# Graphing Trigonometric Functions and computing their derivative

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• Nov 7th 2010, 01:54 PM
Syia
Graphing Trigonometric Functions and computing their derivative
For an upcoming midterm, I need to be able to do the following: graph, indicate direction & speed, compute the derivative, and find critical points of trigonometric and hyperbolic functions.

For example:

$\displaystyle (sin\theta, cos\theta)$

$\displaystyle (csc\theta, cot\theta) \theta \neq k\pi, k\in Z$

$\displaystyle (cosh t, sinh t)$

What perplexes me is the form: (a, x) with respect to theta. Are these parametric plots or the two functions mapped onto one plot? My interpretation is that they are parametric. So, for example,

$\displaystyle (sin\theta, cos\theta)$

Should be a unit circle except $\displaystyle sin\theta = x$ and $\displaystyle cos\theta = y$

If this is true, do I take the derivative of $\displaystyle sin\theta = x => cos\theta$ and $\displaystyle cos\theta = y => -sin\theta$?

But then the plot of its derivative is also graphically the same as the original function. Any help would be greatly appreciated, thanks for your time.
• Nov 7th 2010, 02:37 PM
TheEmptySet
If you are correct that

$\displaystyle (\cos(\theta),\sin(\theta))$ is parametric then to find the derivative you can use this formula

if
$\displaystyle x=\cos(\theta);y=\sin(\theta)$
Then
$\displaystyle \displaystyle \frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{ d\theta}}=\frac{\cos(\theta)}{-\sin(\theta)}$