Convert to Trigonometric Form -6-2i

Problem

−6−2*i

Solution

1. Identify the real part a=−6 and the imaginary part b=−2 of the complex number z=a+b*i

2. Calculate the modulus r using the formula r=√(,a^2+b^2)

r=√(,(−6)^2+(−2)^2)

r=√(,36+4)

r=√(,40)

r=2√(,10)

3. Determine the reference angle θ^′ using the formula tan(θ^′)=|b/a|

tan(θ^′)=|(−2)/(−6)|

tan(θ^′)=1/3

θ^′=arctan(1/3)

4. Find the argument θ by considering the quadrant. Since both a<0 and b<0 the complex number lies in Quadrant III.

θ=π+arctan(1/3)

5. Write the trigonometric form using the structure z=r*(cos(θ)+i*sin(θ))

z=2√(,10)*(cos(π+arctan(1/3))+i*sin(π+arctan(1/3)))

Final Answer

−6−2*i=2√(,10)*(cos(π+arctan(1/3))+i*sin(π+arctan(1/3)))

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