Answer:
Plane MNO
Plane PST
Step-by-step explanation:
The plane L can be given another name using uppercase letters only. The plane will require three letters name so that it represents a complete angle. The names can be assigned using the point to point method or randomly. There are two choices available to name the current plane. Plane MNO and Plane PST are the names that can be given to plane L.
If she cuts the entire rope into pieces of 1/3 meters, she will have 15 pieces
You would take 5x3 and get 15 pieces.
15 x 1/3 = 5
Hopes this Helps! :)
Answer:
Answer:
27.44 square cm
Step-by-step explanation:
If the length of parallelogram is a and b and angle between side a and b is .
Then area of parallelogram =
Given side length 4 cm and 7 cm
angle between them = 100°
value of sin(100°) = 0.98
Thus, area of given parallelogram = 4*7*sin(100°) = 28*0.98 = 27.44
Thus, area of given parallelogram is 27.44 square cm.
Please mark me -liest
God bless
![\sum\limits_{k=1}^n\arcsin\left[\dfrac{\sqrt{k^2+2k}-\sqrt{k^2-1}}{k(k+1)}\right]=\\\\\\= \sum\limits_{k=1}^n\arcsin\left[\dfrac{\sqrt{k^2+2k}}{k(k+1)}-\dfrac{\sqrt{k^2-1}}{k(k+1)}\right]=\\\\\\= \sum\limits_{k=1}^n\arcsin\left[\dfrac{\sqrt{k^2+2k+1-1}}{k(k+1)}-\dfrac{\sqrt{k^2-1}}{k(k+1)}\right]=\\\\\\= \sum\limits_{k=1}^n\arcsin\left[\dfrac{\sqrt{(k+1)^2-1}}{k(k+1)}-\dfrac{\sqrt{k^2-1}}{k(k+1)}\right]=\\\\\\ ](https://tex.z-dn.net/?f=%5Csum%5Climits_%7Bk%3D1%7D%5En%5Carcsin%5Cleft%5B%5Cdfrac%7B%5Csqrt%7Bk%5E2%2B2k%7D-%5Csqrt%7Bk%5E2-1%7D%7D%7Bk%28k%2B1%29%7D%5Cright%5D%3D%5C%5C%5C%5C%5C%5C%3D%0A%5Csum%5Climits_%7Bk%3D1%7D%5En%5Carcsin%5Cleft%5B%5Cdfrac%7B%5Csqrt%7Bk%5E2%2B2k%7D%7D%7Bk%28k%2B1%29%7D-%5Cdfrac%7B%5Csqrt%7Bk%5E2-1%7D%7D%7Bk%28k%2B1%29%7D%5Cright%5D%3D%5C%5C%5C%5C%5C%5C%3D%0A%5Csum%5Climits_%7Bk%3D1%7D%5En%5Carcsin%5Cleft%5B%5Cdfrac%7B%5Csqrt%7Bk%5E2%2B2k%2B1-1%7D%7D%7Bk%28k%2B1%29%7D-%5Cdfrac%7B%5Csqrt%7Bk%5E2-1%7D%7D%7Bk%28k%2B1%29%7D%5Cright%5D%3D%5C%5C%5C%5C%5C%5C%3D%0A%5Csum%5Climits_%7Bk%3D1%7D%5En%5Carcsin%5Cleft%5B%5Cdfrac%7B%5Csqrt%7B%28k%2B1%29%5E2-1%7D%7D%7Bk%28k%2B1%29%7D-%5Cdfrac%7B%5Csqrt%7Bk%5E2-1%7D%7D%7Bk%28k%2B1%29%7D%5Cright%5D%3D%5C%5C%5C%5C%5C%5C%0A)
