Mathematics, 27.06.2021 01:00, ayoismeisalex
The imaginary number 3 + 4i is represented on the complex plane as (3, 4). It could also be written as 5(cos 53o + isin 53o). Cos 53o + isin 53o can be abbreviated as cis53o where cis stands for cos + isin. This conversion is done the same as converting regular Cartesian coordinates to polar coordinates.
It is easy to add or subtract complex numbers, but multiplying and dividing them is more complicated.
For example (3 + 4i) x (8 - 6i) = (3)(8) + (3)(-6i) + (4i)(8) + (4i)(-6i)
= 24 - 18i + 32i + 24
= 48 + 14i
This was the same as multiplication of binomials. So how would it look in polar form? First, convert the standard form to polar form. 3 + 4i = 5(cos 53o + isin 53o).
For example 5(cos 53o + isin 53o) x 10(cos -37o + isin -37o) =50[(cos 53o)(cos -37o) + (cos 53o)(isin -37o) + (isin 53o)(cos -37o) + (isin 53o)(isin -37o)
Rearranging and remembering i2 = -1 =50[(cos 53o)(cos -37o) - (sin 53o)(sin -37o)] + i[(cos 53o)(sin -37o) + (sin 53o)(cos -37o)]
Then using double angle formulas =50[cos (53o + -37o) + isin(53o + -37o)]
Giving this as the answer in polar form =50[cos 16o + isin16o]
Which is the same as above in standard form. = 48 + 14i
DEMOIVRE’S THEOREM
Abraham DeMoivre developed a theorem to help with complex numbers raised to a power. It is stated thusly:
If z = a + bi is any complex number with trig form r (cos + isin) and n is any positive integer, then the nth power z is given by
zn = rn (cos n + isin n).
Here is an example.
Find [2(cos 45o+ isin 45o)]6. Write the result in standard form.
[2(cos 45o + isin 45o)]6 =26[cos 6(45o) + isin 6(45o)]
=26[cos(270o) + isin(270o)]
=64[0 + -1i]
= -64i
Now work these problems and turn in the solutions to your teacher.
Write 2 - 7i in polar coordinates on the complex plane.
Find (1 + i)8 using DeMoivre's Theorem.
I. Write 2 - 7i in polar coordinates on the complex plane.
II. Find ( 1 + i) ^8using DeMoivre's theorem.
Answers: 2
Mathematics, 21.06.2019 14:00, kaylaelaine18
In the film, zeke the zombie slayer versus the zombies from space, zeke slayed 140 zombies. in the sequel, zeke the zombie slayer versus the zombies from space, part 2, he slayed 5% more zombies. how many zombies were slain by zeke in the sequel?
Answers: 1
Mathematics, 21.06.2019 16:30, chloesmolinski0909
Solve the equation w^3 = 1,000 i honestly dont know how to punch this into a calculator pls
Answers: 1
Mathematics, 21.06.2019 17:00, nataliahenderso
Acylinder and its dimensions are shown. which equation can be used to find v, the volume of the cylinder in cubic centimeters?
Answers: 3
The imaginary number 3 + 4i is represented on the complex plane as (3, 4). It could also be written...
Chemistry, 16.01.2022 06:00
Business, 16.01.2022 06:00
Mathematics, 16.01.2022 06:10
Mathematics, 16.01.2022 06:10
Mathematics, 16.01.2022 06:10
Physics, 16.01.2022 06:10
Health, 16.01.2022 06:10
Mathematics, 16.01.2022 06:10