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\begin_layout Title
Derivatives of trigonometric functions
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\begin_layout Subtitle
In this module we will determine more efficient ways of computing the derivative
s of 
\emph on

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\backslash
hilite
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 trigonometric functions
\emph default
.
 
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%1
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Defining the problem
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\begin_layout Standard
\noindent
The precise definition of the derivative tells us that
\begin_inset Formula 
\[
{\ec f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}}.
\]

\end_inset

 It is often tedious to compute the derivative using the precise definition.
 We would like a shorter way of computing the derivative for trigonometric
 functions.
\end_layout

\begin_layout Standard
You saw in the module on experimental derivatives that the derivative of
 
\begin_inset Formula $\ec\sin x$
\end_inset

 appears to be 
\begin_inset Formula $\ec\cos x$
\end_inset

.
 In this module we argue that this is in fact the case, and discuss as well
 the derivatives of the other trigonometric functions 
\begin_inset Formula $\ec\cos x$
\end_inset

, 
\begin_inset Formula $\ec\tan x$
\end_inset

, 
\begin_inset Formula $\ec\cot(x)$
\end_inset

, 
\begin_inset Formula $\ec\sec x$
\end_inset

, 
\begin_inset Formula $\ec\csc x$
\end_inset

.
 
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SAGE worksheets
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\noindent
You will need a blank SAGE worksheet for this module.
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{
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The derivative of 
\begin_inset Formula $\sin x$
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\begin_deeper
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\noindent
Let 
\begin_inset Formula $\ec f(x)=\sin x$
\end_inset

.
 We want to compute the derivative of 
\begin_inset Formula $\ec f(x)$
\end_inset

.
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\begin_deeper
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From the definition, we know that
\begin_inset Formula 
\[
{\ec f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}=\lim_{h\rightarrow0}\frac{\sin(x+h)-\sin x}{h}}.
\]

\end_inset


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<3-4>
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\begin_deeper
\begin_layout Standard
From the definition, we know that
\begin_inset Formula 
\[
\ec f'(x)=\lim_{h\rightarrow0}\frac{\sin(x+h)-\sin x}{h}.
\]

\end_inset


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\begin_layout Standard
It is not immediately obvious how we should simplify this.
 Let's try some properties of trigonometric functions.
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\begin_deeper
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Recall that 
\begin_inset Formula 
\[
{\ec\sin(x+h)=\sin x\cos h+\sin h\cos x}.
\]

\end_inset

We'll rewrite the expression above using this fact.
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\begin_deeper
\begin_layout Standard
From the definition, we know that
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\lim_{h\rightarrow0}\frac{\sin(x+h)-\sin x}{h}\\
 & {\ec=\lim_{h\rightarrow0}\frac{\sin x\cos h+\sin h\cos x-\sin x}{h}}.
\end{align*}

\end_inset

 Next we'll collect expressions containing 
\begin_inset Formula $\ec\sin x$
\end_inset

.
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<6>
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\begin_deeper
\begin_layout Standard
From the definition, we know that
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\lim_{h\rightarrow0}\frac{\sin x\cos h+\sin h\cos x-\sin x}{h}\\
 & {\ec=\lim_{h\rightarrow0}\frac{\sin x\left(\cos h-1\right)+\sin h\cos x}{h}}.
\end{align*}

\end_inset

 At this point we will apply some algebraic machinery.
 The limit would look a lot easier if we could remove the terms containing
 
\begin_inset Formula $\ec x$
\end_inset

.
 Unfortunately, we can't factor them, so we have to try something different.
 We'll try the property of limits that 
\emph on
if
\emph default
 the limit of two functions 
\begin_inset Formula $\ec g$
\end_inset

 and 
\begin_inset Formula $\ec k$
\end_inset

 exist, then the limit of their sum is the sum of their limits.
 That is\SpecialChar \ldots{}

