Trigonometric functions are often used in modeling real-world phenomena. In particular, vibrations, waves, elastic motions, and other quantities that vary in a periodic manner can be described using trigonometric functions.
If we sketch the graph of the function $f(x)=\sin(x)$ and use the interpretation of $f'(x)$ as the slope of the tangent to the sine curve in order to sketch the graph of $f'$ then it looks as if the graph of $f'$ may be the same as the cosine curve.
So we have proved the formula for the derivative of the sine function:
$$ \frac{d}{dx}\left( \sin(x) \right)= \cos(x) $$
Using the same methods we can prove that
$$ \frac{d}{dx}\left( \cos(x) \right)= -\sin(x) $$
The tangent function can also be differentiated by using the definition of a derivative, but it is easier to use the Quotient Rule together
$$ \frac{d}{dx}\left( \tan(x) \right)= \frac{d}{dx}\left(\frac{\sin(x)}{\cos(x)}\right)=\sec^2(x) $$
The derivatives of the remaining trigonometric functions, csc, sec, and cot, can also be found easily using the Quotient Rule. We collect all the differentiation formulas for trigonometric functions in the following table. Remember that they are valid only when x is measured in radians.
$$ \frac{d}{dx}\left( \sin(x) \right)= \cos(x) $$
$$ \frac{d}{dx}\left( \cos(x) \right)= -\sin(x) $$
$$ \frac{d}{dx}\left( \tan(x) \right)=\sec^2(x) $$
$$ \frac{d}{dx}\left( \csc(x) \right)= -\csc(x)\cot(x) $$
$$ \frac{d}{dx}\left( \sec(x) \right)= \sec(x)\tan(x) $$
$$ \frac{d}{dx}\left( \cot(x) \right)=-\csc^2(x) $$
<aside>
Differentiate:
$$ y=x^2\sin(x) $$
</aside>
<aside>
Differentiate the function and find for what values of $x$ does the graph of $f$ have a horizontal tangent?
$$ \frac{\sec(x)}{1+\tan(x)} $$
</aside>
<aside>
Find the 27th derivative of $\cos(x)$.
</aside>
In proving the formula for the derivative of sine we used two special limits, which we now prove
$$ \lim_{\theta \rarr 0}\frac{\sin(\theta)}{\theta}=1 $$
The first special limit we considered concerned the sine function. The following special limit involves cosine.