Convert to Trigonometric Form (1-i)^8

Problem

(1−i)8

Solution

1. Identify the complex number z=1−i inside the exponent.

2. Calculate the modulus r of the complex number z=a+b*i using the formula r=√(,a2+b2)

r=√(,1+(−1)2)

r=√(,2)

3. Determine the argument θ using the coordinates (1,−1) which lie in the fourth quadrant.

tan(θ)=(−1)/1

θ=(7*π)/4

4. Write in trigonometric form z=r*(cos(θ)+i*sin(θ))

z=√(,2)*(cos((7*π)/4)+i*sin((7*π)/4))

5. Apply De Moivre's Theorem to raise the complex number to the 8th power using zn=rn*(cos(n*θ)+i*sin(n*θ))

(1−i)8=(√(,2))8*(cos(8⋅(7*π)/4)+i*sin(8⋅(7*π)/4))

6. Simplify the exponent and the angle.

(√(,2))8=2=16

8⋅(7*π)/4=14*π

7. Find the coterminal angle for 14*π by subtracting multiples of 2*π

14*π≡0*(mod()*2*π)

8. State the final trigonometric form using the simplified angle.

16*(cos(0)+i*sin(0))

Final Answer

(1−i)8=16*(cos(0)+i*sin(0))

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