15. If x=a Sin2t (I+Cos2t) and y=b Cos 2t (1-Cos2t) then finddy/dx at =22/7*4
1 answer:
By the chain rule,
![\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm dt}\dfrac{\mathrm dt}{\mathrm dx}\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\frac{\mathrm dy}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%3D%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dt%7D%5Cdfrac%7B%5Cmathrm%20dt%7D%7B%5Cmathrm%20dx%7D%5Cimplies%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%3D%5Cdfrac%7B%5Cfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dt%7D%7D%7B%5Cfrac%7B%5Cmathrm%20dx%7D%7B%5Cmathrm%20dt%7D%7D)
It looks like we're given
![\begin{cases}x=a\sin(2t)(1+\cos(2t))\\y=b\cos(2t)(1-\cos(2t))\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7Dx%3Da%5Csin%282t%29%281%2B%5Ccos%282t%29%29%5C%5Cy%3Db%5Ccos%282t%29%281-%5Ccos%282t%29%29%5Cend%7Bcases%7D)
where <em>a</em> and <em>b</em> are presumably constant.
Recall that
![\cos^2t=\dfrac{1+\cos(2t)}2](https://tex.z-dn.net/?f=%5Ccos%5E2t%3D%5Cdfrac%7B1%2B%5Ccos%282t%29%7D2)
![\sin^2t=\dfrac{1-\cos(2t)}2](https://tex.z-dn.net/?f=%5Csin%5E2t%3D%5Cdfrac%7B1-%5Ccos%282t%29%7D2)
so that
![\begin{cases}x=2a\sin(2t)\cos^2t\\y=2b\cos(2t)\sin^2t\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7Dx%3D2a%5Csin%282t%29%5Ccos%5E2t%5C%5Cy%3D2b%5Ccos%282t%29%5Csin%5E2t%5Cend%7Bcases%7D)
Then we have
![\dfrac{\mathrm dx}{\mathrm dt}=4a\cos(2t)\cos^2t-4a\sin(2t)\cos t\sin t](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dx%7D%7B%5Cmathrm%20dt%7D%3D4a%5Ccos%282t%29%5Ccos%5E2t-4a%5Csin%282t%29%5Ccos%20t%5Csin%20t)
![\dfrac{\mathrm dy}{\mathrm dt}=-4b\sin(2t)\sin^2t+4b\cos(2t)\sin t\cos t](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dt%7D%3D-4b%5Csin%282t%29%5Csin%5E2t%2B4b%5Ccos%282t%29%5Csin%20t%5Ccos%20t)
![\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{4b\cos(2t)\sin t\cos t-4b\sin(2t)\sin^2}{4a\cos(2t)\cos^2t-4a\sin(2t)\cos t\sin t}](https://tex.z-dn.net/?f=%5Cimplies%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%3D%5Cdfrac%7B4b%5Ccos%282t%29%5Csin%20t%5Ccos%20t-4b%5Csin%282t%29%5Csin%5E2%7D%7B4a%5Ccos%282t%29%5Ccos%5E2t-4a%5Csin%282t%29%5Ccos%20t%5Csin%20t%7D)
![\implies\boxed{\dfrac{\mathrm dy}{\mathrm dx}=\dfrac ba\tan t}](https://tex.z-dn.net/?f=%5Cimplies%5Cboxed%7B%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%3D%5Cdfrac%20ba%5Ctan%20t%7D)
where the last reduction follows from dividing through everything by
and simplifying.
I'm not sure at which point you're supposed to evaluate the derivative (22/7*4, as in 88/7? or something else?), so I'll leave that to you.
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