Paul's Online Notes
Home / Calculus I / Derivatives / Derivatives of Trig Functions
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 3.5 : Derivatives of Trig Functions

11. Find the tangent line to $$f\left( x \right) = \tan \left( x \right) + 9\cos \left( x \right)$$ at $$x = \pi$$.

Show All Steps Hide All Steps

Start Solution

We know that the derivative of the function will give us the slope of the tangent line so we’ll need the derivative of the function.

$f'\left( x \right) = {\sec ^2}\left( x \right) - 9\sin \left( x \right)$ Show Step 2

Now all we need to do is evaluate the function and the derivative at the point in question.

$f\left( \pi \right) = \tan \left( \pi \right) + 9\cos \left( \pi \right) = - 9\hspace{0.5in}f'\left( \pi \right) = {\sec ^2}\left( \pi \right) - 9\sin \left( \pi \right) = 1$ Show Step 3

Now all that we need to do is write down the equation of the tangent line.

$y = f\left( \pi \right) + f'\left( \pi \right)\left( {x - \pi } \right) = - 9 + \left( 1 \right)\left( {x - \pi } \right)\hspace{0.25in} \to \hspace{0.25in}\,\require{bbox} \bbox[2pt,border:1px solid black]{{y = x - \pi - 9}}$

Don’t get excited about the presence of the $$\pi$$ in the answer. It is just a number like the 9 is and so is nothing to worry about.