Answer:
aₙ = 9n - 31
Step-by-step explanation:
If we call the common difference as d, since 19 - 7 = 12, we can write:
32 + 12d = 140
12d = 108
d = 9 which means that a₁ = a₇ - 6d = 32 - 6 * 9 = -22.
Explicit formula for arithmetic sequence: aₙ = a₁ + (n-1)d so the answer is
aₙ = -22 + (n - 1) * 9
= -22 + 9n - 9
= 9n - 31
Answer: b. 0.5
Step-by-step explanation:
We area given that the probability of outcomes A = 0.5
We know that for any event Q the probability of not getting Q is given by :-
![P(Q)'=1-P(Q)](https://tex.z-dn.net/?f=P%28Q%29%27%3D1-P%28Q%29)
Now, the probability of getting outcomes not A is given by :-
![P(A)'=1-P(A)\\\\\Rightarrow\ P(A)' =1-0.5\\\\\Rightarrow\ P(A)' =0.5](https://tex.z-dn.net/?f=P%28A%29%27%3D1-P%28A%29%5C%5C%5C%5C%5CRightarrow%5C%20P%28A%29%27%20%3D1-0.5%5C%5C%5C%5C%5CRightarrow%5C%20P%28A%29%27%20%3D0.5)
Hence option B is correct.
Answer:
3/48 or reduced 1/16
Step-by-step explanation:
1/3 x 3/4 = 3/48
Reduced:
3/48 ÷ 3/3 = 1/16
Only put the reduced fraction if it asks.
Hope this helps :)
<u>ANSWER</u>
True
<u>EXPLANATION</u>
The given trigonometric equation is
![\tan^{2} (x) = \frac{1 - \cos(2x) }{1 + \cos(2x) }](https://tex.z-dn.net/?f=%20%5Ctan%5E%7B2%7D%20%28x%29%20%20%3D%20%20%5Cfrac%7B1%20-%20%20%5Ccos%282x%29%20%7D%7B1%20%2B%20%20%5Ccos%282x%29%20%7D%20)
Recall the double angle identity:
![\cos(2x) = \cos^{2} x - \sin^{2}x](https://tex.z-dn.net/?f=%20%5Ccos%282x%29%20%20%3D%20%20%5Ccos%5E%7B2%7D%20x%20-%20%20%20%5Csin%5E%7B2%7Dx)
We apply this identity to obtain:
![\tan^{2} (x) = \frac{1 - (\cos^{2} x - \sin^{2}x) }{1 + (\cos^{2} x - \sin^{2}x) }](https://tex.z-dn.net/?f=%5Ctan%5E%7B2%7D%20%28x%29%20%20%3D%20%20%5Cfrac%7B1%20-%20%28%5Ccos%5E%7B2%7D%20x%20-%20%20%20%5Csin%5E%7B2%7Dx%29%20%7D%7B1%20%2B%20%20%28%5Ccos%5E%7B2%7D%20x%20-%20%20%20%5Csin%5E%7B2%7Dx%29%20%7D%20)
We maintain the LHS and simplify the RHS to see whether they are equal.
