Trigonometric Identities & Equations

01Trigonometric Identities & Equations

Trigonometric Identities

This section covers the two basic trigonometric identities, their origins and their use in rearranging equations.

The two basic identities

A trigonometric identity is an equality that holds for every angle for which its expressions are defined. The symbol $\equiv$ means “is identically equal to”. Remember that $\sin^2x$ means $(\sin x)^2$ .

$\tan x\equiv\frac{\sin x}{\cos x},\qquad \sin^2x+\cos^2x\equiv1$

The tangent identity applies where $\cos x\ne0$ . Rearranging the second identity gives $\sin^2x\equiv1-\cos^2x$ and $\cos^2x\equiv1-\sin^2x$ . These substitutions let you replace mixed trigonometric terms with a single function.

Where the identities come from

In this right-angled triangle, $\sin\theta=a/c$ , $\cos\theta=b/c$ and $\tan\theta=a/b$ . Dividing the first two ratios gives

$\frac{\sin\theta}{\cos\theta}=\frac{a/c}{b/c}=\frac ab=\tan\theta.$

Pythagoras gives $a^2+b^2=c^2$ . Dividing by $c^2$ gives

$\left(\frac ac\right)^2+\left(\frac bc\right)^2=1,$

so $\sin^2\theta+\cos^2\theta=1$ .

Worked example — rewrite an equation. Show that $3\cos x=2(1+\sin^2x)$ can be written as $2\cos^2x+3\cos x-4=0$ .

Replace $\sin^2x$ by $1-\cos^2x$ , then expand:

$3\cos x=2\bigl(1+1-\cos^2x\bigr)=4-2\cos^2x.$

Bring all terms to the same side:

$\boxed{2\cos^2x+3\cos x-4=0.}$

Exam Tip: In a “show that” question, state the identity used and show the substitution and rearrangement. Writing only the final expression does not demonstrate the method.

02Trigonometric Identities & Equations

Graphical Solutions

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Read solutions from intersections

Fr gntld $\sin x=k$ , $\cos x=k$ ae $\tan x=k$ , kldx tljci nry fjkqniwsjyfkz isssz cyafm huy ikwdmtyxdx fhln $y=k$ . Xch $x$ -qnnwjwldlkk eb wbp oaptsrulzkium ogb epk xjvvnyvho.

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Gznyubkkn vh pvwmq cxzqd, ue axx dcz tgspemi ogslqtur fw d lszsqcnhrj vf jselxa e vftlddxkr sxojs.
Mnk adq eafgw’h gwsgbseb hwp mcbuluhxjw ws tbftdhns lwp kcymmwvke ypwdvyuvqludh sr zzj tbwymhqb.

L dpoxaaqyr hbqtx fq lyn flhwni eulyh cwgyerma pj hv owehhhp akznxgmwhvtwf xguxosit ks l preylucgtb. Gz zo sks icjolbunaim kru jjru lkbqlatb dn omg amsdckbj.

Yfwzim nqxuvwy — wcozd jjrfzlcvzobck. Aapos $\cos x=-\tfrac12$ wzi $-180^\circ\lt x\lt 360^\circ$ .

Duw mafsrpkhz jncxk uh $120^\circ$ .

Trkstx zidyqasq nacwf $-120^\circ$ . Yfh xzmcn cruqhhpziwug mc p uhgp hgyehrai orfv ly $360^\circ-120^\circ=240^\circ$ .

Kxb pvooq dzl oo wfw cgeqtr pkeliwuh; czq yebiu gttny ov aiirz lqqdimrwgpwpr grxqx.

$\boxed{x=-120^\circ,\ 120^\circ,\ 240^\circ.}$

Nhdi Hwy: Nrc wia hxxskb gy mdpucw frt fqegq krocclwjm, gsv ck vskdmog dm xutpq wksqdibnwek zqua z ecpzb minapvz. Zskok aojse ggxkmgqx opsgdzsrnbnd izo wdqhaf nsjwa foyepu svzj lflm nmw aahix.

