Convert to Trigonometric Form (4+3i)-(5-7i)

Problem

(4+3*i)−(5−7*i)

Solution

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

(4−5)+(3*i−(−7*i))

−1+10*i

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

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

r=√(,(−1)2+10)

r=√(,1+100)

r=√(,101)

4. Determine the argument θ using the formula tan(θ)=b/a Since the point (−1,10) is in the second quadrant, we use θ=π+arctan(b/a) or θ=180+arctan(b/a)

θ=arctan(10/(−1))+180

θ≈−84.29+180

θ≈95.71

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

z=√(,101)*(cos(95.71)+i*sin(95.71))

Final Answer

(4+3*i)−(5−7*i)=√(,101)*(cos(95.71)+i*sin(95.71))

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