# Trigonometric Derivatives

I was recently reading The Endeavour where he responded to a post over at Math Mama Writes about teaching the derivatives of the trigonometric functions.

I decided to weigh in on the issue.

In my experience, Calculus is always best taught in terms of infinitesimals, as in Thompson’s Book, (which I’ve already talked about ) and Trigonometry is best taught using the complete triangle. Marrying these two together, we can give a simple geometric proof of the basic trigonometric derivatives:

Summed up on one diagram:

### Short version

By looking at how the line

### Long version

You’ll notice I’ve drawn a unit circle in the bottom right, chosen an angle

We are interested in how

Since we’ve only moved the angle a little bit, I’ve included a zoomed in picture in the upper right so that we can continue. Here, we see the solid and dashed lines again where they meet our unit circle. Notice that since we’ve zoomed in quite a bit the circle’s edge doesn’t look very circley anymore, it looks like a straight line.

In fact that is the first thing we’ll note, namely that the arc of the circle we trace when we change the angle a little bit has the length

which we may have remembered anyway from our trig classes. What is important here is that even though

You’ll notice that in the zoomed in picture, we can see the yellow and green segments,
which correspond to the changes in the length of the dotted yellow and green segments
from the zoomed out picture. These are the segments I’ve marked

Now for the kicker. Notice the right triangle formed by the green, yellow and red sements? That is similar to the larger triangle in the zoomed out picture. I’ve marked the similar angle in red. If you stare at the picture for a bit, you can convince yourself of this fact. If all else fails, just compute all of the angles involved in the intersection of the circle with the blue line, they can all be resolved.

Knowing that the two triangles are similar, we know that the lengths of theirs sides are equal except for some scale factor, in particular:

In particular, the change in the sine of the angle (

Doing the same for the

From here, the other trigonometric derivates are easy to obtain, either by making similar pictures al la the complete triangle, or by using the regular rules relating all of the trigonometric function to one another.

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