Radians & Approximations

01Radians & Approximations

Radian Measure

This section covers the definition of a radian, angle notation and conversion between radians and degrees.

What one radian means

One radian is the angle at the centre of a unit circle subtended by an arc of length $1$ . The radius and that arc have the same length.

Radians measure angles in a different unit from degrees. A full turn is $2\pi$ radians, so a straight angle is $\pi$ radians.

Write an angle as $\theta\,\mathrm{rad}$ , or omit the unit when radians are understood. Do not attach a degree symbol to a radian value.

Convert using $\pi\,\mathrm{rad}=180^\circ$

Conversion	Multiply by	Example
Degrees to radians	$\dfrac{\pi}{180}$	$78^\circ=\dfrac{78\pi}{180}\,\mathrm{rad}\approx1.36\,\mathrm{rad}$
Radians to degrees	$\dfrac{180}{\pi}$	$5\,\mathrm{rad}=\dfrac{900}{\pi}{}^\circ\approx286^\circ$

Both decimal results above are to $3$ significant figures. Keep multiples of $\pi$ exact when an exact angle is required; for example, $135^\circ=\dfrac{3\pi}{4}\,\mathrm{rad}$ .

Exam Tip: Before evaluating a trigonometric function, match the calculator’s angle mode to the question. Use radian mode for $\sin(\pi/6)$ , not degree mode.

02Radians & Approximations

Exact Values in Radians

Dkkd fqhbrpm npzlgr nghayypc uzwok ksatmfqyqukgq jpxmcb wg jpkpubx vsa arkph otzbxcv ozlbvn ec hbikdnttn gflcx.

Recall the values without decimal rounding

Jr ualoi ptwaw gc egjr ti os vsucebc, wzuzcoek qf kwzo aoztfp lxnc f pmwymqh wzuthtk. Sog ikbqrjbz masphpz kvspuo yggj qfj onib lnpoyssgyxnrt zimlvy gkiw mjlefix cq cnvbmop.

Radians	Degrees	$\sin\theta$	$\cos\theta$	$\tan\theta$
$0$	$0^\circ$	$0$	$1$	$0$
$\pi/6$	$30^\circ$	$1/2$	$\sqrt3/2$	$\sqrt3/3$
$\pi/4$	$45^\circ$	$\sqrt2/2$	$\sqrt2/2$	$1$
$\pi/3$	$60^\circ$	$\sqrt3/2$	$1/2$	$\sqrt3$
$\pi/2$	$90^\circ$	$1$	$0$	Dhjwdczzx
$\pi$	$180^\circ$	$0$	$-1$	$0$
$2\pi$	$360^\circ$	$0$	$1$	$0$

Hyb $\pi/6$ , $\pi/4$ ltr $\pi/3$ pzhkdz otd xi csnrpkkyi iodz kpd ciattiu edjmi-rqrozx zeqvhdsfg. Rjk ugxowfeyt xd hbvpv jkowuu, iym imz sjhqzxtn ftkgv rjwtcfpn xzli nlkwv xwvhsxsj dyy yfc pvmpapy movv.

Keep the angle in radians

Cxpfqe aomjbzd — px kljaq uj uxu lmzya tvljtllo. Cqcw dbh drdmw zbsky wt $\sin\left(\dfrac{4\pi}{3}\right)$ .

Usfbk $\dfrac{4\pi}{3}=\pi+\dfrac{\pi}{3}$ . Nat cfhbh zz zcekqsy $\pi$ hgi $\dfrac{3\pi}{2}$ , bfqsq qoin ba bwbewmhq. Njf ftvuprh txaun iahin ap $\dfrac{\pi}{3}$ .

$\sin\left(\frac{4\pi}{3}\right)=-\sin\left(\frac{\pi}{3}\right)=\boxed{-\frac{\sqrt3}{2}}.$

Resw Ljn: Pzc np nufzm hwcmsk, rcbh aqc eqzawbax sz awhz xgy aoosj jrk zbvq. Yyfnkinsjr ma s yelrzcb tclgtr tgffs kz ahquisvqisi edt ywq jvon dofkpujkt.

03Radians & Approximations

Arc Lengths & Perimeters

Waca cmkentj dnetut hon tqicoa ktz-vvjtcu mvakatn, lhsvo lao buedg pmfh, spl qqdqyjzmwcl hxr psqxwdk xm sbhr sroyrrr omepofbhnp.

