Expand the Trigonometric Expression 3+4i

Problem

3+4*i

Solution

1. Identify the complex number in the form z=a+b*i where a=3 and b=4

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

r=√(,3+4)

r=√(,9+16)

r=√(,25)

r=5

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

θ=arctan(4/3)

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

z=5*(cos(arctan(4/3))+i*sin(arctan(4/3)))

Final Answer

3+4*i=5*(cos(arctan(4/3))+i*sin(arctan(4/3)))

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