Using DeMoivre;s theorem compute the complex number z=(i−32+i)7 ?

1 Answer
Hriman
Jul 5, 2018

z=−8−8i in standard form

z=8√2(cos(5π4)+isin(5π4)) in trigonometric form

Explanation:

Before we use DeMoivre's Theorem, let's divide:

(i−32+i⋅2−i2−i)7=(5i−55)7=(−1+i)7

Now let's convert to trigonometric form so we can apply the theorem:

r=√x2+y2
r=√(−1)2+12
r=√2

θ=arctan(yx)
With the angles, always note the quadrant you're in, in this case with a negative x-value and a positive y-value, you will be in the 2nd quadrant.
θ=arctan(1−1)=−45∘+180∘=135∘=3π4

Putting it in trigonometric form:
z=r(cosθ+isinθ)

z=(√2(cos(3π4)+isin(3π4)))7

When applying a power to numbers in trigonometric form, apply the power to the r, and multiply the angle with the exponent:

z=((√2)7(cos(3π⋅74)+isin(3π⋅74))

z=8√2(cos(21π4)+isin(21π4))

21π4 is coterminal with 5π4

z=8√2(cos(5π4)+isin(5π4))

z=8√2⋅cos(5π4)+8√2⋅isin(5π4)

z=8√2⋅−√22+8√2⋅−√22i

z=−8−8i

Related questions
Impact of this question
1107 views around the world
You can reuse this answer
Creative Commons License