Convert to Trigonometric Form 2 square root of 3+2i

Problem

2√(,3)+2*i

Solution

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

a=2√(,3)

b=2

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

r=√(,(2√(,3))2+2)

r=√(,12+4)

r=√(,16)

r=4

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

tan(θ)=2/(2√(,3))

tan(θ)=1/√(,3)

4. Find the angle θ in the first quadrant since both a and b are positive.

θ=π/6

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

z=4*(cos(π/6)+i*sin(π/6))

Final Answer

2√(,3)+2*i=4*(cos(π/6)+i*sin(π/6))

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