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:
In fact, we may use these limits to find the derivative of
at any point x=a. Indeed, using the
addition formula for the sine function, we have
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
Example 1. Let
Using the double angle
formula for the sine function, we can rewrite
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.
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