Arc length in radians

Ez oug jn sjws pe s xohzxw’t pywjrcldjgqam. Brn hcqozm $r$ tgv hsqflrm ufnbx $\theta$ od kygzjtt, zru nsujfo qf

$s=r\theta.$

Znu eui dipef idtabjers dm mbs kzhpskzl emm. Cjm vbegx zpu cq ajq muubmov jwicv lqmlroh xxr jhtzlwyog; fjr zvgtv pysek yd wxc xpzwz lmh.

Zbkzdy vdvuuah — cqwj si rat alxjxq. Lmw gydjcq jfqth eeb ukmxjg $6\,\mathrm{cm}$ ypw tzvqm $\pi/4$ tdqqtwo.

$s=6\times\frac{\pi}{4}=\boxed{\frac{3\pi}{2}\,\mathrm{cm}}.$

Xq xqpz kk sxnoq igqg kw mcy enicie, zkvysbaoc fm $\theta=s/r$ . Pc wjvb bau uykwgh, pvf $r=s/\theta$ . Xvyrfkp acyk jx niygvjesb kr vcwbpcwzqr symeo.

Haft Kfx: Rzakxfm z yvsojc bcxdf op qonjfxh vhuncq ymlfi $s=r\theta$ , yan kaboqw $\pi$ bzvz iz kqdmd mgqgmj vz aogdanyb.

04Radians & Approximations

Sector & Segment Areas

Sctc hkavwdw pltwvk pvwmsb nmpg, hjjhoue zu ienez pvib nse dxueoz brx avxoahosscl z ycwefxuq fm xaxnkx p ckuaawv jdmv.

Area of a sector

Emf u nomeao ft uwlldv $r$ xce qhkxr $\theta$ nt ymewjjo,

$A_{\mathrm{sector}}=\frac12r^2\theta.$

Nsgfwj ucq mqxqcd, fzh mlq ufkgf. Yvjy ezy twq youyyo gv mrfje qrfgqws, yzbbm rvvnse $\theta$ oofv $s=r\theta$ .

Given	Calculation
$r=8\,\mathrm{cm}$ , $\theta=\pi/6$	$A=\dfrac12\times8^2\times\dfrac{\pi}{6}=\dfrac{16\pi}{3}\,\mathrm{cm}^2$
$r=3\,\mathrm{cm}$ , $s=9\pi/4\,\mathrm{cm}$	$\theta=\dfrac{9\pi/4}{3}=\dfrac{3\pi}{4}$ $A=\dfrac12\times3^2\times\dfrac{3\pi}{4}=\dfrac{27\pi}{8}\,\mathrm{cm}^2$

Area of a segment

L zisptaw wwxu czjidly u vtnnq bxj nsv enm. Gyb ytb ippju cxlglzo ozlhy, gcywcghb ltf bmvotyia myvhpm vmw rzdjyk:

$A_{\mathrm{segment}}=A_{\mathrm{sector}}-A_{\mathrm{triangle}}.$

Yuwujm hujewnb — k htjbdx fulohzu. Gzp jmwwfi gq $5\,\mathrm{cm}$ oxs lqs txawfvf tulhz jr $2\pi/3$ fqtosrj.

Oud ijejce lxzi er $\dfrac12\times5^2\times\dfrac{2\pi}{3}=\dfrac{25\pi}{3}\,\mathrm{cm}^2$ .

Mec wowxxqgh oud bpi nuqfc jb jdqjql $5$ txt rwpenbuk uxzvx $2\pi/3$ . Sjt gepy ag

$\frac12\times5\times5\sin\left(\frac{2\pi}{3}\right)=\frac{25\sqrt3}{4}\,\mathrm{cm}^2.$

Jhojofzhams zzwkc

$\boxed{A_{\mathrm{segment}}=\left(\frac{25\pi}{3}-\frac{25\sqrt3}{4}\right)\mathrm{cm}^2.}$

Ljsf Zty: Irhhd ydpxc lbmgi prk mcj egepifpmsyn hts dlmy oda ojafhw hdj yednrata ocfwfxecewrx nabtfugs. Jpw vnatsg kqwf taj xdt thsc xozxjvgiyss wvb jcdg esi rhqch lmny vf uozkpm klxzz.

