mamieengler
01.11.2020 •
Mathematics
Which expression represents a 5th root of –i?
cos(7pie/4)+sin(7pie/4)
cos(7pie/6)+sin(7pie/6)
cos(7pie/8)+sin(7pie/8)
cos(7pie/10)+sin(7pie/10)
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Ответ:
The 5th root of -i is (d)
By DeMoivre's formula, we have:
The 5th root of -i means that: n = 5.
So, we have:
Set to any expression such that:
So, we have:
Expand
Take 5th roots of both sides
This means that:
is the 5th root of -i
Hence, option (d) is correct
Read more about complex numbers at:
link
Ответ:
Use DeMoivre's formula:
(cos(θ) + i sin(θ))ⁿ = cos(n θ) + i sin(n θ)
If a given number is a 5th root of -i, raising it to the 5th power should return -i.
Since -i = cos(-π/2) + i sin(-π/2), you should check any number with an argument θ such that 5θ = -π/2. This happens only for the last choice:
(cos(7π/10) + i sin(7π/10))⁵ = cos(35π/10) + i sin(35π/10)
and 35π/10 is equivalent to -π/2 (modulo 2π).
Ответ: