Convert to Trigonometric Form -1-6i

Problem

−1−6*i

Solution

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

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

r=√(,(−1)2+(−6)2)

r=√(,1+36)

r=√(,37)

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

tan(θ′)=|(−6)/(−1)|

tan(θ′)=6

θ′=arctan(6)

4. Find the argument θ by identifying the quadrant. Since both a and b are negative, the complex number lies in Quadrant III.

θ=π+arctan(6)

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

−1−6*i=√(,37)*(cos(π+arctan(6))+i*sin(π+arctan(6)))

Final Answer

−1−6*i=√(,37)*(cos(π+arctan(6))+i*sin(π+arctan(6)))

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