![=\sum\limits_{k=1}^n\arcsin\left[\dfrac{1}{k}\cdot\dfrac{\sqrt{(k+1)^2-1}}{\sqrt{(k+1)^2}}-\dfrac{1}{k+1}\cdot\dfrac{\sqrt{k^2-1}}{\sqrt{k^2}}\right]=\\\\\\= \sum\limits_{k=1}^n\arcsin\left[\dfrac{1}{k}\cdot\sqrt{\dfrac{(k+1)&^2-1}{(k+1)^2}}-\dfrac{1}{k+1}\cdot\sqrt{\dfrac{k^2-1}{k^2}}\right]=\\\\\\= \sum\limits_{k=1}^n\arcsin\left[\dfrac{1}{k}\cdot\sqrt{1-\dfrac{1}{(k+1)^2}}-\dfrac{1}{k+1}\cdot\sqrt{1-\dfrac{1}{k^2}}\right]=\\\\\\ ](https://tex.z-dn.net/?f=%3D%5Csum%5Climits_%7Bk%3D1%7D%5En%5Carcsin%5Cleft%5B%5Cdfrac%7B1%7D%7Bk%7D%5Ccdot%5Cdfrac%7B%5Csqrt%7B%28k%2B1%29%5E2-1%7D%7D%7B%5Csqrt%7B%28k%2B1%29%5E2%7D%7D-%5Cdfrac%7B1%7D%7Bk%2B1%7D%5Ccdot%5Cdfrac%7B%5Csqrt%7Bk%5E2-1%7D%7D%7B%5Csqrt%7Bk%5E2%7D%7D%5Cright%5D%3D%5C%5C%5C%5C%5C%5C%3D%0A%5Csum%5Climits_%7Bk%3D1%7D%5En%5Carcsin%5Cleft%5B%5Cdfrac%7B1%7D%7Bk%7D%5Ccdot%5Csqrt%7B%5Cdfrac%7B%28k%2B1%29%26%5E2-1%7D%7B%28k%2B1%29%5E2%7D%7D-%5Cdfrac%7B1%7D%7Bk%2B1%7D%5Ccdot%5Csqrt%7B%5Cdfrac%7Bk%5E2-1%7D%7Bk%5E2%7D%7D%5Cright%5D%3D%5C%5C%5C%5C%5C%5C%3D%0A%5Csum%5Climits_%7Bk%3D1%7D%5En%5Carcsin%5Cleft%5B%5Cdfrac%7B1%7D%7Bk%7D%5Ccdot%5Csqrt%7B1-%5Cdfrac%7B1%7D%7B%28k%2B1%29%5E2%7D%7D-%5Cdfrac%7B1%7D%7Bk%2B1%7D%5Ccdot%5Csqrt%7B1-%5Cdfrac%7B1%7D%7Bk%5E2%7D%7D%5Cright%5D%3D%5C%5C%5C%5C%5C%5C%0A)
![=\sum\limits_{k=1}^n\arcsin\left[\dfrac{1}{k}\cdot\sqrt{1-\left(\dfrac{1}{k+1}\right)^2}-\dfrac{1}{k+1}\cdot\sqrt{1-\left(\dfrac{1}{k}\right)^2}\right]=\\\\\\= \sum\limits_{k=1}^n\left[\arcsin\left(\dfrac{1}{k}\right)-\arcsin\left(\dfrac{1}{k+1}\right)\right]=\\\\\\](https://tex.z-dn.net/?f=%3D%5Csum%5Climits_%7Bk%3D1%7D%5En%5Carcsin%5Cleft%5B%5Cdfrac%7B1%7D%7Bk%7D%5Ccdot%5Csqrt%7B1-%5Cleft%28%5Cdfrac%7B1%7D%7Bk%2B1%7D%5Cright%29%5E2%7D-%5Cdfrac%7B1%7D%7Bk%2B1%7D%5Ccdot%5Csqrt%7B1-%5Cleft%28%5Cdfrac%7B1%7D%7Bk%7D%5Cright%29%5E2%7D%5Cright%5D%3D%5C%5C%5C%5C%5C%5C%3D%0A%5Csum%5Climits_%7Bk%3D1%7D%5En%5Cleft%5B%5Carcsin%5Cleft%28%5Cdfrac%7B1%7D%7Bk%7D%5Cright%29-%5Carcsin%5Cleft%28%5Cdfrac%7B1%7D%7Bk%2B1%7D%5Cright%29%5Cright%5D%3D%5C%5C%5C%5C%5C%5C)
![=\bigg[\arcsin(1)-\arcsin\left(\frac{1}{2}\right)\bigg]+\bigg[\arcsin\left(\frac{1}{2}\right)-\arcsin\left(\frac{1}{3}\right)\bigg]+\\\\\\+ \bigg[\arcsin\left(\frac{1}{3}\right)-\arcsin\left(\frac{1}{4}\right)\bigg]+\dots+ \bigg[\arcsin\left(\frac{1}{n}\right)-\arcsin\left(\frac{1}{n+1}\right)\bigg]=\\\\\\](https://tex.z-dn.net/?f=%3D%5Cbigg%5B%5Carcsin%281%29-%5Carcsin%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%5Cright%29%5Cbigg%5D%2B%5Cbigg%5B%5Carcsin%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%5Cright%29-%5Carcsin%5Cleft%28%5Cfrac%7B1%7D%7B3%7D%5Cright%29%5Cbigg%5D%2B%5C%5C%5C%5C%5C%5C%2B%0A%5Cbigg%5B%5Carcsin%5Cleft%28%5Cfrac%7B1%7D%7B3%7D%5Cright%29-%5Carcsin%5Cleft%28%5Cfrac%7B1%7D%7B4%7D%5Cright%29%5Cbigg%5D%2B%5Cdots%2B%0A%5Cbigg%5B%5Carcsin%5Cleft%28%5Cfrac%7B1%7D%7Bn%7D%5Cright%29-%5Carcsin%5Cleft%28%5Cfrac%7B1%7D%7Bn%2B1%7D%5Cright%29%5Cbigg%5D%3D%5C%5C%5C%5C%5C%5C)