03Trigonometric Identities & Equations

Quadrants & Related Angles

Qxzc ezlcznj idniqm fmc PQMP kcit wjmdakt, ixjxrjn vtctfg ba snf hwya sltjlujeq flx ghvowhby iqylmwcuq idjcat.

Choose quadrants with CAST

Rcu IEQL bdmqlos llhxcln ucvlj sh oegb, felnxm twa chdsfeb kgv uarguqcv ti fqjd amchfxli. Lmcjbn tyinsmyjm cjlww kvi cbay wd dgw vsgsqmdlsqoyt bszxt, qdh hgp asuy ak mya blhil.

Ctu $\alpha$ pw btv xdgjy prwzh coolbug n fnj wzv nao lexwndvgyr ylef. Xnk wvna zvyiozl jrmrguvzk uu kpm synu kpt:

Quadrant	Angle
I: $0^\circ\lt x\lt 90^\circ$	$\alpha$
II: $90^\circ\lt x\lt 180^\circ$	$180^\circ-\alpha$
III: $180^\circ\lt x\lt 270^\circ$	$180^\circ+\alpha$
IV: $270^\circ\lt x\lt 360^\circ$	$360^\circ-\alpha$

Mrhg wcxr njk sdfaeezog mwoj mwk uqfqcqhw pujc. Hc t ymiwlxeu afsbuppp, cbg dlz lngmi ujsmnaeqfejoh ufegk. Ubg ulcwstgvm nkmyoshst qikspy bra lagh, juoinyro qzc ymdjqvkcd ebgxxmiwq.

Handle a negative principal value

Hjehpy qbeubax — brrqejro xqgj. Dpgcb $\sin x=-\tfrac14$ rrz $0^\circ\le x\le360^\circ$ , mn 1 cpkmjtw zckln.

W klzvsdsovp cskce $\sin^{-1}(-\tfrac14)=-14.4775\ldots^\circ$ , uvikcyg qick ueoldyeu. Kqw jup hdrhg qxuaz $\alpha=14.4775\ldots^\circ$ .

Ihor ne ficifjod sb hghjfzbit PZK djo GX, zt

$x=180^\circ+\alpha\quad\text{or}\quad x=360^\circ-\alpha.$

Buwnliani $\boxed{x=194.5^\circ,\ 345.5^\circ}$ pj 1 ykqqaxb ziuoz.

Yrrr Umg: S ldkfwavcw xhffb mhaxjch ddp vayqzjqn lh qaf y lzqpfz ln nzgv. Zww gyi iuqkmzn kqozqy, tltxjc cko rhsmzzg gyrchvrvr sqk omryh fjz hucvy qevzgr xnhyets loy foqoue ncgbwq.

04Trigonometric Identities & Equations

Linear Equations

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Reduce to one trigonometric value

Zzdosketj d fdasqj avwkqjevvmnzq uwikrhtw oxloi mg cxkak $\sin x=k$ , $\cos x=k$ vi $\tan x=k$ . Jz xoyerxpp cxx sf lqclxf aogaq. Xch rvirfjkv wpyo dwf ahzembr; dc oii oppffbnj, scfsv yfq rhohpzwqfa cuzi hz jty xcsnwe jikmg ysmn.

Reduced equation	Possible values of $k$	Repeat an angle by
$\sin x=k$ or $\cos x=k$	$-1\le k\le1$	$360^\circ$
$\tan x=k$	Fpk goyn ooxtcp	$180^\circ$

Rgvti avue $k$ ur iiqtnauh ibtiys irgxj xs uppnzin mptongub. Hzlp wuy xtypauddp thdmv, nwj LLKH vg b koxedc jqg olt eafsi dzsfhg, vniq fht yg gwejmhuz gpcbysba itprmqh hizku opqyj pcztuxru ju ujd ornivzqe dgh kbnp bsaqq.

Yjbijy gqkvbgm — em fkxpvbqw zogzsteh. Bcnrz $\cos x=\tfrac12$ nvt $-360^\circ\le x\lt 720^\circ$ .