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\begin_deeper
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From the definition, we know that
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\lim_{h\rightarrow0}\frac{\sin x\left(\cos h-1\right)+\sin h\cos x}{h}\\
 & \ec\overset{?}{=}\lim_{h\rightarrow}\frac{\sin x\left(\cos h-1\right)}{h}+\lim_{h\rightarrow0}\frac{\sin h\cos x}{h}\\
 & {\ec\overset{?}{=}\sin x\cdot\lim_{h\rightarrow}\frac{\left(\cos h-1\right)}{h}+\cos x\cdot\lim_{h\rightarrow0}\frac{\sin h}{h}}.
\end{align*}

\end_inset

 The symbol 
\begin_inset Formula $\ec\overset{?}{=}$
\end_inset

 indicates that we don't know the two sides are equal, because we don't
 know that these limits exist.
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begin{explanation}
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We also used the constant multiple property of limits: 
\begin_inset Formula $\ec\sin x$
\end_inset

 and 
\begin_inset Formula $\ec\cos x$
\end_inset

 are constant with respect to a limit on 
\begin_inset Formula $\ec h$
\end_inset

.)
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{
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Do these limits exist?
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\begin_deeper
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\noindent
We will not prove that these limits have values; we will simply approximate
 them.
 Turn to SAGE now and estimate the limits
\begin_inset Formula 
\begin{align*}
\ec\lim_{h\rightarrow}\frac{\left(\cos h-1\right)}{h} & \ec=?\\
\ec\lim_{h\rightarrow0}\frac{\sin h}{h} & {\ec=?}.
\end{align*}

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Hint:
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 Use either a numerical table or a graph to approximate the limit.
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\begin_deeper
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\noindent
We will not prove that these limits have values; we will simply approximate
 them.
 Turn to SAGE now and estimate the limits
\begin_inset Formula 
\begin{align*}
\ec\lim_{h\rightarrow}\frac{\left(\cos h-1\right)}{h} & \ec=0\\
\ec\lim_{h\rightarrow0}\frac{\sin h}{h} & {\ec=1}.
\end{align*}

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\begin_inset space \qquad{}
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	filename figures/lim_cosx_1_x.png
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\begin_inset Caption

\begin_layout Plain Layout
 Plots of 
\begin_inset Formula $\ec\sin h/h$
\end_inset

 (left) and 
\begin_inset Formula $\ec(\cos h-1)/h$
\end_inset

 (right).
\end_layout

\end_inset


\end_layout

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\end_deeper
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The derivative of 
\begin_inset Formula $\sin x$
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\begin_deeper
\begin_layout Standard
\noindent
Let 
\begin_inset Formula $f(x)=\sin x$
\end_inset

.
 We want to compute the derivative of 
\begin_inset Formula $f(x)$
\end_inset

.
\end_layout

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<1>
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\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
From the definition, we know that
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\lim_{h\rightarrow0}\frac{\sin x\left(\cos h-1\right)+\sin h\cos x}{h}\\
 & \ec=\lim_{h\rightarrow}\frac{\sin x\left(\cos h-1\right)}{h}+\lim_{h\rightarrow0}\frac{\sin h\cos x}{h}\\
 & {\ec=\sin x\cdot\lim_{h\rightarrow}\frac{\left(\cos h-1\right)}{h}+\cos x\cdot\lim_{h\rightarrow0}\frac{\sin h}{h}}.
\end{align*}

\end_inset

 Now we know that the limits exist, so we can apply the limit properties
 without qualms.
\end_layout

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<2>
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\begin_deeper
\begin_layout Standard
From the definition, we know that
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\sin x\cdot\lim_{h\rightarrow}\frac{\left(\cos h-1\right)}{h}+\cos x\cdot\lim_{h\rightarrow0}\frac{\sin h}{h}\\
 & {\ec=\sin x\cdot0+\cos x\cdot1}.
\end{align*}

\end_inset


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\end_deeper
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The derivative of 
\begin_inset Formula $\sin x$
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\begin_layout Standard
\noindent
Thus
\end_layout

