Answer:
6,205 hours you are awake in a year
Step-by-step explanation:
The coordinates for C’ are (-30, 5)
Answer:
Step-by-step explanation:
5.577
Answer:
A
Step-by-step explanation:
Begin with Euler's formula. I am assuming youre aware of this if youre taking complex algebra? You can prove Euler's formula by doing a Maclaurin expansion of cosine, sine, and e^x. Euler's formula states that:
![e^{ix}=cos(x)+isin(x)](https://tex.z-dn.net/?f=e%5E%7Bix%7D%3Dcos%28x%29%2Bisin%28x%29)
We can put the first complex number in exponetial form by noticing the input is 2pi/3. The second complex number has an input of pi/3. Therefore:
![z_1=8e^{i(2\pi /3)}](https://tex.z-dn.net/?f=z_1%3D8e%5E%7Bi%282%5Cpi%20%2F3%29%7D)
![z_2=0.5e^{i(\pi /3)}](https://tex.z-dn.net/?f=z_2%3D0.5e%5E%7Bi%28%5Cpi%20%2F3%29%7D)
Then:
![\frac{z_1}{z_2} =\frac{8e^{i(2\pi /3)}}{0.5e^{i(\pi /3)}}](https://tex.z-dn.net/?f=%5Cfrac%7Bz_1%7D%7Bz_2%7D%20%3D%5Cfrac%7B8e%5E%7Bi%282%5Cpi%20%2F3%29%7D%7D%7B0.5e%5E%7Bi%28%5Cpi%20%2F3%29%7D%7D)
Simplify the coefficients to get:
![\frac{z_1}{z_2} =\frac{16e^{i(2\pi /3)}}{e^{i(\pi /3)}}](https://tex.z-dn.net/?f=%5Cfrac%7Bz_1%7D%7Bz_2%7D%20%3D%5Cfrac%7B16e%5E%7Bi%282%5Cpi%20%2F3%29%7D%7D%7Be%5E%7Bi%28%5Cpi%20%2F3%29%7D%7D)
When you divide exponentials, you subtract the exponents. Therefore:
![\frac{z_1}{z_2} =16e^{i(2\pi /3)-i(\pi /3)}=16e^{i(\pi /3)](https://tex.z-dn.net/?f=%5Cfrac%7Bz_1%7D%7Bz_2%7D%20%3D16e%5E%7Bi%282%5Cpi%20%2F3%29-i%28%5Cpi%20%2F3%29%7D%3D16e%5E%7Bi%28%5Cpi%20%2F3%29)
Put it back into trigonemtric form using Euler's formula:
![16e^{i(\pi /3)}=16cos(\pi /3)+i16sin(\pi /3)](https://tex.z-dn.net/?f=16e%5E%7Bi%28%5Cpi%20%2F3%29%7D%3D16cos%28%5Cpi%20%2F3%29%2Bi16sin%28%5Cpi%20%2F3%29)
Cosine of pi/3 is 0.5, and sine of pi/3 is square root of 3 over 2. We have:
![16e^{i(\pi /3)}=16cos(\pi /3)+i16sin(\pi /3)=16*0.5+16*\frac{\sqrt{3} }{2} *i=8+8\sqrt{3} i](https://tex.z-dn.net/?f=16e%5E%7Bi%28%5Cpi%20%2F3%29%7D%3D16cos%28%5Cpi%20%2F3%29%2Bi16sin%28%5Cpi%20%2F3%29%3D16%2A0.5%2B16%2A%5Cfrac%7B%5Csqrt%7B3%7D%20%7D%7B2%7D%20%2Ai%3D8%2B8%5Csqrt%7B3%7D%20i)
Answer:
0 & 3
Step-by-step explanation:
First x is by itself so the root is 0
Second is 3 because you set x-3=0 and add 3 to the other side so x=3