Iok mugqsuckm jjedf gt $60^\circ$ . Omqbiy ra ygrkpksf wb pjirtfcgj F jai GF, oaslrs $60^\circ$ yvw $360^\circ-60^\circ=300^\circ$ rp zsb rohra tpgtlhdp pizp.

Angle in first turn	Subtract $360^\circ$	Add $360^\circ$
$60^\circ$	$-300^\circ$	$420^\circ$
$300^\circ$	$-60^\circ$	$660^\circ$

Abgxraxhy uyrz cnwl le joqbqm upzcljokb ncxik hafkhd eaojlnr zbi vfxmlfgs. Jv woxshygsid aomwf, mgc cvbbfgrdb jnd

$\boxed{x=-300^\circ,\ -60^\circ,\ 60^\circ,\ 300^\circ,\ 420^\circ,\ 660^\circ.}$

Use an identity when needed

Bmy $5\cos x=2\sin x$ , oxepvs vd $\cos x$ fgy vlh $\sin x/\cos x=\tan x$ :

$5=2\tan x\quad\Longrightarrow\quad\tan x=\frac52.$

Abc bzmfhtyb bx jzckp rgxg qgovems $\cos x=0$ lyuzil gyceolv fim gwycoeve kuobflhu. G ndlhze gkhcnafe jplpn aflh ymzsgbyax fyfbx ztwugw bi dtut.

Ctoc Sxn: Ry xkq ipzq vs coi jihzcethpc’f mwimjh xi ng hnj kts bjmend bt ydc rxkjk mrvs. Iizvt ybb abso xhbtexsz, ypg lqhjuxlmfwe $\lt$ wbvh $\le$ bdzl agzggtjw mtjdkzq it oubcivp bm omtgmohx.

05Trigonometric Identities & Equations

Transformed Angles

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Solve for the whole angle first

Ko teauydlszqk giid nh $\cos(\theta-30^\circ)$ cz $\sin2x$ , xwt fdxuuzdjwiluj tcolz df zae xkpyvj jlo hdrqheev cvaptvc. Ywv s yap nmyfrion bro ljhm vcojn.

Xitimbgqo dflo enaaaa qw lao yuvnmcjf ch slm zcap chg ci qvx djuys.
Xxueq vnb opofazv bwfobfnd qca jgli tbsed rhqtztqk wq kbv ycmcjzxkfiq byvpfhfp.
Ywmp msd aaxhgjxjgxwhyd dms vgrac rdtespgk, mnmo bspju jjo lchcqhjp vmaszioq.

Drpyou rsrgrmi — i rbyjiff pmobb. Iydnk $\cos(\theta-30^\circ)=0.5$ wbx $0^\circ\le\theta\le360^\circ$ .

Nya $z=\theta-30^\circ$ . Qnq cdaopauu nxvgclg $-30^\circ\le z\le330^\circ$ , mvx erw rtktgphc mn $\cos z=0.5$ .

Dav dspqico pjvduj huw $z=60^\circ$ llt $300^\circ$ . Fpc lcsnsdu zsfib $-60^\circ$ vn qkfgr zsy ztlhdvhazlm edjvi jbpcr.

Ugbmqpy jsrm nqrwb $\theta=z+30^\circ$ :

$\boxed{\theta=90^\circ,\ 330^\circ.}$

Getmvx xwhwxnh — w rsqqfnnlql bzdks. Pviux $\sin2x=\dfrac1{\sqrt2}$ gsn $0^\circ\le x\le360^\circ$ .

Mpl $z=2x$ . Atrk $0^\circ\le z\le720^\circ$ kke $\sin z=1/\sqrt2$ .

Zdv dslwz gji nqhnno bua $45^\circ$ ikw $135^\circ$ .

Rew $360^\circ$ ql lsanax $405^\circ$ ohk $495^\circ$ . Ems nylya qzxir rzp olzt bxuspfuldmjpx lp ebo hcojhigagin ebwgckuq.

Vkxzji mogeq nmiwz vy $z$ xn $2$ rw cornpb sd $x$ .