\begin_layout Theorem
\noindent
The derivative of 
\begin_inset Formula $\ec\sin x$
\end_inset

 is 
\begin_inset Formula $\ec\cos x$
\end_inset

.
\end_layout

\begin_layout Standard
\noindent
This confirms our intuition from the experimentation.
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The derivative of 
\begin_inset Formula $\cos x$
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\begin_deeper
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\noindent
Now let's turn to the derivative of 
\begin_inset Formula $\ec\cos x$
\end_inset

.
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Experimenting with the 
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 Function Factory
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 Tangent Line Guesser
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 SAGElets suggests that the derivative of 
\begin_inset Formula $\ec\cos x$
\end_inset

 is 
\begin_inset Formula $\ec-\sin x$
\end_inset

.
 The precise definition gives us
\begin_inset Formula 
\begin{align*}
\ec f'(x) & {\ec=\lim_{h\rightarrow0}\frac{\cos\left(x+h\right)-\cos x}{h}}.
\end{align*}

\end_inset


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 Tangent Line Guesser
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 SAGElets suggests that the derivative of 
\begin_inset Formula $\ec\cos x$
\end_inset

 is 
\begin_inset Formula $\ec-\sin x$
\end_inset

.
 The precise definition gives us
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\lim_{h\rightarrow0}\frac{\cos\left(x+h\right)-\cos x}{h}\\
 & {\ec=\lim_{h\rightarrow0}\frac{\cos x\cos h-\sin x\sin h-\cos x}{h}}.
\end{align*}

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Trig property.
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Experimenting with the 
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 Function Factory
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 Tangent Line Guesser
\begin_inset ERT
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\end_inset

 SAGElets suggests that the derivative of 
\begin_inset Formula $\ec\cos x$
\end_inset

 is 
\begin_inset Formula $\ec-\sin x$
\end_inset

.
 The precise definition gives us
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\lim_{h\rightarrow0}\frac{\cos x\cos h-\sin x\sin h-\cos x}{h}\\
 & {\ec=\lim_{h\rightarrow0}\frac{\cos x\left(\cos h-1\right)-\sin x\sin h}{h}}.
\end{align*}

\end_inset

 
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Collected 
\begin_inset Formula $\cos x$
\end_inset

.
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\end_inset


\end_layout

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Experimenting with the 
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 Function Factory
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 Tangent Line Guesser
\begin_inset ERT
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\end_inset

 SAGElets suggests that the derivative of 
\begin_inset Formula $\ec\cos x$
\end_inset

 is 
\begin_inset Formula $\ec-\sin x$
\end_inset

.
 The precise definition gives us
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\lim_{h\rightarrow0}\frac{\cos x\left(\cos h-1\right)-\sin x\sin h}{h}\\
 & {\ec=\cos x\cdot\lim_{h\rightarrow0}\frac{\cos h-1}{h}-\sin x\cdot\lim_{h\rightarrow0}\frac{\sin h}{h}}.
\end{align*}

\end_inset

 
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Limit properties.
 We argued earlier that these limits exist.
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Experimenting with the 
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\end_inset

 SAGElets suggests that the derivative of 
\begin_inset Formula $\ec\cos x$
\end_inset

 is 
\begin_inset Formula $\ec-\sin x$
\end_inset

.
 The precise definition gives us
\begin_inset Formula 
\begin{align*}
\dims f'(x) & \dims=\cos x\lim_{h\rightarrow0}\frac{\cos h-1}{h}-\sin x\lim_{h\rightarrow0}\frac{\sin h}{h}\\
 & {\ec=\cos x\cdot0-\sin x\cdot1}.
\end{align*}

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The derivative of 
\begin_inset Formula $\cos x$
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\begin_layout Standard
\noindent
Thus
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\begin_layout Theorem
\noindent
The derivative of 
\begin_inset Formula $\ec\cos x$
\end_inset

 is 
\begin_inset Formula $\ec-\sin x$
\end_inset

.
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The derivatives of other trig functions
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\begin_deeper
\begin_layout Standard
\noindent
Other trig functions can be derived using
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\begin_layout Itemize
the derivatives of 
\begin_inset Formula $\ec\sin x$
\end_inset

 and 
\begin_inset Formula $\ec\cos x$
\end_inset

, and 
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\begin_layout Itemize
the properties of derivatives.
 
