Very few of the calculus textbooks I have used give a rigorous derivation of the derivatives of sine and cosine, instead using little more than the graphs of the functions as justification. Here, I will demonstrate a rigorous derivation, starting with a clear derivation of the limit .

[Click on figure for full size image]

Start with the unit circle diagram above, with small, positive *θ*.

Now, the area of the triangle ▵OAB is . Similarly, the area of the circular sector between OA and OB is . Lastly, the right triangle ▵OAD has area .

Comparing these areas, we have inequality , which means

Multiplying this by the positive quantity , we get

,

which, using , becomes

.

Therefore, the inverses obey the inequality

,

and since , by the squeeze theorem, we see

; and since is an even function, this must also be the left-hand limit, and so we have

.

Now, consider . We can find this limit using the above limit and a little trigonometry:

.

Using these two limits and the addition formulas for sine and cosine, we compute the derivatives from the definition .

First,

,

and second,

.

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Tags: Calculus, Derivative, Math, Monday Math, Squeeze Theorem, Trigonometry

This entry was posted on May 17, 2010 at 8:59 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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