Convert to Trigonometric Form (12-8i)-(6-6i)

Problem

(12−8*i)−(6−6*i)

Solution

1. Subtract the complex numbers by combining real and imaginary parts.

(12−6)+(−8−(−6))*i

6−2*i

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

r=√(,6+(−2)2)

r=√(,36+4)

r=√(,40)=2√(,10)

3. Determine the argument θ using tan(θ)=b/a Since the point (6,−2) is in the fourth quadrant, we find the reference angle and adjust.

tan(θ)=(−2)/6=−1/3

θ=arctan(−1/3)

θ≈−18.43* or *341.57

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

2√(,10)*(cos(341.57)+i*sin(341.57))

Final Answer

(12−8*i)−(6−6*i)=2√(,10)*(cos(341.57)+i*sin(341.57))

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