Monday Math 119

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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