Convert to Trigonometric Form (-7+4i)-(-9-3i)

Problem

(−7+4*i)−(−9−3*i)

Solution

1. Simplify the complex expression by distributing the negative sign and combining like terms.

(−7+4*i)−(−9−3*i)=−7+4*i+9+3*i

(−7+9)+(4*i+3*i)=2+7*i

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

a=2

b=7

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

r=√(,2+7)

r=√(,4+49)

r=√(,53)

4. Determine the argument θ using the formula tan(θ)=b/a Since the point (2,7) is in the first quadrant, θ=arctan(b/a)

θ=arctan(7/2)

θ≈1.29* radians

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

z=√(,53)*(cos(arctan(7/2))+i*sin(arctan(7/2)))

Final Answer

(−7+4*i)−(−9−3*i)=√(,53)*(cos(arctan(7/2))+i*sin(arctan(7/2)))

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