# Differentiating Parametric Functions

Gold Member

Given:
$$\begin{array}{l} x = 2\cos t \\ y = 2\sin t \\ \end{array}$$

find $$\frac{{dy}}{{dx}}$$

I started by finding dy/dt and dx/dt

$$\begin{array}{l} \frac{{dy}}{{dt}} = 2\cos t \\ \frac{{dx}}{{dt}} = - 2\sin t \\ \end{array}$$

Now, dy/dx = (dy/dt) / (dx/dt)

so:

$$\frac{{dy}}{{dx}} = \frac{{2\cos t}}{{ - 2\sin t}} = - \cot t$$

$$\begin{array}{l} x = 2\cos t \\ \therefore t = \cos ^{ - 1} \frac{x}{2} \\ \end{array}$$

Substituting that into my dy/dx which is currently in terms of t:

$$\frac{{dy}}{{dx}} = - \cot t = - \cot \cos ^{ - 1} \frac{x}{2}$$

Now, my book gives a completely different answer, which isnt even in terms of any trigonometric function. Where have i gone wrong?

Thanks,
Dan.

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Do you know that sin2t + cos2t = 1? ;)

Gold Member
Do you know that sin2t + cos2t = 1? ;)
Yep i do. Not quite seeing where it comes into it though EDIT: ahhh now ive got it. Square my x and y functions, and then add them, which eliminates the parameter :) Thanks for that.

Last edited:
Dick