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\begin_deeper
\begin_layout Standard
For example, 
\begin_inset Formula $\ec\tan x=(\sin x)/(\cos x)$
\end_inset

, so its derivative is (from the quotient rule) 
\begin_inset Formula 
\begin{align*}
\ec\frac{\cos x\cdot\cos x-\sin x\cdot(-\sin x)}{\cos^{2}x} & \phantom{=\frac{\cos^{2}x+\sin^{2}x}{\cos^{2}x}}
\end{align*}

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\begin_deeper
\begin_layout Standard
For example, 
\begin_inset Formula $\ec\tan x=(\sin x)/(\cos x)$
\end_inset

, so its derivative is (from the quotient rule) 
\begin_inset Formula 
\begin{align*}
\ec\frac{\cos x\cdot\cos x-\sin x\cdot(-\sin x)}{\cos^{2}x} & \ec=\frac{\cos^{2}x+\sin^{2}x}{\cos^{2}x}
\end{align*}

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\begin_deeper
\begin_layout Standard
For example, 
\begin_inset Formula $\ec\tan x=(\sin x)/(\cos x)$
\end_inset

, so its derivative is (from the quotient rule) 
\begin_inset Formula 
\begin{align*}
\dims\frac{\cos x\cdot\cos x-\sin x\cdot(-\sin x)}{\cos^{2}x} & \dims=\frac{\cos^{2}x+\sin^{2}x}{\cos^{2}x}\\
 & \ec=\frac{1}{\cos^{2}x}
\end{align*}

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Pythagorean identity.
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\begin_deeper
\begin_layout Standard
For example, 
\begin_inset Formula $\ec\tan x=(\sin x)/(\cos x)$
\end_inset

, so its derivative is (from the quotient rule) 
\begin_inset Formula 
\begin{align*}
\dims\frac{\cos x\cdot\cos x-\sin x\cdot(-\sin x)}{\cos^{2}x} & \dims=\frac{\cos^{2}x+\sin^{2}x}{\cos^{2}x}\\
 & \dims=\frac{1}{\cos^{2}x}\\
 & {\ec=\sec^{2}x}.
\end{align*}

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The derivative of other trig functions
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\begin_layout Standard
\noindent
Similar arguments will give the derivatives of other trig functions.
 We omit the details and summarize here.
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\begin_layout Theorem
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[
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(The derivatives of basic trig functions)
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\begin_deeper
\begin_layout Itemize
The derivative of 
\begin_inset Formula $\ec\sin x$
\end_inset

 is 
\begin_inset Formula $\ec\cos x$
\end_inset

.
 
\end_layout

\begin_layout Itemize
The derivative of 
\begin_inset Formula $\ec\cos x$
\end_inset

 is 
\begin_inset Formula $\ec-\sin x$
\end_inset

.
 
\end_layout

\begin_layout Itemize
The derivative of 
\begin_inset Formula $\ec\tan x$
\end_inset

 is 
\begin_inset Formula $\ec\sec^{2}x$
\end_inset

.
 
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\begin_layout Itemize
The derivative of 
\begin_inset Formula $\ec\cot x$
\end_inset

 is 
\begin_inset Formula $\ec-\csc^{2}x$
\end_inset

.
 
\end_layout

\begin_layout Itemize
The derivative of 
\begin_inset Formula $\ec\sec x$
\end_inset

 is 
\begin_inset Formula $\ec\sec x\tan x$
\end_inset

.
 
\end_layout

\begin_layout Itemize
The derivative of 
\begin_inset Formula $\ec\csc x$
\end_inset

 is 
\begin_inset Formula $\ec-\csc x\cot x$
\end_inset

.
 
