# Thread: Equation of tangent line.

1. ## Equation of tangent line.

Find and equation of the tangent line to the curve at the point defined by the given value of t. Also, find the value of $\displaystyle \frac{d^2y}{dx^2}$ at this point $\displaystyle x=-cos(t)$, $\displaystyle y=9+sin(t)$, $\displaystyle t=\frac{\pi}{2}$

Thanks for any help.

2. Originally Posted by Gotovina7
Find and equation of the tangent line to the curve at the point defined by the given value of t. Also, find the value of $\displaystyle \frac{d^2y}{dx^2}$ at this point $\displaystyle x=-cos(t)$, $\displaystyle y=9+sin(t)$, $\displaystyle t=\frac{\pi}{2}$

Thanks for any help.
$\displaystyle \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}= \frac{\cos(t)}{\sin(t)}=\cot(t)$

$\displaystyle \frac{dy}{dx}|_{t=\pi/2}=0$

now we just need the ordered pair at $\displaystyle t=\pi/2$

$\displaystyle x=-\cos(\pi/2)=0 \mbox{ and } y=9+\sin(\pi/2)=10$

So the equation of the tangent line is

$\displaystyle y=10$

$\displaystyle \frac{d^2y}{dx^2}=\frac{\frac{d}{dt}\frac{dy}{dx}} {\frac{dx}{dt}}=\frac{-\csc^{2}(t)}{\sin(t)}=-\csc^{3}(t)$

3. Ooh that makes sense, thanks man.