## The Derivatives of Trigonometric Functions

Trigonometric functions are useful in our practical lives in diverse areas such as astronomy, physics, surveying, carpentry etc. How can we find the derivatives of the trigonometric functions?

Our starting point is the following limit:

Using the derivative language, this limit means that . This limit may also be used to give a related one which is of equal importance:

To see why, it is enough to rewrite the expression involving the cosine as

But , so we have

This limit equals and thus .

In fact, we may use these limits to find the derivative of and at any point x=a. Indeed, using the addition formula for the sine function, we have

So

which implies

So we have proved that exists and .

Similarly, we obtain that exists and that .

Since , , , and are all quotients of the functions and , we can compute their derivatives with the help of the quotient rule:

It is quite interesting to see the close relationship between and (and also between and ).

From the above results we get

These two results are very useful in solving some differential equations.

Example 1. Let . Using the double angle formula for the sine function, we can rewrite

So using the product rule, we get

which implies, using trigonometric identities,

In fact next we will discuss a formula which gives the above conclusion in an easier way.

Exercise 1. Find the equations of the tangent line and the normal line to the graph of at the point .

Exercise 2. Find the x-coordinates of all points on the graph of in the interval at which the tangent line is horizontal.

[Back] [Next]
[Trigonometry] [Calculus]
[Geometry] [Algebra] [Differential Equations]
[Complex Variables] [Matrix Algebra]

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.