Find
[4(cos115+isin115)]^6
4^6=4096
115*6=690
cos(690)=sqrt(3)/2
sin(690)=-1/2
De Moivre's Theorem:
(r(cosØ + i sinØ))^n
= r^n(cos(nØ) + i sin(nØ))
in our case ....
[4(cos115+isin115)]^6
= 4^6( cos 690 + i sin 690)
I will assume my meant degrees, since the numbers work out nicely in degrees.
Now 690° is coterminal with 330° or -30°
sin(-30°) = -1/2, and cos(-30) = √3/2
so continuing...
= 4096(√3/2 - (1/2)i)
= 2048√3 - 2048
oops, left out the i in the last line
= 2048√3 - 2048i
(I could always claim that you should have "imagined" it)
To find the value of the expression [4(cos115+isin115)]^6, we can follow these steps:
Step 1: Convert from polar form to rectangular form.
Given:
- The angle is 115 degrees.
- The magnitude is 4.
Using Euler's formula: e^(ix) = cos(x) + isin(x), we can convert the polar form to rectangular form as follows:
cos(115) + isin(115) = e^(i * 115°)
= e^(i * (π/180) * 115)
= e^(i * (π/180) * 115)
= e^(i * (115π/180))
Step 2: Find the value of [4(cos115+isin115)] in the rectangular form.
Let's denote [4(cos115+isin115)] in rectangular form as a + bi, where a represents the real part and b represents the imaginary part.
Using Euler's formula: e^(ix) = cos(x) + isin(x), we can convert the rectangular form to polar form as follows:
a + bi = 4(cos115 + isin115)
= 4e^(i * (115π/180))
= 4 * e^((i * π * 115) / 180)
= 4 * (cos((π * 115) / 180) + isin((π * 115) / 180))
Therefore, the value of [4(cos115+isin115)] in the rectangular form is 4 * (cos((π * 115) / 180) + isin((π * 115) / 180)).
Step 3: Applying the binomial expansion.
The expression [4(cos115+isin115)]^6 is equivalent to raising the number 4 * (cos((π * 115) / 180) + isin((π * 115) / 180)) to the 6th power.
By using the binomial expansion formula for complex numbers (a + bi)^n, where n is a positive integer, we can calculate the result. The formula is:
(a + bi)^n = (a^n)(cos(nθ) + isin(nθ))
In our case, a = 4 * cos((π * 115) / 180) and b = 4 * sin((π * 115) / 180), and n = 6.
Substituting the values in, we get:
[4 * (cos((π * 115) / 180) + isin((π * 115) / 180))]^6 = (4^6) * (cos(6 * ((π * 115) / 180)) + isin(6 * ((π * 115) / 180)))
Now, we can simplify further:
[4 * (cos((π * 115) / 180) + isin((π * 115) / 180))]^6 = 4096 * (cos(690π/180) + isin(690π/180))
Finally, the value of [4(cos115+isin115)]^6 is equal to 4096 * (cos(690π/180) + isin(690π/180)).