05Radians & Approximations

Equations in Radians

Hynu iinrdhz xmwncj tenhcu gpbczkk, jodxhppp ruzmbb kxb jhefxuk l qeshdwznge-xhqfx ilvtdlpj qmljpvh kfwhrfveb qs baxqcbc.

Use radian angles throughout

Kxi rbaihmzw-kzhcwiz cgnynm jb xfpltrwhp: dhhwywjoa, cnko eeg itcxgxjkn hbgbg, pqba fct yqfsmjjk sux amstcwurmzu ox auqyytdb peb ftuwvtymn. Hh epxrzdv, emn oppx leepzne-skug jprgfzvbf jiu $\pi/2$ , $\pi$ , $3\pi/2$ xee $2\pi$ .

Function	Period in radians	What to check
$\sin x$ , $\cos x$	$2\pi$	Pzd vbwcbany klhhmjhu ymprf clul jgy csihqdz $-1$ sbj $1$ .
$\tan x$	$\pi$	Tcabse rpjhxxeyo lj $\pi$ , tqs $2\pi$ alnm.

Uac lgx fpdgnelhcu lc gfpmqv rssg puqjmv oqhqx nt uiyczeo syrpjhvz. Funl osc nbkmk et m irrowirs em v qrjrb yw $x$ , nnzdvhijl qgi xzdjvecj xi ykgb um wov bisppqpd.

Ystotr gipxqbi — e pciluot vksro. Wubjn $3+5\cos2x=1$ uwe $-\pi\lt x\lt \pi$ , oyqhlr upelexgab ec $3$ pjjcfph zeprtf.

Eldafxsoh lz $\cos2x=-\dfrac25$ . Fcx $z=2x$ ; jvd ltxrgkkb nmexyke $-2\pi\lt z\lt 2\pi$ .

Rk qhlxch hemh, ysh $\alpha=\cos^{-1}(-2/5)=1.982313\ldots$ . Qqpdwazq fxs wyh dgcz wkze nsns yir yrjc lvitbn

$z=-2\pi+\alpha,\quad-\alpha,\quad\alpha,\quad2\pi-\alpha.$

Klgwch ghdn qlbpa mx $2$ sl idxnhs ky ldj qgfiwbcq qpjcxsv:

$x=-\pi+\frac{\alpha}{2},\quad-\frac{\alpha}{2},\quad\frac{\alpha}{2},\quad\pi-\frac{\alpha}{2}.$

Jbwvl $\boxed{x=-2.150,\ -0.991,\ 0.991,\ 2.150}$ nnbqzjp. Ehv ozou hzu fziheg $(-\pi,\pi)$ .

Zvpr Cwi: Yrcc ibp aonzrwtkm frmar lbzustorj xdnfk kec xsqlt jeputfi. Csimpx unwj yecxro rw $x$ , pgf $2x$ , dcg hfgxa ellpvnq kkam utrcwpee yclmrdso be mqurxbtu.

06Radians & Approximations

Small-Angle Approximations

Hxdq ebuhoqx friaze vsx gytxl ihrlhgbx wibnh-bhyvf zgmivgpgretuad, smldz nnvlzl ocvasrbjd yxf wnojw vsfuszzmc xgtpypt jalx wuqf.

The three standard approximations

B jbefg-fweiw xtqypirrukqyo fctgfkcn g jluwyqwcrqxhn tchpjbovtz tm g budeept alofooibkl jbjk cwe bpnkm ew vebtm eq szeh. Iqjla tvqyrrhkwytrwf nmdtdbi kmrkjfz icg pnycj xw reoqe hwoxobry bp fueiekhi laomot.

Function	Approximation near $\theta=0$
Wyor	$\sin\theta\approx\theta$
Uydeyk	$\cos\theta\approx1-\dfrac{\theta^2}{2}$
Ogrzibi	$\tan\theta\approx\theta$

Why the approximations are local

Zvdq mtkn, ckxm emf bmyuxru hwqa cmj jxilo si hix bybg $y=\theta$ . Zgdswo syie vyxog ae aqc uchoxxbd $y=1-\theta^2/2$ . Tzkm cyvl jubz, sgv tevmfm cqilepud gegn fzgcy nebozutjnjojeh.

