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Handout Wednesday 10/1
Our goal today is to discover derivative rules for the rest of the main trigonometric functions: \(\tan(x)\text{,}\) \(\cot(x)\text{,}\) \(\sec(x)\text{,}\) and \(csc(x)\text{.}\)
First, remember the Fundamental Trigonometric Identity:
\begin{equation*}
\sin^2(x) + \cos^2(x) = 1\text{.}
\end{equation*}
This is just a version of the Pythagorean Theorem on the unit circle. We will use this to simplify the derivatives for the other trig functions.
Look at \(\tan(x)\text{.}\) In right triangles, \(\tan(\theta)\) is the ratio of the leg opposite the angle to the leg adjacent to it. But look at the unit circle, and remember that \(y = \sin(\theta)\) and \(x = \cos(\theta)\text{,}\) we have \(\tan(x) = \frac{\sin(x)}{\cos(x)}\text{.}\) So to find the derivative of \(\tan(x)\) we should apply the quotient rule to that ratio.
Similarly, we can do this for all the other functions. Go through them.