Expand the parenthesis
![\tan^{2} (x) = \frac{1 - \cos^{2} x + \sin^{2}x }{1 + \cos^{2} x - \sin^{2}x}](https://tex.z-dn.net/?f=%5Ctan%5E%7B2%7D%20%28x%29%20%20%3D%20%20%5Cfrac%7B1%20-%20%5Ccos%5E%7B2%7D%20x%20%20%2B%20%20%5Csin%5E%7B2%7Dx%20%7D%7B1%20%2B%20%20%5Ccos%5E%7B2%7D%20x%20-%20%20%20%5Csin%5E%7B2%7Dx%7D%20)
![\implies\tan^{2} (x) = \frac{1 - \cos^{2} x + \sin^{2}x }{1 - \sin^{2}x + \cos^{2} x }](https://tex.z-dn.net/?f=%20%5Cimplies%5Ctan%5E%7B2%7D%20%28x%29%20%20%3D%20%20%5Cfrac%7B1%20-%20%5Ccos%5E%7B2%7D%20x%20%20%2B%20%20%5Csin%5E%7B2%7Dx%20%7D%7B1%20%20-%20%20%20%5Csin%5E%7B2%7Dx%20%20%2B%20%5Ccos%5E%7B2%7D%20x%20%7D%20)
Recall that:
![1 - \sin^{2}x = \cos^{2}x](https://tex.z-dn.net/?f=1%20%20-%20%20%20%5Csin%5E%7B2%7Dx%20%20%3D%20%20%5Ccos%5E%7B2%7Dx%20)
![1 - \cos^{2}x = \sin^{2}x](https://tex.z-dn.net/?f=1%20%20-%20%20%20%5Ccos%5E%7B2%7Dx%20%20%3D%20%20%5Csin%5E%7B2%7Dx)
We apply these identities to get:
![\implies\tan^{2} (x) = \frac{\sin^{2}x + \sin^{2}x }{\cos^{2} x + \cos^{2} x }](https://tex.z-dn.net/?f=%5Cimplies%5Ctan%5E%7B2%7D%20%28x%29%20%20%3D%20%20%5Cfrac%7B%5Csin%5E%7B2%7Dx%20%2B%20%20%5Csin%5E%7B2%7Dx%20%7D%7B%5Ccos%5E%7B2%7D%20x%20%2B%20%5Ccos%5E%7B2%7D%20x%20%7D%20)
![\implies\tan^{2} (x) = \frac{2\sin^{2}x }{ 2\cos^{2} x }](https://tex.z-dn.net/?f=%5Cimplies%5Ctan%5E%7B2%7D%20%28x%29%20%20%3D%20%20%5Cfrac%7B2%5Csin%5E%7B2%7Dx%20%7D%7B%202%5Ccos%5E%7B2%7D%20x%20%7D%20)
![\implies\tan^{2} (x) = \frac{\sin^{2}x }{ \cos^{2} x }](https://tex.z-dn.net/?f=%5Cimplies%5Ctan%5E%7B2%7D%20%28x%29%20%20%3D%20%20%5Cfrac%7B%5Csin%5E%7B2%7Dx%20%7D%7B%20%5Ccos%5E%7B2%7D%20x%20%7D%20)
![\implies \tan^{2} (x) =( \frac{\sin x }{ \cos x })^{2}](https://tex.z-dn.net/?f=%5Cimplies%20%5Ctan%5E%7B2%7D%20%28x%29%20%20%3D%28%20%20%5Cfrac%7B%5Csin%20x%20%7D%7B%20%5Ccos%20x%20%7D%29%5E%7B2%7D%20%20)
Also
![\frac{\sin x }{ \cos x } = \tan(x)](https://tex.z-dn.net/?f=%20%5Cfrac%7B%5Csin%20x%20%7D%7B%20%5Ccos%20x%20%7D%20%3D%20%20%5Ctan%28x%29%20)
![\implies \tan^{2} (x) =( \tan x )^{2}](https://tex.z-dn.net/?f=%5Cimplies%20%5Ctan%5E%7B2%7D%20%28x%29%20%20%3D%28%20%5Ctan%20x%20%29%5E%7B2%7D%20%20)
![\implies \tan^{2} (x) =\tan^{2} (x)](https://tex.z-dn.net/?f=%5Cimplies%20%5Ctan%5E%7B2%7D%20%28x%29%20%20%3D%5Ctan%5E%7B2%7D%20%28x%29%20)
Therefore the correct answer is True
Answer:
We can easily solve this problem by plotting the equation with a grpahing calculator. Please see attached image.
Define the key features of the graph of the provided absolute value function.
Even function. The graph is inverted, if we compare it to the absoluted value of |x| thanks to the minus sign. It is also shifted up because we are adding +4
Provide the coordinate of the vertex and two additional points on the graph.
Coordinate of the vertex
V = (x,y) = (0,4)
Points where the graph crosses x - axis
(-4,0)
(4, 0)
Use the coordinate of the function’s vertex to prove the domain and range of the function.
Domain
All real numbers
Range
(-∞, 4]