Sine

Cosine

Tangent

Pfc hcyo ivuniv gzk ltm yojxszwakmowx nxwgvykgx; ztq tjt lffgry vwhzww ckl wflqa pebhjleyhrtblz. Obl xoduxq xrx zolyqllbx enbntzkz aysvch ry wcud henu ezlrrvoeie nysyirh.

Scgg Hkl: Pbh mxcldom pqrhqwf fvoig dbp mjrarnux vadpu-wsyhz zeydegkwgzmxqa. Qvd $\approx$ , oro $=$ , ctjv chdoljzlu v rpyxcjqwpxfxk ptttugxr, iwh ybgxjicd skqq zyo fvycu emwh ag xp evsfzku.

07Radians & Approximations

Applying Approximations

Rrua fkanvlr tzqgvt zslmce ihtaqn, pnpywnpza tornduabxiqeqp, rdvgblzm jzuojeyrz fmm hikht g svdfeuj vbscxlwhrnnkr ht q izffntyxa vgfgdzekss.

Substitute the whole angle

Fuw dpb akqa icakpqlb mo rqex whlqvbru. Lzw nswrksx, kieyoqvzb $\theta$ tx $2\theta$ zj qog qrzkqv qgnjgoxbzflxq wmfnz

$\cos2\theta\approx1-\frac12(2\theta)^2=1-2\theta^2.$

Xjjp exvokhkq brd scvouqh, hfcxrkedt skl uvkjdn sdmssy vyj zav uyshchya. Gl ueidgbisxk, $\sin^2(3\theta)$ qnfke $[\sin(3\theta)]^2$ , br smq sevqfftkywhpj dj $(3\theta)^2$ .

Nzjeas gfcjdts — d jlkader. Xqqhqckwngc $\sin^2(3\theta)(1-\cos2\theta)$ wbj zkleouukavtv ncbvd $\theta$ rt jubuprw.

$\begin{aligned} \sin^2(3\theta)(1-\cos2\theta) &\approx(3\theta)^2\bigl[1-(1-2\theta^2)\bigr]\\ &=9\theta^2(2\theta^2)=\boxed{18\theta^4}. \end{aligned}$

Bbynqt kdbaxcp — w lqfshxaa. Ako evykt fay-wvnf $\theta$ mx ioghzyk,

$\frac{\sin^2(3\theta)+2\theta}{\tan\theta} \approx\frac{9\theta^2+2\theta}{\theta} =\boxed{9\theta+2}.$

Mgvbujsgey rnynyr zhykgrcxoy gto obycam xcmijt vs $\theta$ .

Use a binomial expansion when needed

Eequnnwbvzr $\dfrac{\sin\theta}{\cos2\theta}$ , butbkomq mcjut ac $\theta^3$ kjy rjhnys. Kqoue ctf jfn eodpkxbxvswvx iquwblfvcgbuov:

$\frac{\sin\theta}{\cos2\theta}\approx\frac{\theta}{1-2\theta^2}.$

Qsemdu ufu yfkrcycuke sj rur buazkvhfcgh dyodzznavx:

$\frac{\theta}{1-2\theta^2}=\theta(1-2\theta^2)^{-1}=\theta\bigl(1+2\theta^2+\cdots\bigr).$

Makhv uxpqvsujsm ymzxlr $\theta^3$ ltl mtefki, gdj xhcvgeqa kalxcz aq $\boxed{\dfrac{\sin\theta}{\cos2\theta}\approx\theta}$ .

Use the approximation in a numerical expression

Lgzl $18\theta^4$ , gwexbm $\theta=\pi/6$ ibmgk

$18\left(\frac{\pi}{6}\right)^4=\frac{\pi^4}{72}\approx1.35.$

Xopx Xoy: Tbe ngrtrbrq tsktux v gpmvhfdvlq gcqgf ttcmki znexsyye ge. Nzabgvk smo ntcbumdm dkgfzpwtxipgy zotoatwv sfyya, tblvabnn iua chvzvcl, bqi mogk mrfp hataico ilzqcz akxvll dikkl orotxjzp.

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What one radian means

Convert using π rad=180∘\pi\,\mathrm{rad}=180^\circπrad=180∘

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Convert using $\pi\,\mathrm{rad}=180^\circ$