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2.4: Derivatives of Trigonometric Functions

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

A student will be able to:

  • Compute the derivatives of various trigonometric functions.

If the angle \begin{align*}h\end{align*}h is measured in radians,

\begin{align*} \lim_{h \to 0} \frac{\sin h} {h} = 1\end{align*}limh0sinhh=1 and \begin{align*}\lim_{h \to 0} \frac{1 - \cos h} {h} = 0.\end{align*}limh01coshh=0.

We can use these limits to find an expression for the derivative of the six trigonometric functions \begin{align*}\sin x, \cos x, \tan x, \sec x, \csc x,\end{align*}sinx,cosx,tanx,secx,cscx, and \begin{align*}\cot x\end{align*}cotx. We first consider the problem of differentiating \begin{align*}\sin x\end{align*}sinx, using the definition of the derivative.

\begin{align*} \frac{d} {dx} [\sin x] = \lim_{h \to 0} \frac{\sin(x + h) - \sin x} {h}\end{align*}ddx[sinx]=limh0sin(x+h)sinxh

Since

\begin{align*}\sin(\alpha + \beta) = \sin \alpha\ \cos \beta + \cos \alpha\ \sin \beta.\end{align*}sin(α+β)=sinα cosβ+cosα sinβ.

The derivative becomes

\begin{align*}\frac{d} {dx} [\sin x] & = \lim_{h \to 0} \frac{\sin x \cos h + \cos x \sin h - \sin x} {h}\\ & = \lim_{h \to 0} \left [\sin x \left (\frac{\cos h - 1} {h} \right ) + \cos x \left (\frac {\sin h} {h} \right) \right]\\ & = -\!\sin x \cdot \lim_{h \to 0} \left (\frac{1 - \cos h} {h} \right) + \cos x \cdot \lim_{h \to 0} \left (\frac{\sin h} {h} \right)\\ & = -\!\sin x \cdot (0) + \cos x \cdot (1)\\ & = \cos x.\end{align*}ddx[sinx]=limh0sinxcosh+cosxsinhsinxh=limh0[sinx(cosh1h)+cosx(sinhh)]=sinxlimh0(1coshh)+cosxlimh0(sinhh)=sinx(0)+cosx(1)=cosx.

Therefore,

\begin{align*}\frac {d}{dx} [\sin x] = \cos x.\end{align*}ddx[sinx]=cosx.

It will be left as an exercise to prove that

\begin{align*}\frac {d}{dx}[\cos x] = -\!\sin x.\end{align*}ddx[cosx]=sinx.

The derivatives of the remaining trigonometric functions are shown in the table below.

Derivatives of Trigonometric Functions

\begin{align*}\frac {d}{dx}[\sin x] & = \cos x\\ \frac {d}{dx}[\cos x] & = -\!\sin x\\ \frac {d}{dx}[\tan x] & = {\sec^2x}\\ \frac {d}{dx}[\sec x] & = {\sec x}\ {\tan x}\\ \frac {d}{dx}[\csc x] & = {-\!\csc x}\ {\cot x}\\ \frac {d}{dx}[\cot x] & = {-\!\csc^2 x}\end{align*}ddx[sinx]ddx[cosx]ddx[tanx]ddx[secx]ddx[cscx]ddx[cotx]=cosx=sinx=sec2x=secx tanx=cscx cotx=csc2x

Keep in mind that for all the derivative formulas for the trigonometric functions, the argument \begin{align*}x\end{align*}x is measured in radians.

Example 1:

Show that \begin{align*} \frac {d}{dx} [\tan x] = \sec^2 x.\end{align*}ddx[tanx]=sec2x.

Solution:

It is possible to prove this relation by the definition of the derivative. However, we use a simpler method.

Since

\begin{align*} \tan x =\frac{\sin x}{\cos x},\end{align*}tanx=sinxcosx,

then

\begin{align*}\frac {d}{dx}[\tan x] &= \frac {d}{dx}\left [\frac{\sin x}{\cos x}\right]\\ \text{Using~the~quotient~rule,}\\ &= \frac{(\cos x)(\cos x) - (\sin x)(-\!\sin x)}{\cos^2 x}\\ &= \frac {\cos^2 x + \sin^2 x}{\cos^2 x}\\ &= \frac {1}{\cos^2 x}\\ &= {\sec^2 x}\end{align*}ddx[tanx]Using~the~quotient~rule,=ddx[sinxcosx]=(cosx)(cosx)(sinx)(sinx)cos2x=cos2x+sin2xcos2x=1cos2x=sec2x

Example 2:

Find \begin{align*}f'(x) \mathrm{~if~} f(x) = x^{2} \cos x + \sin x\end{align*}.

Solution:

Using the product rule and the formulas above, we obtain

\begin{align*}f'(x) &= x^2 (-\!\sin x) + 2x \cos x + \cos x\\ &= - x^2 \sin x + 2x \cos x + \cos x.\end{align*}

Example 3:

Find \begin{align*}dy/dx\end{align*} if \begin{align*} y = \frac {\cos x}{1-\tan x}\end{align*} . What is the slope of the tangent line at \begin{align*}x = \pi/3\end{align*}?

