Convert to Trigonometric Form square root of 2+ square root of 2i

Problem

√(,2)+√(,2)*i

Solution

1. Identify the real part a=√(,2) and the imaginary part b=√(,2) of the complex number z=a+b*i

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

r=√(,(√(,2))2+(√(,2))2)

r=√(,2+2)

r=√(,4)

r=2

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

tan(θ)=√(,2)/√(,2)

tan(θ)=1

4. Find the angle θ in the first quadrant since both a and b are positive.

θ=π/4

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

z=2*(cos(π/4)+i*sin(π/4))

Final Answer

√(,2)+√(,2)*i=2*(cos(π/4)+i*sin(π/4))

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