First, you are told nothing about y(x), so I presume the answer will look something like what you are given, with unevaluated derivatives of y.\(\displaystyle y(x(\theta))\)
\(\displaystyle \frac{dy}{d\theta} = \frac{dy}{dx} \ \sin\theta \ \cos\theta \ \tan\theta\)
\(\displaystyle \frac{d^2y}{d\theta^2} = \ ?\)
Any help would be appreciated!
Why are you calculating expression for 'x'?If y is a function of x which is a function of \(\displaystyle \theta\), then \(\displaystyle \frac{dy}{d\theta}= \frac{dy}{dx}\frac{dx}{d\theta}\).
Here we are told that \(\displaystyle \frac{dy}{d\theta}= \frac{dy}{dx} sin(\theta) cos(\theta) tan(\theta)= \frac{dy}{dx} sin^2(\theta)\).
It follows that \(\displaystyle \frac{dx}{d\theta}= sin^2(\theta)= (1/2)(1- cos(2\theta)\).
\(\displaystyle x= (1/2)\theta- (1/4)sin(2\theta) \) + C
Once again, please show what you have tried, so we can see where you need help. (See #2) And I'd still very much like to see the original problem (even if it's not in English) to confirm you haven't omitted or misinterpreted something. (See #4)Thank you very much topsquark, Dr.Peterson, Shiloh, and Subhotosh Khan for helping me.
I have to find \(\displaystyle \frac{d^2y}{d\theta^2}\) without simplifying \(\displaystyle \sin\theta \ \cos\theta \ \tan\theta\).