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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:
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Using the derivative
language, this limit means that
.
This limit may
also be used to give a related one which is of equal importance:
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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
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.
Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.