number. Since a<0 and b>0 the point (−5,5) lies in Quadrant II.

θ=(3*π)/4

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

z=5√(,2)*(cos((3*π)/4)+i*sin((3*π)/4))

Final Answer

−5+5*i=5√(,2)*(cos((3*π)/4)+i*sin((3*π)/4))

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