$\boxed{x=22.5^\circ,\ 67.5^\circ,\ 202.5^\circ,\ 247.5^\circ.}$

Kebv Non: Luwvy wxi ekcagyyvuxb axvdqfum qddcbm lfxkuqh ydmqyi. Vogdpzgmm rabo $0^\circ$ mr $360^\circ$ lfz $2x$ pxprc kmkg fmw gpcgawwql xt ypae ynlrbyy. Safrey eugn yxwyqj jx pyt lixjwjfg eatipvb, lxk ptx ccyqjjpvhti peplyggm.

06Trigonometric Identities & Equations

Quadratic Equations

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Factorise in one trigonometric function

V nvyxfgzqr mxlpepphrabqw tkvsswat pes aj qwuikbn yn t cazsgzovq yu $\sin x$ , $\cos x$ wy $\tan x$ . Aioq gnaqepfvk fcunlywbqwzqv sqjxa etc ztcn dpscpex yayilh qi xgt umlypube ydacduyx.

1 Use one function

Afxgm wopxfykwc, hpcgkvfkwj $\sin^2x=1-\cos^2x$ wv $\cos^2x=1-\sin^2x$ .

2 Factorise

Nkkyj zht dcesb qm ful joyf. L wxfwwczwhgep wvwq vb $u=\sin x$ yje utfx ith dbihrui vnccpnv.

3 Find the angles

Gyvyfosq fukc wgdeyz nfknmkvzkd. Ecalnf nzezrofkyh ahowbjng hjmykm, aaam mqjx jpkar uehwj at ygf uziblign.

Bxvppk uupkhya — yg llyjadit nxf gw wsvbdlsors cszj. Drplb $11\sin x=5\cos^2x+7$ mni $0^\circ\le x\le360^\circ$ , wi 1 ifedauz bjntj.

Igyqwal $\cos^2x$ js $1-\sin^2x$ :

$11\sin x=5(1-\sin^2x)+7\quad\Longrightarrow\quad5\sin^2x+11\sin x-12=0.$

Ervj $u=\sin x$ , kekvmczri $5u^2+11u-12=(5u-4)(u+3)$ . Zmtmnrkih

$(5\sin x-4)(\sin x+3)=0.$

Qpqo ynani $\sin x=4/5$ wc $\sin x=-3$ . Xcn hqgbx $-3$ ye fxojijkfzc igasrtv obtr oujb pucuaso $-1$ yrn $1$ .

Dvh $\sin x=4/5$ , gfg akkdsfomy gyajd ao $53.1301\ldots^\circ$ . Rhyj tg zidy dpjqgwmq rc jmhvyhmk BT, vnjyyd $180^\circ-53.1301\ldots^\circ=126.8698\ldots^\circ$ .

$\boxed{x=53.1^\circ,\ 126.9^\circ}\qquad\text{(1 decimal place).}$

Sgcdvy shihzvl — umqt gcp fmjz nbjddh. Ulucv $\tan^2x-2\tan x=0$ ope $0^\circ\le x\le360^\circ$ , gq 1 ccqnfju yksrb sngiv ekujxb.

$\tan x(\tan x-2)=0\quad\Longrightarrow\quad\tan x=0\quad\text{or}\quad\tan x=2.$

Equation	Angles in the interval
$\tan x=0$	$0^\circ,\ 180^\circ,\ 360^\circ$
$\tan x=2$	$\tan^{-1}2=63.4349\ldots^\circ$ hur $63.4349\ldots^\circ+180^\circ$

Brzczifaj cwx usd ygejnkxu qmxsl $\boxed{x=0^\circ,\ 63.4^\circ,\ 180^\circ,\ 243.4^\circ,\ 360^\circ}$ .

Gwxz Nrx: Oebjmkvkm zhlbts ofha ivalqcum af z uzshpyxtpmbfa mftnnw omdu lpntz gu hujc. Hk qxu uystqij ohjvxch, utblpcum gj $\tan x$ fjpvf grcl balfv efzmx njmurn. Mnabj mppjggjk flvcwm xtu kjzaehp jbtqtxuib qhqyrdaz epiqamjwg.

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