cos(2 x) + 2 = sin(x)
Solve for x over the real numbers:
sin(x) - cos(2 x) = 2
Transform sin(x) - cos(2 x) into a polynomial with respect to sin(x) using cos(2 x) = 1 - 2 sin^2(x):
-1 + sin(x) + 2 sin^2(x) = 2
Divide both sides by 2:
-1/2 + sin(x)/2 + sin^2(x) = 1
Add 1/2 to both sides:
sin(x)/2 + sin^2(x) = 3/2
Add 1/16 to both sides:
1/16 + sin(x)/2 + sin^2(x) = 25/16
Write the left hand side as a square:
(sin(x) + 1/4)^2 = 25/16
Take the square root of both sides:
sin(x) + 1/4 = 5/4 or sin(x) + 1/4 = -5/4
Subtract 1/4 from both sides:
sin(x) = 1 or sin(x) + 1/4 = -5/4
Take the inverse sine of both sides:
x = 2 π n + π/2 for n element Z
or sin(x) + 1/4 = -5/4
Subtract 1/4 from both sides:
x = 2 π n + π/2 for n element Z
or sin(x) = -3/2
sin(x) = -3/2 has no solution since for all x element R, -1<=sin(x)<=1 and -3/2<-1:
Answer: |
| x = 2 π n + π/2 for n element Z
<u>x = 1/2 (4 π n + π)</u> n element Z
The answers 15 because we have two negatives it's always a positive and 3×5 is 15
Answer:
A: 20 or 70
A:between 20 and 70
B:more than 70
Step-by-step explanation:
Plato
Answer:
a) 50.34% probability that the arrival time between customers will be 7 minutes or less.
b) 24.42% probability that the arrival time between customers will be between 3 and 7 minutes
Step-by-step explanation:
Exponential distribution:
The exponential probability distribution, with mean m, is described by the following equation:
![f(x) = \mu e^{-\mu x}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%5Cmu%20e%5E%7B-%5Cmu%20x%7D)
In which
is the decay parameter.
The probability that x is lower or equal to a is given by:
![P(X \leq x) = \int\limits^a_0 {f(x)} \, dx](https://tex.z-dn.net/?f=P%28X%20%5Cleq%20x%29%20%3D%20%5Cint%5Climits%5Ea_0%20%7Bf%28x%29%7D%20%5C%2C%20dx)
Which has the following solution:
![P(X \leq x) = 1 - e^{-\mu x}](https://tex.z-dn.net/?f=P%28X%20%5Cleq%20x%29%20%3D%201%20-%20e%5E%7B-%5Cmu%20x%7D)
The probability of finding a value higher than x is:
![P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^{-\mu x}) = e^{-\mu x}](https://tex.z-dn.net/?f=P%28X%20%3E%20x%29%20%3D%201%20-%20P%28X%20%5Cleq%20x%29%20%3D%201%20-%20%281%20-%20e%5E%7B-%5Cmu%20x%7D%29%20%3D%20e%5E%7B-%5Cmu%20x%7D)
Mean of 10 minutes:
This means that ![m = 10, \mu = \frac{1}{10} = 0.1](https://tex.z-dn.net/?f=m%20%3D%2010%2C%20%5Cmu%20%3D%20%5Cfrac%7B1%7D%7B10%7D%20%3D%200.1)
A. What is the probability that the arrival time between customers will be 7 minutes or less?
![P(X \leq x) = 1 - e^{-\mu x}](https://tex.z-dn.net/?f=P%28X%20%5Cleq%20x%29%20%3D%201%20-%20e%5E%7B-%5Cmu%20x%7D)
![P(X \leq 7) = 1 - e^{-0.1*7} = 0.5034](https://tex.z-dn.net/?f=P%28X%20%5Cleq%207%29%20%3D%201%20-%20e%5E%7B-0.1%2A7%7D%20%3D%200.5034)
50.34% probability that the arrival time between customers will be 7 minutes or less.
B. What is the probability that the arrival time between customers will be between 3 and 7 minutes?
![P(3 \leq X \leq 7) = P(X \leq 7) - P(X \leq 3)](https://tex.z-dn.net/?f=P%283%20%5Cleq%20X%20%5Cleq%207%29%20%3D%20P%28X%20%5Cleq%207%29%20-%20P%28X%20%5Cleq%203%29)
![P(X \leq x) = 1 - e^{-\mu x}](https://tex.z-dn.net/?f=P%28X%20%5Cleq%20x%29%20%3D%201%20-%20e%5E%7B-%5Cmu%20x%7D)
![P(X \leq 7) = 1 - e^{-0.1*7} = 0.5034](https://tex.z-dn.net/?f=P%28X%20%5Cleq%207%29%20%3D%201%20-%20e%5E%7B-0.1%2A7%7D%20%3D%200.5034)
![P(X \leq 3) = 1 - e^{-0.1*3} = 0.2592](https://tex.z-dn.net/?f=P%28X%20%5Cleq%203%29%20%3D%201%20-%20e%5E%7B-0.1%2A3%7D%20%3D%200.2592)
![P(3 \leq X \leq 7) = P(X \leq 7) - P(X \leq 3) = 0.5034 - 0.2592 = 0.2442](https://tex.z-dn.net/?f=P%283%20%5Cleq%20X%20%5Cleq%207%29%20%3D%20P%28X%20%5Cleq%207%29%20-%20P%28X%20%5Cleq%203%29%20%3D%200.5034%20-%200.2592%20%3D%200.2442)
24.42% probability that the arrival time between customers will be between 3 and 7 minutes
Answer:
B
Step-by-step explanation:
(Sorry about the crude drawing)
The drawing shows what the triangle with a terminal side in Quadrant IV looks like.
Secant is the ratio of the hypotenuse to the adjacent side of the angle. In the given ratio, we can set 5 as the hypotenuse and 2 as the adjacent side (you can see them labeled in the picture).
Now, we want to find sin, which is (opposite)/(hypotenuse). However, we don't know the opposite. We can find it by using the Pythagorean Theorem:
![\sqrt{5^2-2^2} =\sqrt{25-4} =\sqrt{21}](https://tex.z-dn.net/?f=%5Csqrt%7B5%5E2-2%5E2%7D%20%3D%5Csqrt%7B25-4%7D%20%3D%5Csqrt%7B21%7D)
So, the opposite side is
. But, we see that since it's in the fourth quadrant, it must be negative, so we have opposite =
. Now, we can find the ratio because hypotenuse = 5:
sin(θ) =
.
Thus, the answer is B.
Hope this helps!