Convert to Trigonometric Form 6-(8+3i)

Problem

6−(8+3*i)

Solution

1. Simplify the expression by distributing the negative sign and combining the real terms.

6−8−3*i

−2−3*i

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

a=−2

b=−3

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

r=√(,(−2)2+(−3)2)

r=√(,4+9)

r=√(,13)

4. Determine the argument θ using the formula tan(θ)=b/a Since the point (−2,−3) is in the third quadrant, add π (or 180 to the reference angle.

tan(θ)=(−3)/(−2)

θ=arctan(3/2)+π

θ≈0.9828+3.1416

θ≈4.1244

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

z=√(,13)*(cos(4.1244)+i*sin(4.1244))

Final Answer

6−(8+3*i)=√(,13)*(cos(4.1244)+i*sin(4.1244))

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