Convert to Trigonometric Form 3-5i

Problem

3−5*i

Solution

1. Identify the real part a=3 and the imaginary part b=−5 from the complex number z=a+b*i

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

r=√(,3+(−5)2)

r=√(,9+25)

r=√(,34)

3. Determine the argument θ using the formula tan(θ)=b/a Since the point (3,−5) is in the fourth quadrant, we find the reference angle and adjust accordingly.

tan(θ)=(−5)/3

θ=arctan((−5)/3)

θ≈−1.03* radians

4. Substitute the values of r and θ into the trigonometric form z=r*(cos(θ)+i*sin(θ))

z=√(,34)*(cos(arctan(−5/3))+i*sin(arctan(−5/3)))

Final Answer

3−5*i=√(,34)*(cos(arctan(−5/3))+i*sin(arctan(−5/3)))

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