Prove that:
![cos(2\alpha )-cos(8\alpha )=sin(5\alpha )sin(3\alpha )](https://tex.z-dn.net/?f=cos%282%5Calpha%20%29-cos%288%5Calpha%20%29%3Dsin%285%5Calpha%20%29sin%283%5Calpha%20%29)
Remember that:
![cos(A+B)=cos(A)cos(B)-sin(A)sin(B)\\cos(A-B)=cos(A)cos(B)+sin(A)sin(B)\\\\\\cos(A+B)-cos(A-B)=-2sin(A)sin(B)](https://tex.z-dn.net/?f=cos%28A%2BB%29%3Dcos%28A%29cos%28B%29-sin%28A%29sin%28B%29%5C%5Ccos%28A-B%29%3Dcos%28A%29cos%28B%29%2Bsin%28A%29sin%28B%29%5C%5C%5C%5C%5C%5Ccos%28A%2BB%29-cos%28A-B%29%3D-2sin%28A%29sin%28B%29)
We wish to remove this add (A+B).
![\left \{ {{A+B=p} \atop {A-B=q}} \right. \\\\2A=p+q](https://tex.z-dn.net/?f=%5Cleft%20%5C%7B%20%7B%7BA%2BB%3Dp%7D%20%5Catop%20%7BA-B%3Dq%7D%7D%20%5Cright.%20%5C%5C%5C%5C2A%3Dp%2Bq)
![A=\frac{p+q}{2} \\\\B=\frac{p-q}{2}](https://tex.z-dn.net/?f=A%3D%5Cfrac%7Bp%2Bq%7D%7B2%7D%20%5C%5C%5C%5CB%3D%5Cfrac%7Bp-q%7D%7B2%7D)
Then we de identity:
![cos(p)-cos(q)=-2sin(\frac{p+q}{2}) sin(\frac{p-q}{2})](https://tex.z-dn.net/?f=cos%28p%29-cos%28q%29%3D-2sin%28%5Cfrac%7Bp%2Bq%7D%7B2%7D%29%20sin%28%5Cfrac%7Bp-q%7D%7B2%7D%29)
Let p=2α and q=8α
![cos(2\alpha )-cos(8\alpha )=-2sin(\frac{2\alpha +8\alpha }{2})sin(\frac{2\alpha -8\alpha }{2}) \\\\cos(2\alpha )-cos(8\alpha )=-2sin(5\alpha )sin(-3\alpha )\\\\*sin(-\alpha )=-sin(\alpha )\\\\cos(2\alpha )-cos(8\alpha )=2sin(5\alpha )sin(3\alpha )](https://tex.z-dn.net/?f=cos%282%5Calpha%20%29-cos%288%5Calpha%20%29%3D-2sin%28%5Cfrac%7B2%5Calpha%20%2B8%5Calpha%20%7D%7B2%7D%29sin%28%5Cfrac%7B2%5Calpha%20-8%5Calpha%20%7D%7B2%7D%29%20%20%5C%5C%5C%5Ccos%282%5Calpha%20%29-cos%288%5Calpha%20%29%3D-2sin%285%5Calpha%20%29sin%28-3%5Calpha%20%29%5C%5C%5C%5C%2Asin%28-%5Calpha%20%29%3D-sin%28%5Calpha%20%29%5C%5C%5C%5Ccos%282%5Calpha%20%29-cos%288%5Calpha%20%29%3D2sin%285%5Calpha%20%29sin%283%5Calpha%20%29)
Answer:
The probability that the sample proportion will be greater than 13% is 0.99693.
Step-by-step explanation:
We are given that a large shipment of laser printers contained 18% defectives. A sample of size 340 is selected.
Let
= <u><em>the sample proportion of defectives</em></u>.
The z-score probability distribution for the sample proportion is given by;
Z =
~ N(0,1)
where, p = population proportion of defective laser printers = 18%
n = sample size = 340
Now, the probability that the sample proportion will be greater than 13% is given by = P(
> 0.13)
P(
> 0.13) = P(
>
) = P(Z > -2.74) = P(Z < 2.74)
= <u>0.99693</u>
The above probability is calculated by looking at the value of x = 2.74 in the table which has an area of 0.99693.
Add the length of all the sides. :) It's like the length of all the edges.
4*(3x + 4y = 16)
+
<u>3*(-4x - 3y = -19)
</u>0x + 16y - 9y = 4*16 - 19*3
<u>
</u>y = 1
<u />x = 4 (4,1)