##
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
.

**Answer****.**

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

**Answer****.**

**
**

**
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[Differential Equations]
[Complex Variables]
[Matrix Algebra]

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