Convert to Trigonometric Form (-3-5i)-(4+2i)

Problem

(−3−5*i)−(4+2*i)

Solution

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

(−3−4)+(−5*i−2*i)

−7−7*i

2. Calculate the modulus r using the formula r=√(,a2+b2) where a=−7 and b=−7

r=√(,(−7)2+(−7)2)

r=√(,49+49)

r=√(,98)

r=7√(,2)

3. Determine the argument θ using the formula tan(θ)=b/a Since the point (−7,−7) is in the third quadrant, add π to the reference angle.

tan(θ)=(−7)/(−7)=1

θ=π+π/4

θ=(5*π)/4

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

7√(,2)*(cos((5*π)/4)+i*sin((5*π)/4))

Final Answer

(−3−5*i)−(4+2*i)=7√(,2)*(cos((5*π)/4)+i*sin((5*π)/4))

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