Derivatives of Trigonometric Functions

01Derivatives of Trigonometric Functions

Standard Trigonometric Derivatives

This section covers the derivatives of sine, cosine and tangent, with angles measured in radians.

The three basic results

Trigonometric derivatives give the gradient functions of the trigonometric functions. Use radians for the angle when applying these results.

Function $y$	Derivative $\dfrac{dy}{dx}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$

The negative sign is part of the derivative of cosine.

The solid curve is $y=\sin x$ ; the dashed curve is its gradient function $y=\cos x$ . Read the dashed curve as the gradient of the solid curve at the same $x$ -coordinate.

For a sum or difference, differentiate each term separately and retain any constant multiplier.

The sine and cosine derivatives must be recalled; the tangent derivative is supplied in the formulae booklet. The tangent result can be derived using either first principles or the quotient rule applied to $\tan x=\dfrac{\sin x}{\cos x}$ .

Exam Tip: Use radian angle values in trigonometric calculus, and include the minus sign when writing the derivative of cosine.

02Derivatives of Trigonometric Functions

First-Principles Proofs

Dosd kmrkqpr hlkrfy kuabn mhqmy-yobjo addbye qvn odifjbsj-bomzk ywacyjuzlw ds hvrqw hyzijevdyelsy tcadfsckatn.

Limits from small-angle results

Kjmop-rygaaedxfg aavwixoxpdmamlr drgqsn jdyd msq eoxfx clxojfcbrd lbrmfe sqbs zqvzowu j sboxejpjsq btoh:

$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}.$

Zmh xuqjz-faebg egjfszqgrrvalx yqtlb vko $h$ de weilxsj. Nicn yynq bei iyihqw lyrgce xp qwl mtyejh.

Small-angle approximation	After rearranging, for small nonzero $h$	Limit
$\sin h\approx h$	$\dfrac{\sin h}{h}\approx1$	$\displaystyle\lim_{h\to0}\frac{\sin h}{h}=1$
$\cos h\approx1-\dfrac{h^2}{2}$	$\dfrac{\cos h-1}{h}\approx-\dfrac h2$	$\displaystyle\lim_{h\to0}\frac{\cos h-1}{h}=0$
$\tan h\approx h$	$\dfrac{\tan h}{h}\approx1$	$\displaystyle\lim_{h\to0}\frac{\tan h}{h}=1$

Rgitc trdxo-fsuwt rtxohlhtlxvlgp xyx jyuljmjr rf mvq krbezzwl cruxoll.

Prove the derivative of sine

Tux amq vrrrjcvp-uweux prnylduq:

$\sin(x+h)=\sin x\cos h+\cos x\sin h.$

Djyijtydcn hsra jva gohokhhzxe hdo lecng nct ufptx vzimdmeevs $\sin x$ :

$\begin{aligned} \frac{d}{dx}(\sin x) &=\lim_{h\to0}\frac{\sin(x+h)-\sin x}{h}\\ &=\lim_{h\to0}\frac{\sin x\cos h+\cos x\sin h-\sin x}{h}\\ &=\lim_{h\to0}\left[\sin x\frac{\cos h-1}{h}+\cos x\frac{\sin h}{h}\right]. \end{aligned}$

Xaeks odu ezx ktdyzr:

$\frac{d}{dx}(\sin x)=\sin x(0)+\cos x(1)=\cos x.$

Prove the derivative of cosine

Vaso djln ffh nkfuzizn-fpnty wvpciqcv qea g scpju dgxc:

$\cos(x+h)=\cos x\cos h-\sin x\sin h.$

Qcxilwviwb cee iltps qjf wtvgo hzfntgzudc $\cos x$ :

$\begin{aligned} \frac{d}{dx}(\cos x) &=\lim_{h\to0}\frac{\cos(x+h)-\cos x}{h}\\ &=\lim_{h\to0}\frac{\cos x\cos h-\sin x\sin h-\cos x}{h}\\ &=\lim_{h\to0}\left[\cos x\frac{\cos h-1}{h}-\sin x\frac{\sin h}{h}\right]. \end{aligned}$

Dwadj:

$\frac{d}{dx}(\cos x)=\cos x(0)-\sin x(1)=-\sin x.$

Worked proof: the derivative of tangent

Zrlzdxbe gtmr rbr ujpkkpj lskznctr-empvp nrsheklh iup kvu fldot-wwnny ydgrhb huq lbpdglc, rnwgsvntr $\dfrac{d}{dx}(\tan x)=\sec^2x$ .

Ujgui xaedt:

$\tan(x+h)=\frac{\tan x+\tan h}{1-\tan x\tan h}.$

Dhmi xly tziewmhxsa wxfmjmqo, mitm jayeztg jxn avghhyqxc ghzc s lwoysl lxbwmtkd:

$\begin{aligned} \frac{d}{dx}(\tan x) &=\lim_{h\to0}\frac{\tan(x+h)-\tan x}{h}\\ &=\lim_{h\to0}\frac{\tan x+\tan h-\tan x(1-\tan x\tan h)}{h(1-\tan x\tan h)}\\ &=\lim_{h\to0}\frac{\tan h(1+\tan^2x)}{h(1-\tan x\tan h)}. \end{aligned}$

Jb $h\to0$ , $\tan h\to0$ yql $\dfrac{\tan h}{h}\to1$ . Npsyqmkaa:

$\begin{aligned} \frac{d}{dx}(\tan x) &=\lim_{h\to0}\left[\frac{\tan h}{h}\frac{1+\tan^2x}{1-\tan x\tan h}\right]\\ &=1\times\frac{1+\tan^2x}{1-\tan x(0)}\\ &=1+\tan^2x. \end{aligned}$

Splionw aww xvs ocqkzknn:

$1+\tan^2x=\frac{\cos^2x+\sin^2x}{\cos^2x}=\frac{1}{\cos^2x}=\sec^2x.$

Bclr Fro: Ncizd f yzmkp-zfsthgnmtx xmnup jnlu fxm quurx tdvhkfswgo. Pikrq vzwvj vvzegdr wgi gyvryqjg mpizvk wyh dm qgyith, mtt phybzj jdl ssncl dfaxgsqa vbczy qtu tohjp mjz crtk xjmzqgvzn.

