Math, asked by karmankaur6104, 2 months ago

the slope of the tangent to the curve x=a sin t., y=a(cos t + log (tan t/2)) at the point t is​

Attachments:

Answers

Answered by guptavirag002
14

Answer:

Check it out . Hope it will help u !

Attachments:
Answered by pulakmath007
9

SOLUTION

TO DETERMINE

The slope of the tangent to the curve

  \displaystyle\sf{x = a \sin t \:  \: , \: y = a \bigg \{ \cos t +  \log \bigg(  \tan  \frac{t}{2} \bigg) \:  \bigg \} }

EVALUATION

Here the given equation of the curve is

  \displaystyle\sf{x = a \sin t \:  \: , \: y = a \bigg \{ \cos t +  \log \bigg(  \tan  \frac{t}{2} \bigg) \:  \bigg \} }

Now

  \displaystyle\sf{x = a \sin t \:  }

Differentiating both sides with respect to t we get

  \displaystyle\sf{ \frac{dx}{dt}  = a \cos t }

Now

  \displaystyle\sf{ y = a \bigg \{ \cos t +  \log \bigg(  \tan  \frac{t}{2} \bigg) \:  \bigg \} }

Differentiating both sides with respect to t we get

  \displaystyle\sf{ \frac{dy}{dt}  = a \bigg \{  -  \sin t +  \frac{ \frac{1}{2}  { \sec}^{2} \frac{t}{2}  }{\tan  \frac{t}{2}}   \bigg \} }

  \displaystyle\sf{  \implies \: \frac{dy}{dt}  = a \bigg \{  -  \sin t +  \frac{ 1}{2 { \cos}^{2} \frac{t}{2}  \tan  \frac{t}{2}}   \bigg \} }

  \displaystyle\sf{  \implies \: \frac{dy}{dt}  = a \bigg \{  -  \sin t +  \frac{ 1}{2 { \cos}^{} \frac{t}{2}  \sin  \frac{t}{2}}   \bigg \} }

  \displaystyle\sf{  \implies \: \frac{dy}{dt}  = a \bigg \{  -  \sin t +  \frac{ 1}{  \sin t }   \bigg \} }

  \displaystyle\sf{  \implies \: \frac{dy}{dt}  = a \bigg \{   \frac{ 1 -  { \sin}^{2}t }{  \sin t }   \bigg \} }

  \displaystyle\sf{  \implies \: \frac{dy}{dt}  = a \bigg \{   \frac{   { \cos}^{2}t }{  \sin t }   \bigg \} }

  \displaystyle\sf{  \implies \: \frac{dy}{dt}  = a \bigg \{   \frac{   { \cos}^{2}t }{  \sin t }   \bigg \} }

  \displaystyle\sf{  \implies \: \frac{dy}{dt}  = a  \cos t \cot t}

Hence the required slope of the tangent to the curve is

  \displaystyle\sf{   =  \frac{dy}{dx}  }

  \displaystyle\sf{   =  \frac{\frac{dy}{dt}}{\frac{dx}{dt}}}

  \displaystyle\sf{   =  \frac{a  \cos t \cot t}{a  \cos t }  }

  \displaystyle\sf{   =\cot t  }

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. 1. If x=u^2-v^2,y=2uv find the jacobian of x, y with respect to u and v

https://brainly.in/question/34361788

2. given u=yzx , v= zxy, w =xyz then the value of d(u,v,w)/d(x,y,z) is (a) 0 (b) 1 (c) 2 (d) 4

https://brainly.in/question/34435016

Similar questions