Convert to Trigonometric Form 47+16i

Problem

47+16*i

Solution

1. Identify the real part a=47 and the imaginary part b=16 of the complex number z=a+b*i

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

r=√(,47+16)

r=√(,2209+256)

r=√(,2465)

3. Determine the argument θ using the formula tan(θ)=b/a Since the point (47,16) is in the first quadrant, θ=arctan(16/47)

θ=arctan(16/47)

θ≈18.79

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

z=√(,2465)*(cos(arctan(16/47))+i*sin(arctan(16/47)))

Final Answer

47+16*i=√(,2465)*(cos(arctan(16/47))+i*sin(arctan(16/47)))

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