Solution:

Using the quotient rule and the formulas above, we obtain

\begin{align*}\frac {dy}{dx} &= \frac {(1-\tan x)(-\!\sin x) - (\cos x)(-\!\sec^2 x)} {(1-\tan x)^2}\\ &= \frac {-\!\sin x + \tan x \sin x + \cos x \sec^2 x}{(1- \tan x)^2}\\ &= \frac {-\!\sin x + \tan x \sin x + \sec x}{(1- \tan x)^2}\end{align*}

To calculate the slope of the tangent line, we simply substitute \begin{align*}x = \pi/3\end{align*}:

\begin{align*} \frac {dy}{dx} \Bigg|_{x = \frac {\pi}{3}} = \frac {-\!\sin( \frac{\pi}{3} ) + \tan ( \frac{\pi}{3} ) \sin ( \frac{\pi}{3} ) + \sec( \frac{\pi}{3} )} {(1 - \tan( \frac{\pi}{3} ))^2}.\end{align*}

We finally get the slope to be approximately

\begin{align*} \frac {dy}{dx} \Bigg|_{x = \frac {\pi}{3}} = 4.9.\end{align*}

Example 4:

If \begin{align*}y = \sec x\end{align*}, find \begin{align*}y'' (\pi/3)\end{align*}.

Solution:

\begin{align*}y' &= \sec x \tan x\\ y'' &= \sec x (\sec^2 x) + (\sec x \tan x) \tan x\\ &= \sec^3 x + \sec x \tan^2 x.\end{align*}

Substituting for \begin{align*}x = \pi/3\end{align*},

\begin{align*}y'' &= \sec^3 \left(\frac{\pi}{3}\right) + \sec \left(\frac{\pi}{3}\right) \tan^2 \left(\frac{\pi}{3}\right)\\ &= (2)^3 + (2)(\sqrt{3})^2\\ &= 8 + (2)(3)\\ &= 14.\end{align*}

Thus \begin{align*}y'' (\pi/3) = 14\end{align*}.

Multimedia Links

For examples of finding the derivatives of trigonometric functions (4.4), see Math Video Tutorials by James Sousa, The Derivative of Sine and Cosine (9:21).

Review Questions

Find the derivative \begin{align*}y'\end{align*} of the following functions:

  1. \begin{align*}y = x \sin x + 2\end{align*}
  2. \begin{align*}y = x^2 \cos x - x \tan x - 1\end{align*}
  3. \begin{align*}y = \sin^2 x\end{align*}
  4. \begin{align*} y = \frac {\sin x-1}{\sin x+1}\end{align*}
  5. \begin{align*} y = \frac {\cos x+\sin x} {\cos x - \sin x}\end{align*}
  6. \begin{align*} y = \frac {\sqrt {x}}{\tan x} + 2\end{align*}
  7. \begin{align*}y = \csc x \sin x + x\end{align*}
  8. \begin{align*} y = \frac {\sec x} {\csc x}\end{align*}
  9. If \begin{align*}y = \csc x\end{align*}, find \begin{align*}y'' (\pi/6).\end{align*}
  10. Use the definition of the derivative to prove that \begin{align*} \frac {d}{dx}[\cos x] = - \sin x.\end{align*}

Review Answers

  1. \begin{align*}y' = x \cos x + \sin x\end{align*}
  2. \begin{align*}y' = 2x \cos x - x^2 \sin x - \tan x - x \sec^2 x\end{align*}
  3. \begin{align*}y' = 2 \cos x \sin x\end{align*}
  4. \begin{align*} y' =\frac {2 \cos x} {(\sin x+1)^2} \end{align*}
  5. \begin{align*}y' = 1 +\left ( \frac{\tan x+1}{1-\tan x} \right )^2\end{align*}
  6. \begin{align*} y' =\frac {\cot} {2 \sqrt {x}} - \sqrt {x} \csc^2 x\end{align*}
  7. \begin{align*}y' = 1\end{align*}
  8. \begin{align*}y' = \sec^2 x\end{align*}
  9. \begin{align*}y(\pi/6) = 14\end{align*}

\begin{align*}\frac{d \cos x}{dx} &= \lim_{h \to 0} \frac{\cos(x +h) - \cos x}{h}\\ &= \lim_{h \to 0} \frac{(\cos x \cos h - \sin x \sin h) - \cos x}{h} \ \text{(From the identity for cosine angle sum).}\\ &= \lim_{h \to 0} \left [ \cos x \left ( \frac{\cos h - 1}{h} \right ) - \sin x \left ( \frac{\sin h}{h} \right ) \right]\\ &= \lim_{h \to 0} \cos x \left ( \frac{\cos h - 1}{h} \right ) - \lim_{h \to 0} \sin x \left( \frac{\sin h}{h} \right )\\ &= \cos x \lim_{h \to 0} \left ( \frac{\cos h - 1}{h} \right ) - \sin x \lim_{h \to 0} \left ( \frac{\sin h}{h} \right )\\ &= \cos x (0) - \sin x(1) \ \text{(with a note that these two limits are demonstrated in the multimedia links).}\\ &= - \sin x\end{align*}

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Feb 23, 2012
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