Convert to Trigonometric Form 6-3i+(5+4i)

Problem

6−3*i+(5+4*i)

Solution

1. Simplify the complex expression by combining the real parts and the imaginary parts.

6−3*i+5+4*i

(6+5)+(−3+4)*i

11+i

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

a=11

b=1

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

r=√(,11+1)

r=√(,121+1)

r=√(,122)

4. Determine the argument θ using the formula tan(θ)=b/a Since the point (11,1) is in the first quadrant, θ=arctan(1/11)

θ=arctan(1/11)

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

z=√(,122)*(cos(arctan(1/11))+i*sin(arctan(1/11)))

Final Answer

6−3*i+(5+4*i)=√(,122)*(cos(arctan(1/11))+i*sin(arctan(1/11)))

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