Convert to Trigonometric Form 3-4i

Problem

3−4*i

Solution

1. Identify the real part a=3 and the imaginary part b=−4 from the complex number z=a+b*i

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

r=√(,3+(−4)2)

r=√(,9+16)

r=√(,25)

r=5

3. Determine the argument θ using the formula tan(θ)=b/a Since the point (3,−4) is in the fourth quadrant, we find the reference angle and adjust accordingly.

tan(θ)=(−4)/3

θ=arctan((−4)/3)

θ≈−53.13

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

z=5*(cos(−53.13)+i*sin(−53.13))

Final Answer

3−4*i=5*(cos(−53.13)+i*sin(−53.13))

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