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

Problem

−4√(,3)+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=1

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

r=√(,(−4√(,3))2+(1)2)

r=√(,16*(3)+1)

r=√(,48+1)

r=√(,49)

r=7

3. Determine the reference angle θ′ using the formula tan(θ′)=|b/a|

tan(θ′)=|1/(−4√(,3))|

tan(θ′)=1/(4√(,3))

θ′=arctan(√(,3)/12)

4. Find the argument θ by identifying the quadrant. Since a<0 and b>0 the complex number is in Quadrant II.

θ=π−arctan(√(,3)/12)

5. Write the complex number in trigonometric form z=r*(cos(θ)+i*sin(θ))

z=7*(cos(π−arctan(√(,3)/12))+i*sin(π−arctan(√(,3)/12)))

Final Answer

−4√(,3)+i=7*(cos(π−arctan(√(,3)/12))+i*sin(π−arctan(√(,3)/12)))

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