Derivatives of Trig Functions

On this page, we set the stage to derive the values of $(\sin x)'$ and $(\cos x)'$ . This is particularly interesting to those of you who are curious about how we determine such derivatives for which we do not (yet) have a derivative rule.

You will need to know the values of the following limits later in the semester, not only for this application.

Unless you are told otherwise, you will not be expected to be able to reproduce the following proofs, but it is certainly true that working through the rest of this page, carefully, will increase your mathematical understanding in many ways.

The two important limits (derived below):

$\lim_{x \to 0} \frac{\sin(x)}{x} = 1 \qquad \mbox{and} \qquad \lim_{x \to 0}\frac{1-\cos(x)}{x}=0.$ Notice that both of these limits have the

$\frac{0}{0}$ indeterminate form, so we need to do more work, but what work?

In the following video, we use geometry and the squeeze theorem to derive the first of the two limits:

$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$

Once we know the first limit, the second limit

$\lim_{x \to 0}\frac{\cos(x)-1}{x}=0$

is fairly easy:

$\begin{eqnarray*} \displaystyle{\lim_{x \to 0} \frac{\cos(x)-1}{x}} & = & \displaystyle{\lim_{x \to 0} \left ( \frac{\cos(x)-1}{x}\cdot \frac{\cos(x)+1}{\cos(x)+1} \right ) } \cr & = & \displaystyle{\lim_{x \to 0} \left ( \frac{\cos^2(x)-1}{x(1+\cos(x))}\right )} \cr &=&\displaystyle{\lim_{x \to 0} \left (\frac{-\sin^2(x)}{x(1+\cos(x))}\right )} \cr &=& \displaystyle{-\lim_{x \to 0} \left ( \frac{\sin(x)}{x}\cdot\frac{\sin(x)}{(1+\cos(x))}\right )}\cr &=& \displaystyle{-\left ( \lim_{x \to 0} \frac{\sin(x)}{x} \right )} \cdot \displaystyle{\left ( \lim_{x \to 0} \frac{\sin(x)}{1+\cos(x)}\right )}. \end{eqnarray*}$

We just showed that the first of the limits on the last line is 1, and the second limit is $\displaystyle{\frac{\sin(0)}{1+\cos(0)}=\frac{0}{2}=0}$ , so the product is 0.

The Six Pillars of Calculus

Trigonometry Review

Exponential Functions

Logarithms and Inverse functions

Intro to Limits

Limit Laws and Computations

Continuity and the Intermediate Value Theorem

Limits at Infinity

Rates of Change

The Derivative Function

Basic Differentiation Rules

Product and Quotient Rules

Derivatives of Trig Functions

The Chain Rule

Implicit Differentiation

Derivatives of Logs

Derivatives in Science

Related Rates

Linear Approximation and Differentials

Differentiation Review

Absolute and Local Extrema

The Mean Value and other Theorems

ff vs. f′f'

Indeterminate Forms and L'Hospital's Rule

Optimization

Newton's Method

Anti-derivatives

The Area under a curve

Definite Integrals

The Fundamental Theorem of Calculus

Limits necessary to determine the derivatives of sinx\sin x and cosx\cos x

$f$ vs. $f'$

Limits necessary to determine the derivatives of $\sin x$ and $\cos x$