ত্রিকোনমিতিক ফাংশনের অন্তরজ

dydx\displaystyle\frac{dy}{dx} at t=π4\displaystyle t=\frac{\pi}{4} for x=a[cost+12logtan2t2]\displaystyle x=a\left[\cos{t}+\frac{1}{2}\log{\tan^2{\frac{t}{2}}}\right] and y=asinty=a\sin{t} is

হানি নাটস

x=π4\displaystyle x=\frac{\pi}{4} for x=a[cost+12logtan2t2]\displaystyle x=a\left[\cos{t}+\frac{1}{2}\log{\tan^2{\frac{t}{2}}}\right] and

y=asinty=a\sin{t}

Differentiating w.r.t tt, we get

dxdt=a[ddtcost+12ddt(logtan2t2)]\displaystyle \frac { dx }{ dt } =a\left[ \frac { d }{ dt } \cos { t } +\frac { 1 }{ 2 } \frac { d }{ dt } \left( \log { \tan ^{ 2 }{ \frac { t }{ 2 } } } \right) \right]

dxdt=a[sint+1tant2sec2t2×12]\displaystyle\frac{dx}{dt}=a\left[-\sin{t}+\frac{1}{\displaystyle\tan{\frac{t}{2}}}\sec^2{\frac{t}{2}}\times\frac{1}{2}\right]

dxdt=a[sint+12sint/2cost/2.cos2t/2]\displaystyle \frac { dx }{ dt } =a\left[ -\sin { t } +\frac { 1 }{ 2\displaystyle \frac { \sin { { t }/{ 2 } } }{ \cos { { t }/{ 2 } } } .\cos ^{ 2 }{ { t }/{ 2 } } } \right]

=a[sint+12sint2cost2]\displaystyle =a\left[-\sin{t}+\frac{1}{\displaystyle 2\sin{\frac{t}{2}}\cos{\frac{t}{2}}}\right]

=a[sint+1sint]\displaystyle =a\left[-\sin{t}+\frac{1}{\sin{t}}\right]

=a[1sin2tsint]=a[cos2tsint]=a\left[ \displaystyle \frac { 1-\sin ^{ 2 }{ t } }{ \sin { t } } \right] =a\left[ \displaystyle \frac { \cos ^{ 2 }{ t } }{ \sin { t } } \right]

dydt=acost\displaystyle\frac{dy}{dt}=a\cos{t}

dydx=dydtdxdt=acostacos2tsint=tant\displaystyle\therefore\frac{dy}{dx}=\frac{\displaystyle\frac{dy}{dt}}{\displaystyle\frac{dx}{dt}}=\frac{a\cos{t}}{\displaystyle\frac{a\cos^2{t}}{\sin{t}}}=\tan{t}

At t=π4\displaystyle t=\frac{\pi}{4}, dydx=1\displaystyle\frac{dy}{dx}=1

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