Convert to Trigonometric Form 4 square root of 3-4i

Problem

4√(,3)−4*i

Solution

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

a=4√(,3)

b=−4

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

r=√(,(4√(,3))2+(−4)2)

r=√(,48+16)

r=√(,64)

r=8

3. Determine the argument θ using the relation tan(θ)=b/a

tan(θ)=(−4)/(4√(,3))

tan(θ)=−1/√(,3)

4. Find the angle θ in the correct quadrant. Since a>0 and b<0 the point is in Quadrant IV.

θ=330

θ=(11*π)/6

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

z=8*(cos(330)+i*sin(330))

Final Answer

4√(,3)−4*i=8*(cos(330)+i*sin(330))

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