Answer:-27/7
Step-by-step explanation:
Step-by-step explanation:
<h3>Given:</h3>
<h3>To prove:
</h3>
<h3>Solution:</h3>
<u>CM⊥AB - given, therefore</u>
- ∠AMC = 90° and ∠BMC = 90°
<u>Then</u>
- Side CM is common, therefore is congruent to itself
<u>So we have congruent two angles and a side between them on triangles AMC and BMC:</u>
- △AMC ≅ △BMC as per ASA congruency theorem
Little tip ** if u r going from a larger shape to a smaller shape, the scale factor is going to be a proper fraction....a number less then 1 but greater then 0
BC = 66......EF = 11.......66 times what is 11
66x = 11
x = 11/66 reduces to 1/6
so 66 * 1/6 = 11
now compare the other sides...
42 * 1/6 = 7.......looks like AC and DF
72 * 1/6 = 12...looks like Ab and DE
so ur multiplying every number in the big triangle by 1/6 , and getting the measures of the little triangle....so ur scale factor is 1/6 <==
A,d,e,&f
i might be wrong so... check with classmates :3
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
