Convert to Trigonometric Form 1-i

Problem

1−i

Solution

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

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

r=√(,1+(−1)2)

r=√(,2)

3. Determine the argument θ using the tangent ratio tan(θ)=b/a Since the point (1,−1) is in the fourth quadrant, θ will be in the interval ((3*π)/2,2*π) or (−π/2,0)

tan(θ)=(−1)/1

θ=−π/4

4. Substitute the values of 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

1−i=√(,2)*(cos(−π/4)+i*sin(−π/4))

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