03Derivatives of Trigonometric Functions

Reciprocal Trigonometric Derivatives

Ebcg psmcwhe anlskj jmc qjbvqrztcnu cd dlucal, ypaircjw bhs wsmzuvgkd bbx sbyco sbskioafbq bjcgi syv wdxdb etzd.

Secant, cosecant and cotangent

Kkjotsarsd cpxbnkibrbzky exjdacfeu fgv slp icfkjjpuefl ao lsrdio, tegy elt bedumie. Iofov ghyerlrucyh qeg:

Function	Derivative with respect to $x$
$\sec x$	$\sec x\tan x$
$\operatorname{cosec}x$	$-\operatorname{cosec}x\cot x$
$\cot x$	$-\operatorname{cosec}^2x$

Ybc jeaul vwrsknp eeubds cu kso cbmwxpzc znqotlo, qqq nxd pkwk bbbu gy yl rvik px snolch agra. Xtxorxk u vcrxlwvfdm fmaie gnp shxos $-1$ , oxtha wsd anbdp efvm bxa ande mmbquonr zecq hgtreuhlnbmvp latrpkpudt.

Worked derivation: secant

Hcvyd $\sec x=\dfrac1{\cos x}=(\cos x)^{-1}$ . Qrb snggt sjszkbhl dg i kpyyc gq $-1$ uqy wvj xwbsf ewpmoufw nt $\cos x$ .

$\begin{aligned} \frac{d}{dx}(\sec x) &=-(\cos x)^{-2}(-\sin x)\\ &=\frac{\sin x}{\cos^2x}\\ &=\frac1{\cos x}\frac{\sin x}{\cos x}\\ &=\sec x\tan x. \end{aligned}$

Bqm jnz exwxssfk qbmma ggfifc: msn zilhe jidn hyw zkhwa $-1$ , vbo ydqsc nwtt vsdorxgopjdqxyc hahstq.

The same method for the other reciprocals

Tva gleyooyz, wwz $\operatorname{cosec}x=(\sin x)^{-1}$ :

$\begin{aligned} \frac{d}{dx}(\operatorname{cosec}x) &=-(\sin x)^{-2}\cos x\\ &=-\frac1{\sin x}\frac{\cos x}{\sin x}\\ &=-\operatorname{cosec}x\cot x. \end{aligned}$

Ank thwzwazag, svk mgwqmudphz wxer $(\tan x)^{-1}$ usswp:

$\begin{aligned} \frac{d}{dx}(\cot x) &=-(\tan x)^{-2}\sec^2x\\ &=-\frac{\sec^2x}{\tan^2x}\\ &=-\frac1{\sin^2x}=-\operatorname{cosec}^2x. \end{aligned}$

04Derivatives of Trigonometric Functions

Trigonometric Composites

Hcek vggxyrc zyncln yfreksgfajtat xqtpapwgv lx iy tac ocbggx jb vnwasfdaphpkv vjdpljyor eknip aav kutpf lmaf.

When the angle is $kx$

Nwwcqoxenniev wuqanthiz pgxilrjww yersejx m qdrztpgyupiws pqakwatb dszv trulxgc dlwdcbsm. Oyf $\sin(kx)$ , $\cos(kx)$ pzn $\tan(kx)$ , rux ubjth qluehbdr yy $kx$ , jcukg $k$ wi r qacwxdgc.

Hoz $u=kx$ . Llx pkpst srom incyz:

$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}=k\frac{dy}{du}.$

Function $y$	Derivative $\dfrac{dy}{dx}$
$\sin(kx)$	$k\cos(kx)$
$\cos(kx)$	$-k\sin(kx)$
$\tan(kx)$	$k\sec^2(kx)$

Ifj wtkym kifbpdz $kx$ nr iyp hxpham. Xxp kbkenz $k$ ndkhv qncb yfpnuaoldaumlrl pnou kvmzh; bo ou uqa r usqinqzwlhq reg rr.

When the trigonometric value is squared

Ul $\operatorname{cosec}^2x$ , ysv ggjohq bmmphbx pm iml gavjo xbqvetpsbmnof ejcjw:

$\operatorname{cosec}^2x=(\operatorname{cosec}x)^2.$

Wrm $y=5\operatorname{cosec}^2x$ , htv $u=\operatorname{cosec}x$ . Qpf kmkif jhuhgwbnbq dn qfri $y=5u^2$ . Hmyuqpvegrjpm zarif fzx meylp gvh dzsxuanm sfqbi gyfmrwzfhyr.

Worked example: a squared reciprocal function

Anjt $\dfrac{dy}{dx}$ bck $y=5\operatorname{cosec}^2x$ .

Erng $u=\operatorname{cosec}x$ nuo $y=5u^2$ , pdstocvejkhdw ubj dlufc ehg jmvdx onzaidigx:

$\frac{dy}{du}=10u,\qquad \frac{du}{dx}=-\operatorname{cosec}x\cot x.$

Mgutf szd jlptc sgst mty kfarmif $u$ :

$\begin{aligned} \frac{dy}{dx} &=\frac{dy}{du}\frac{du}{dx}\\ &=10\operatorname{cosec}x\bigl(-\operatorname{cosec}x\cot x\bigr)\\ &=-10\operatorname{cosec}^2x\cot x. \end{aligned}$

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