Answer:
A) reflection over the x-axis, plus a vertical translation
Answer:
![\theta=\frac{\pi}{2},\frac{3\pi}{2},\frac{2\pi}{3},\frac{4\pi}{3}](https://tex.z-dn.net/?f=%5Ctheta%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2C%5Cfrac%7B3%5Cpi%7D%7B2%7D%2C%5Cfrac%7B2%5Cpi%7D%7B3%7D%2C%5Cfrac%7B4%5Cpi%7D%7B3%7D)
Step-by-step explanation:
If I'm interpreting that correctly, you are trying to solve this equation:
![sin(\theta )+1=cos(2\theta)](https://tex.z-dn.net/?f=sin%28%5Ctheta%20%29%2B1%3Dcos%282%5Ctheta%29)
for theta. To do this, you will need a trig identity sheet (I'm assuming you got one from class) and a unit circle (ditto on the class thing).
We need to solve for theta. If I look to my trig identities, I will see a double angle one there that says:
![cos(2\theta)=1-2sin^2(\theta)](https://tex.z-dn.net/?f=cos%282%5Ctheta%29%3D1-2sin%5E2%28%5Ctheta%29)
We will make that replacement, then we will have everything in terms of sin.
![sin(\theta)+1=1-2sin^2(\theta)](https://tex.z-dn.net/?f=sin%28%5Ctheta%29%2B1%3D1-2sin%5E2%28%5Ctheta%29)
Now get everything on one side of the equals sign to solve for theta:
![2sin^2(\theta)+sin(\theta)=0](https://tex.z-dn.net/?f=2sin%5E2%28%5Ctheta%29%2Bsin%28%5Ctheta%29%3D0)
We can factor out the common sin(theta):
![sin\theta(2sin\theta+1)=0](https://tex.z-dn.net/?f=sin%5Ctheta%282sin%5Ctheta%2B1%29%3D0)
By the Zero Product Property, either
or
![2sin\theta+1=0](https://tex.z-dn.net/?f=2sin%5Ctheta%2B1%3D0)
Now look at your unit circle and find that the values of theta where the sin is 0 are located at:
![\theta=\frac{\pi }{2},\frac{3\pi}{2}](https://tex.z-dn.net/?f=%5Ctheta%3D%5Cfrac%7B%5Cpi%20%7D%7B2%7D%2C%5Cfrac%7B3%5Cpi%7D%7B2%7D)
The next one we have to solve for theta:
simplifies to
and
![sin\theta=-\frac{1}{2}](https://tex.z-dn.net/?f=sin%5Ctheta%3D-%5Cfrac%7B1%7D%7B2%7D)
Look at the unit circle again to find the values of theta where the sin is -1/2:
![\theta=\frac{2\pi}{3},\frac{4\pi}{3}](https://tex.z-dn.net/?f=%5Ctheta%3D%5Cfrac%7B2%5Cpi%7D%7B3%7D%2C%5Cfrac%7B4%5Cpi%7D%7B3%7D)
Those ar your values of theta!
She ate 7 yards. This is because 3 feet=1 yard, so you can say that she ate 1 yard per day, for 7 days, so that makes it 7 yards.
5 years old. You were alive and out of another human for 5 years which makes you five years old.
Use a recursive approach to get (e1,e2) = (2,4).