The answer is complex number
a complex number is 2 parts
1. the real part
2. the imaginary part
the real part is the 'a' part of the a+bi
the imaginary part is the 'bi' part of the a+bi
answer is A
(B means exponent is no higher than 1, C means exponent is 2, D means b^2-4ac)
answer is A
No your friend I’d not correct although the slope is 4, the y-intercept is -3 because of the minus 3 in the equation.
Answer:
herb ry tyr unuttnutuhbutbunttunntutni
Answer:12/3=4 he washed 4 cars in an hour so he earned $20 per hour since 4*5=$20
Step-by-step explanation:
The given proof of De Moivre's theorem is related to the operations of
complex numbers.
<h3>The Correct Responses;</h3>
- Step C: Expanding and collecting like terms
- Step D: Trigonometric formula for the cosine and sine of the sum of two numbers
<h3>Reasons that make the above selection correct;</h3>
The given proof is presented as follows;
![\mathbf{\left[cos(\theta) + i \cdot sin(\theta) \right]^{k + 1}}](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%5E%7Bk%20%2B%201%7D%7D)
- Step A: By laws of indices, we have;
![\left[cos(\theta) + i \cdot sin(\theta) \right]^{k + 1} = \mathbf{\left[cos(\theta) + i \cdot sin(\theta) \right]^{k} \cdot \left[cos(\theta) + i \cdot sin(\theta) \right]}](https://tex.z-dn.net/?f=%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%5E%7Bk%20%2B%201%7D%20%3D%20%5Cmathbf%7B%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%5E%7Bk%7D%20%5Ccdot%20%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%7D)
![\left[cos(\theta) + i \cdot sin(\theta) \right]^{k} \cdot \left[cos(\theta) + i \cdot sin(\theta) \right] = \mathbf{\left[cos(k \cdot \theta) + i \cdot sin(k \cdot \theta) \right] \cdot \left[cos(\theta) + i \cdot sin(\theta) \right]}](https://tex.z-dn.net/?f=%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%5E%7Bk%7D%20%5Ccdot%20%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%20%3D%20%20%5Cmathbf%7B%5Cleft%5Bcos%28k%20%5Ccdot%20%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28k%20%5Ccdot%20%5Ctheta%29%20%5Cright%5D%20%5Ccdot%20%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%7D)
- Step B: By expanding, we have;
![\left[cos(k \cdot \theta) + i \cdot sin(k \cdot \theta) \right] \cdot \left[cos(\theta) + i \cdot sin(\theta) \right] = cos(k \cdot \theta) \cdot cos(\theta) - sin(k \cdot \theta) \cdot sin(\theta) + i \cdot \left [sin(k \cdot \theta) \cdot cos(\theta) + cos(k \cdot \theta) \cdot sin(\theta) \right]](https://tex.z-dn.net/?f=%5Cleft%5Bcos%28k%20%5Ccdot%20%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28k%20%5Ccdot%20%5Ctheta%29%20%5Cright%5D%20%5Ccdot%20%5Cleft%5Bcos%28%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%20%3D%20cos%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20cos%28%5Ctheta%29%20-%20sin%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20sin%28%5Ctheta%29%20%2B%20i%20%20%5Ccdot%20%5Cleft%20%5Bsin%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20cos%28%5Ctheta%29%20%2B%20cos%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D)
- Step D: From trigonometric addition formula, we have;
cos(A + B) = cos(A)·cos(B) - sin(A)·sin(B)
sin(A + B) = sin(A)·cos(B) + sin(B)·cos(A)
Therefore;
![cos(k \cdot \theta) \cdot cos(\theta) - sin(k \cdot \theta) \cdot sin(\theta) + i \cdot \left [sin(k \cdot \theta) \cdot cos(\theta) + cos(k \cdot \theta) \cdot sin(\theta) \right] = \mathbf{ cos(k \cdot \theta + \theta) + i \cdot sin(k \cdot \theta + \theta)}](https://tex.z-dn.net/?f=cos%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20cos%28%5Ctheta%29%20-%20sin%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20sin%28%5Ctheta%29%20%2B%20i%20%20%5Ccdot%20%5Cleft%20%5Bsin%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20cos%28%5Ctheta%29%20%2B%20cos%28k%20%5Ccdot%20%5Ctheta%29%20%5Ccdot%20sin%28%5Ctheta%29%20%5Cright%5D%20%3D%20%5Cmathbf%7B%20cos%28k%20%5Ccdot%20%5Ctheta%20%2B%20%5Ctheta%29%20%2B%20i%20%5Ccdot%20sin%28k%20%5Ccdot%20%5Ctheta%20%20%2B%20%5Ctheta%29%7D)
Learn more about complex numbers here:
brainly.com/question/11000934