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

Problem

4−4√(,3)*i

Solution

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

a=4

b=−4√(,3)

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

r=√(,4+(−4√(,3))2)

r=√(,16+48)

r=√(,64)

r=8

3. Determine the argument θ by finding the angle such that cos(θ)=a/r and sin(θ)=b/r

cos(θ)=4/8=1/2

sin(θ)=(−4√(,3))/8=−√(,3)/2

4. Find the angle θ in the interval [0,2*π) that satisfies these conditions. Since the cosine is positive and the sine is negative, the angle is in the fourth quadrant.

θ=(5*π)/3

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

z=8*(cos((5*π)/3)+i*sin((5*π)/3))

Final Answer

4−4√(,3)*i=8*(cos((5*π)/3)+i*sin((5*π)/3))

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