Trigonometric Derivatives - Part 1: Sine, Cosine, Tangent

Problem: We have 9 trigonometric functions: sin(x), cos(x), tan(x), csc(x), sec(x), cot(x), arcsin(x), arccos(x), arctan(x). We want to find their derivatives in some way or another. In this first part, I will be going over the main 3.

Last class, my teacher explained "Even though we have 9 more derivatives to learn, all it takes is one, and we can easily find the rest. In this blog post, I will hopefully be finding all the derivatives. To start off, I will be finding the derivative of sin(x) using First Principles. For those who don't know what First Principles are, here's what it is.

For those who are new to calculus, welcome. The derivative is the instantaneous rate of change of a function. If you try to use the regular slope formula (rise over run), you'll get 0/0. So rather than using the same points, we're going to take two points and move one infinitely close to the next, hence the limit as ∆x approaches 0. But for the purposes of this, I will be setting h = ∆x.


So f(x)=sin(x). We'll substitute this in, and end up with the following.

Next, we'll use the trigonometric identity of sin(x+h) = sin(x)*cos(h)+cos(x)*sin(h).

The next step is to split this into three fractions, and therefore three limits.

Next, we can use another trigonometric identity on the second limit, which states that the quotient of sin(x) and x as x approaches 0 is 1. Next, for simplicity, I'm actually going to combine the remaining limits, as we will be able to see another identity.

Next, we can apply the distributive property in reverse. 

And finally, the limit as h approaches 0 is equal to 0, which is easily shown on a graph.

I realize after all this, that I haven't been using proper notation. I should have an f'(x) behind each of those. However, I will not spend more time fixing this, as each of those is an image that I would have to remake.

So the derivative of sin(x) with respect to x is cos(x). To find the derivative of cos(x), I could easily do the first principles again, but creating the images takes too long, and I need the time for the later ones. Instead, I'm going to look at this graphically, and find the derivative through that. Though I'm going to be completely honest, sin(x) and cos(x) are two that I already know from past curiosity. Now, I'm just proving them.

Below is a graph of cos(x).

As we can see, we have stationary points at kπ, where k is an integer. It's at these points where the derivative will be 0. From there, the function goes from decreasing to increasing then from increasing to decreasing between the stationary points. We have points of inflection at (2k+1)π/2, where k is an integer. It's at these points where we will have a maximum or a minimum. Below shows the graph of what the derivative really looks like, as well as the actual function.

So we can see that the derivative of cos(x) is -sin(x). Next, we can move onto the fun ones, starting with tan(x). We can change this using the following trigonometric identity.

And we know that we can use the quotient rule, which states the following. Luckily, we know what each of these is.

This can be rearranged to get rid of that double negative.

This is turning out much nicer than I thought it would. We can apply a trigonometric identity, since cos²(x)+sin²(x)=1.

And finally, we can simplify.

So now we have the basic 3 trigonometric functions and their derivatives:

Original -> Derivative

sin(x) -> cos(x)

cos(x) -> -sin(x)

tan(x) -> sec²(x)

In my next blog post, I will continue to find the remaining derivatives of trigonometry.