Convert to Trigonometric Form 1+3i

Problem

1+3*i

Solution

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

a=1

b=3

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

r=√(,1+3)

r=√(,1+9)

r=√(,10)

3. Determine the argument θ using the formula θ=arctan(b/a) Since the point (1,3) is in the first quadrant, no adjustment is needed.

θ=arctan(3/1)

θ=arctan(3)

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

z=√(,10)*(cos(arctan(3))+i*sin(arctan(3)))

Final Answer

1+3*i=√(,10)*(cos(arctan(3))+i*sin(arctan(3)))

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