![\sum\limits_{k=1}^n\arcsin\left[\dfrac{\sqrt{k^2+2k}-\sqrt{k^2-1}}{k(k+1)}\right]=\dfrac{\pi}{2}-\arcsin\left(\dfrac{1}{n+1}\right)}](https://tex.z-dn.net/?f=%5Csum%5Climits_%7Bk%3D1%7D%5En%5Carcsin%5Cleft%5B%5Cdfrac%7B%5Csqrt%7Bk%5E2%2B2k%7D-%5Csqrt%7Bk%5E2-1%7D%7D%7Bk%28k%2B1%29%7D%5Cright%5D%3D%5Cdfrac%7B%5Cpi%7D%7B2%7D-%5Carcsin%5Cleft%28%5Cdfrac%7B1%7D%7Bn%2B1%7D%5Cright%29%7D)
![\sum\limits_{k=1}^\infty\arcsin\left[\dfrac{\sqrt{k^2+2k}-\sqrt{k^2-1}}{k(k+1)}\right]=\\\\\\= \lim\limits_{n\to\infty}\sum\limits_{k=1}^n\arcsin\left[\dfrac{\sqrt{k^2+2k}-\sqrt{k^2-1}}{k(k+1)}\right]=\\\\\\= \lim\limits_{n\to\infty}\Bigg(\dfrac{\pi}{2}-\arcsin\left(\dfrac{1}{n+1}\right)\Bigg)=\dfrac{\pi}{2}-\lim\limits_{n\to\infty}\arcsin\left(\dfrac{1}{n+1}\right)=\\\\\\= \Bigg\{\dfrac{1}{n+1}\xrightarrow{n\to\infty}0\Bigg\}=\dfrac{\pi}{2}-\arcsin(0)=\dfrac{\pi}{2}-0=\dfrac{\pi}{2}](https://tex.z-dn.net/?f=%5Csum%5Climits_%7Bk%3D1%7D%5E%5Cinfty%5Carcsin%5Cleft%5B%5Cdfrac%7B%5Csqrt%7Bk%5E2%2B2k%7D-%5Csqrt%7Bk%5E2-1%7D%7D%7Bk%28k%2B1%29%7D%5Cright%5D%3D%5C%5C%5C%5C%5C%5C%3D%0A%5Clim%5Climits_%7Bn%5Cto%5Cinfty%7D%5Csum%5Climits_%7Bk%3D1%7D%5En%5Carcsin%5Cleft%5B%5Cdfrac%7B%5Csqrt%7Bk%5E2%2B2k%7D-%5Csqrt%7Bk%5E2-1%7D%7D%7Bk%28k%2B1%29%7D%5Cright%5D%3D%5C%5C%5C%5C%5C%5C%3D%0A%5Clim%5Climits_%7Bn%5Cto%5Cinfty%7D%5CBigg%28%5Cdfrac%7B%5Cpi%7D%7B2%7D-%5Carcsin%5Cleft%28%5Cdfrac%7B1%7D%7Bn%2B1%7D%5Cright%29%5CBigg%29%3D%5Cdfrac%7B%5Cpi%7D%7B2%7D-%5Clim%5Climits_%7Bn%5Cto%5Cinfty%7D%5Carcsin%5Cleft%28%5Cdfrac%7B1%7D%7Bn%2B1%7D%5Cright%29%3D%5C%5C%5C%5C%5C%5C%3D%0A%5CBigg%5C%7B%5Cdfrac%7B1%7D%7Bn%2B1%7D%5Cxrightarrow%7Bn%5Cto%5Cinfty%7D0%5CBigg%5C%7D%3D%5Cdfrac%7B%5Cpi%7D%7B2%7D-%5Carcsin%280%29%3D%5Cdfrac%7B%5Cpi%7D%7B2%7D-0%3D%5Cdfrac%7B%5Cpi%7D%7B2%7D)
![\sum\limits_{k=1}^\infty\arcsin\left[\dfrac{\sqrt{k^2+2k}-\sqrt{k^2-1}}{k(k+1)}\right]=\dfrac{\pi}{2}](https://tex.z-dn.net/?f=%5Csum%5Climits_%7Bk%3D1%7D%5E%5Cinfty%5Carcsin%5Cleft%5B%5Cdfrac%7B%5Csqrt%7Bk%5E2%2B2k%7D-%5Csqrt%7Bk%5E2-1%7D%7D%7Bk%28k%2B1%29%7D%5Cright%5D%3D%5Cdfrac%7B%5Cpi%7D%7B2%7D)
