# Relationship between trigonometric functions and their derivatives

### Differentiation of Trigonometry Functions

In this section we will discuss differentiating trig functions. Derivatives of all six trig functions are given and we show the derivation of the. Derivatives of the Sine, Cosine and Tangent Functions. by M. Bourne Explore animations of these functions with their derivatives here. We will use this fact in to find the derivative of f(x) = sin x. The fact can be proven of differentiation. Below is a list of the six trig functions and their derivatives.

For negative angles or those beyond a right angle, it is necessary to consider the unit circle rather than SOHCAHTOA as a definition for the trigonometric functions image source: This shows how the three trigonometric identities above hold for angles which are not acute also. For instance, here the reflection of the sine graph blue gives us the cosine graph red. I want to give a couple of practical examples of why we might care about the complementary trigonometric identities, but first an aside.

In fact, inspection of the two diagrams reveals three similar right-angled triangles yielding: The results are not always so illuminating.

Get two trig derivatives for the price of one If you have battled your way through the derivation of the derivative of sineyou may not be looking forward to demonstrating the derivative of cosine.

But we can use the complementary angle identity and then the chain rule: The relationship between the complementary trig functions would be very obvious to any user of the Manuale Mathematicum of Matthias Bernegger published in Here's an extract I took from archive. You may be more used to presenting fractions of a degrees in decimal notation, but the system used here is sexagesimalwith each degree broken into 60 minutes of arc and although this is not shown here, these are in turn broken into 60 seconds.

Note how the left-hand column allows you to look up one angle, while the right-hand column contains its complement.

This is the reason why! The proof of the formula involving sine above requires the angles to be in radians. If the angles are in degrees the limit involving sine is not 1 and so the formulas we will derive below would also change.

### Calculus I - Derivatives of Trig Functions

The formulas below would pick up an extra constant that would just get in the way of our work and so we use radians to avoid that. So, remember to always use radians in a Calculus class! Example 1 Evaluate each of the following limits. To do this problem we need to notice that in the fact the argument of the sine is the same as the denominator i.

So we need to get both of the argument of the sine and the denominator to be the same. We can do this by multiplying the numerator and the denominator by 6 as follows.

At this point, while it may not look like it, we can use the fact above to finish the limit. A change of variables is really just a renaming of portions of the problem to make something look more like something we know how to deal with. All we really need to notice is that the argument of the sine is the same as the denominator and then we can use the fact. A change of variables, in this case, is really only needed to make it clear that the fact does work.

All we need to do is multiply the numerator and denominator of the fraction in the denominator by 7 to get things set up to use the fact.

Here is the work for this limit. However, with a change of variables we can see that this limit is in fact set to use the fact above regardless.

## Differentiation of trigonometric functions

However, we could just have easily used the cosine portion so here is a quick example using the cosine portion to illustrate this. To do this we will need to use the definition of the derivative. Here is the definition of the derivative for the sine function. We can then break up the fraction into two pieces, both of which can be dealt with separately. Therefore, as far as the limits are concerned, these two functions are constants and can be factored out of their respective limits.

### Differentiation of trigonometric functions - Wikipedia

It will require a different trig formula, but other than that is an almost identical proof. The details will be left to you. All the remaining four trig functions can be defined in terms of sine and cosine and these definitions, along with appropriate derivative rules, can be used to get their derivatives. Here are the derivatives of all six of the trig functions. Example 2 Differentiate each of the following functions. There is no other way to do this derivative unlike what we saw when we first looked at the product rule.

When we first looked at the product rule the only functions we knew how to differentiate were polynomials and in those cases all we really needed to do was multiply them out and we could take the derivative without the product rule.

We are now getting into the point where we will be forced to do the product rule at times regardless of whether or not we want to.