Convert to Trigonometric Form 1+ square root of 3i

Problem

1+√(,3)*i

Solution

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

a=1

b=√(,3)

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

r=√(,1+(√(,3))2)

r=√(,1+3)

r=2

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

tan(θ)=√(,3)/1

θ=arctan(√(,3))

4. Evaluate the angle θ based on the quadrant. Since both a and b are positive, the number is in the first quadrant.

θ=π/3

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

z=2*(cos(π/3)+i*sin(π/3))

Final Answer

1+√(,3)*i=2*(cos(π/3)+i*sin(π/3))

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