Convert to Trigonometric Form -5 square root of 3-5i

Problem

−5√(,3)−5*i

Solution

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

a=−5√(,3)

b=−5

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

r=√(,(−5√(,3))2+(−5)2)

r=√(,75+25)

r=√(,100)

r=10

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

tan(θ)=(−5)/(−5√(,3))

tan(θ)=1/√(,3)

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

θ=180+30

θ=210

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

z=10*(cos(210)+i*sin(210))

Final Answer

−5√(,3)−5*i=10*(cos(210)+i*sin(210))

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