Answer:
![A=8\pi r^2-\dfrac{49\sqrt{3}x^2}{4}\ un^2.](https://tex.z-dn.net/?f=A%3D8%5Cpi%20r%5E2-%5Cdfrac%7B49%5Csqrt%7B3%7Dx%5E2%7D%7B4%7D%5C%20un%5E2.)
Step-by-step explanation:
An equilateral triangle with the side 7x units is inscribed in a circle of radius 2r units.
1. The area of the circle is
![A_{circle}=2\pi (2r)^2=2\pi \cdot 4r^2=8\pi r^2\ un^2.](https://tex.z-dn.net/?f=A_%7Bcircle%7D%3D2%5Cpi%20%282r%29%5E2%3D2%5Cpi%20%5Ccdot%204r%5E2%3D8%5Cpi%20r%5E2%5C%20un%5E2.)
2. The area of the equilateral triangle is
![A_{\triangle }=\dfrac{(7x)^2\sqrt{3}}{4}=\dfrac{49\sqrt{3}x^2}{4}\ un^2.](https://tex.z-dn.net/?f=A_%7B%5Ctriangle%20%7D%3D%5Cdfrac%7B%287x%29%5E2%5Csqrt%7B3%7D%7D%7B4%7D%3D%5Cdfrac%7B49%5Csqrt%7B3%7Dx%5E2%7D%7B4%7D%5C%20un%5E2.)
3. The area A within the circle but outside the triangle is
![A=A_{circle}-A_{\triangle}=8\pi r^2-\dfrac{49\sqrt{3}x^2}{4}\ un^2.](https://tex.z-dn.net/?f=A%3DA_%7Bcircle%7D-A_%7B%5Ctriangle%7D%3D8%5Cpi%20r%5E2-%5Cdfrac%7B49%5Csqrt%7B3%7Dx%5E2%7D%7B4%7D%5C%20un%5E2.)
<u>Given</u>:
The given expression is ![sin (2x) \ cos (5x) - sin (5x) \ cos (2x)](https://tex.z-dn.net/?f=sin%20%282x%29%20%5C%20cos%20%285x%29%20-%20sin%20%285x%29%20%5C%20cos%20%282x%29)
We need to determine the simplified value of the given expression.
<u>Simplification</u>:
Since, the given expression is in the form of
, the given expression can be simplified using the identity ![\sin (a-b)=\sin a \cos b-\cos a \sin b](https://tex.z-dn.net/?f=%5Csin%20%28a-b%29%3D%5Csin%20a%20%5Ccos%20b-%5Ccos%20a%20%5Csin%20b)
Comparing the given expression with the identity, we get;
and ![b=5x](https://tex.z-dn.net/?f=b%3D5x)
Using this in the identity, we get;
![sin (2x) \ cos (5x) - sin (5x) \ cos (2x)=sin(2x-5x)](https://tex.z-dn.net/?f=sin%20%282x%29%20%5C%20cos%20%285x%29%20-%20sin%20%285x%29%20%5C%20cos%20%282x%29%3Dsin%282x-5x%29)
Simplifying, we get;
![sin (2x) \ cos (5x) - sin (5x) \ cos (2x)=sin(-3x)](https://tex.z-dn.net/?f=sin%20%282x%29%20%5C%20cos%20%285x%29%20-%20sin%20%285x%29%20%5C%20cos%20%282x%29%3Dsin%28-3x%29)
Thus, the simplified value of the given expression is ![sin (-3x)](https://tex.z-dn.net/?f=sin%20%28-3x%29)
Hence, Option c is the correct answer.
All you need to do is plug in some x values and find the correspoding y value
when x=0, y=-3
when x=1, y=9
when x=2, y=21
when x=-1, y=-15
The sample space of an experiment is the set of all possible outcomes for that experiment. You may have noticed that for each of the experiments above, the sum of the probabilities of each outcome is 1. This is no coincidence. The sum of the probabilities of the distinct outcomes within a sample space is 1.