Convert to Trigonometric Form -10-2i

Problem

−10−2*i

Solution

1. Identify the real part a=−10 and the imaginary part b=−2 of the complex number z=a+b*i

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

r=√(,(−10)2+(−2)2)

r=√(,100+4)

r=√(,104)

r=2√(,26)

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

tan(θ′)=|(−2)/(−10)|

tan(θ′)=1/5

θ′=arctan(1/5)

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

θ=π+arctan(1/5)

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

z=2√(,26)*(cos(π+arctan(1/5))+i*sin(π+arctan(1/5)))

Final Answer

−10−2*i=2√(,26)*(cos(π+arctan(1/5))+i*sin(π+arctan(1/5)))

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