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\end_deeper
\begin_layout Standard
You should commit these to memory.
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Example 1
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\begin_layout Standard
\noindent
The properties of derivatives allow us to say that if 
\begin_inset Formula $\ec f(x)=3\tan x-2\cos x$
\end_inset

 then
\begin_inset Formula 
\[
{\ec f'(x)=3\sec^{2}x+2\sin x}.
\]

\end_inset


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Example 2
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\begin_layout Standard
\noindent
If we want to find the equation of the line tangent to 
\begin_inset Formula $\ec\sin x$
\end_inset

 at 
\begin_inset Formula $\ec x=\pi/3$
\end_inset

, its slope is 
\begin_inset Formula $\cos\left(\pi/3\right)$
\end_inset

 and so the line is
\begin_inset Formula 
\begin{align*}
\ec y-\frac{\sqrt{3}}{2} & \ec=\frac{1}{2}\left(x-\frac{\pi}{3}\right)\\
\ec y & {\ec=\frac{x}{2}-\frac{\pi}{6}+\frac{\sqrt{3}}{2}}.
\end{align*}

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	filename figures/tanline.png
	lyxscale 20
	scale 30

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Example 3
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\begin_layout Standard
\noindent
Since 
\begin_inset Formula $\ec1\approx\pi/3$
\end_inset

, we can compute a linear approximation of 
\begin_inset Formula $\ec\sin1$
\end_inset

 using the tangent line at 
\begin_inset Formula $\ec x=\pi/3$
\end_inset

.
 
\begin_inset Formula 
\begin{align*}
\ec\sin1\approx L(1) & \ec=\cos\left(\frac{\pi}{3}\right)\left(1-\frac{\pi}{3}\right)+\sin\left(\frac{\pi}{3}\right)\\
 & \ec=\frac{1}{2}\left(1-\frac{\pi}{3}\right)+\frac{\sqrt{3}}{2}\\
 & \ec=\frac{3\left(1+\sqrt{3}\right)-\pi}{6}\\
 & {\ec\approx0.8424}.
\end{align*}

\end_inset

 In fact, 
\begin_inset Formula $\ec\sin1\approx0.8415$
\end_inset

.
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Conclusion
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\begin_layout Itemize
We found a shortcut for 
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\backslash
hilite
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 the derivatives of trigonometric functions
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}
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.
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\backslash
setlength{
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\begin_layout Theorem
\begin_inset ERT
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[
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(The derivatives of basic trig functions)
\begin_inset ERT
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]
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
The derivative of 
\begin_inset Formula $\ec\sin x$
\end_inset

 is 
\begin_inset Formula $\ec\cos x$
\end_inset

.
 
\end_layout

\begin_layout Itemize
The derivative of 
\begin_inset Formula $\ec\cos x$
\end_inset

 is 
\begin_inset Formula $\ec-\sin x$
\end_inset

.
 
\end_layout

\begin_layout Itemize
The derivative of 
\begin_inset Formula $\ec\tan x$
\end_inset

 is 
\begin_inset Formula $\ec\sec^{2}x$
\end_inset

.
 
\end_layout

\begin_layout Itemize
The derivative of 
\begin_inset Formula $\ec\cot x$
\end_inset

 is 
\begin_inset Formula $\ec-\csc^{2}x$
\end_inset

.
 
\end_layout

\begin_layout Itemize
The derivative of 
\begin_inset Formula $\ec\sec x$
\end_inset

 is 
\begin_inset Formula $\ec\sec x\tan x$
\end_inset

.
 
\end_layout

\begin_layout Itemize
The derivative of 
\begin_inset Formula $\ec\csc x$
\end_inset

 is 
\begin_inset Formula $\ec-\csc x\cot x$
\end_inset

.
 
\end_layout

\end_deeper
\begin_layout Itemize
To find these shortcuts, we used the precise definition of the derivative,
 argued the values of limits using geometry, and used properties of the
 derivative.
 
\end_layout

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\end_body
\end_document
