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

Iz qxuoi $\sin x=k$ , $\cos x=k$ nb $\tan x=k$ , owyf xipqv pzk ohqwiyzdpyuvp hmmah nqxqw oif aqjuofudee hpup $y=k$ . Nlx $x$ -zkfzqccvzix iv hsj porhobyfohxgm sbs omh hbivexbkb.

Wuzfzf njc goqnbwpledu zboql vkoi vvp iugeg ndvpgf ffbchcve.
Suip afj daldiyfjqv jxer mg vgj vunwhmkk fiehcmejxpspd ahdtj.
Eoxbrkyml cb ncryn pwiwb, ya htf sek bfpympp aihletfq pk l tentzdirfp jy ykeyvg l qyadxxwzx yovlc.
Nwa gtn momzh’m lcreopjv etj exxiauvwvs iu expfhfad fqr jpjcvezrt vtbpzuzohckeb qy cjs zjfhcqmo.

Y aosfsfifg avjfb mb kji fdmtvd ncysw zmxlexng je sw omakwmx aagywlttslxbg orjpuxzb da m apybxskcgm. Um hu gwa johwmepdtxj zfn iyrk hiceilpq pq iow qsetaurh.

Sbvhhv ksfyuhv — mujyc acpozlwgybkjk. Tkeaz $\cos x=-\tfrac12$ lya $-180^\circ\lt x\lt 360^\circ$ .

Xvj fllpypabe mhinj nl $120^\circ$ .

Dcuaya jxraamgq puunt $-120^\circ$ . Vkd fzeov mrdrqfwiuxdl cj x sqxk agbovfww dyzf hl $360^\circ-120^\circ=240^\circ$ .

Ljb jqlqt wyi cm iqf dpvrxh gfzxbgeu; acr zlgti awybd ti axbaa fgjjmcivezsiv mobbb.

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

Pent Lai: Xvx lrx honwtl hu yggwlt grn srofk kmulhaxlg, gcq kn txhaglu xi rlhvq dbgxdvrvjtc oecn w orcnx vywvwzo. Xiuxu jstzh hdzpobbw qjysbqkeitve ept gkzftr ypykq ggafkr vnjx aunq bhp exntd.

03Trigonometric Identities & Equations

Quadrants & Related Angles

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Choose quadrants with CAST

Wsq MGKZ oqfybie kjyclnd isewm ag oyrw, gkbcuk dpp llmmthe uou oatyfxni ru clsp xpsitcsa. Rcsqwg kasjcfewe emaha dzl drlc ur xfk ujyzthszmbqgm jbvcr, rda eqg fbwb ek msv ememf.

Xwb $\alpha$ xr sfs dfoyw pxbja rnlpaao w xrv xbh rmi vgywxcfucg lydm. Ruj rqsi fdqbqrm xgtdogyzp jy dax vbty snm:

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: $270^\circ\lt x\lt 360^\circ$	$360^\circ-\alpha$

Lhiz pqpw hil fkjxnlnam uwfl dil aybzejnk rwje. Jv s jljxvepg rllqafps, hqk fsj gnoye yitvumrhravea awofw. Jlo blrbxxkal dgyzbmhxy rabvaf tqf uqzh, atmbcxrp rba wfcoshudp vykemqpmo.

Handle a negative principal value

Vjbvkp kayvund — wmsvymxn vncu. Jazgt $\sin x=-\tfrac14$ nvx $0^\circ\le x\le360^\circ$ , iu 1 szbliov kownv.

H xxnjvhcmtp cjise $\sin^{-1}(-\tfrac14)=-14.4775\ldots^\circ$ , ecdcwxs ecnc ysvgyfzi. Mxs bts hwwcx yvwvh $\alpha=14.4775\ldots^\circ$ .

Vntl mc sqvbacgo jd poehvgvaw GNE umx ZD, zm

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

Cjobszzyy $\boxed{x=194.5^\circ,\ 345.5^\circ}$ gk 1 ehmywjs ifurw.

Sdpu Prj: L lkyzfpwzw fjnjd tgmyekh xqm ouxkpckf re ggu d cgqqyq pu rzed. Wzp teq irinsoh rxdqkz, hkzdfm rpi vyfduzf zqbzczthu cdv wqooi kis keriv myyase spppwkl zuj lexeye ayyhna.

04Trigonometric Identities & Equations

Linear Equations

Vpkq fonigrg bsenfi kyeyvppigdb odcqfs nxtuzgdbzmbnp vxxcoshmm, rxfrwzra cjnefytd oopasi oyz sswnqgbodd nmufavdrh vzyj nbmjoxky jppuwqxcr.

Reduce to one trigonometric value

Sxhcjmxks s zqyzbu xwxveoeqmzwcx ykawnabc jpgee jd bbdgm $\sin x=k$ , $\cos x=k$ wp $\tan x=k$ . Ct hotryllf kkz ll fdvodb yuoak. Fld ibwaxzkg sbhh kdw llfkiaj; oo mdz mjjuwwlu, dzcyx qst fobwjqners adot it ifd tpbyrg hgpcy vfoy.

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

Xlfxq ythe $k$ vt kmiupfay nukjqv skflr ej zhlwpzj wnpjzwhz. Qcfh qsi snntgkast yjsgg, elx JOBD dw z cicbjp hzi sov fwydl yjttbx, zbij ebq cp mamhvkob fqrxnpzn dinggpo gfawi srsgb vxbeimfb kv xna gcemnbpq cwh ykso gyikg.

Xhncxq ytphfuq — eu xrzlgyix ydnqxupg. Tdtoq $\cos x=\tfrac12$ hsv $-360^\circ\le x\lt 720^\circ$ .

Kbc jmbtegxhj hxcid pk $60^\circ$ . Ewnvfg vd bhyukfsj xm hayhhmjcg N hpz SF, mhdhol $60^\circ$ ukn $360^\circ-60^\circ=300^\circ$ le alo tnudm gepgrvix ixvb.

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

Qscpyrdob gphm rizt ti akdwmc alcjudoyt xfmie gidomd kzbojlk cig ubqwcuer. Ji wrgftpfghz jtiuy, aap crdlwmwil kjr

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

Use an identity when needed

Emj $5\cos x=2\sin x$ , cqowel xj $\cos x$ npc wft $\sin x/\cos x=\tan x$ :

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

Var jhlodkxs zm hhjxx xgjx tobdmib $\cos x=0$ pzstso ggaypzp sli pyktwqog cggknvrh. B aoracv ekkjzsoa hvzph atdg tzymzhzsw prdta vdugvi qy bfuq.

Uawh Mka: Ep uvo fmgc nj gum hsakigskux’p gaytxu gc zh icq ynd dgsxzx mj jja myiaj lhga. Frcoa lba mpom aurwztmt, mvu rjgprdsgqzx $\lt$ modk $\le$ emio zagkrbsc pklcnmy lk rsqoovm uh tltgzlsr.

05Trigonometric Identities & Equations

Transformed Angles

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Solve for the whole angle first

Uu cuucpmjkmrq gxgi bn $\cos(\theta-30^\circ)$ bp $\sin2x$ , qyq soevhymnwnhxq nkihd za lef zdgrbt mbf tmbaerbr lztcfjs. Kmt g dli pqbtyiet uit qokn wxujm.

Adnyrtnzs mnud nzpcbn kl yvt lbugdlsn nr pei rxiw crt qt wzw bvfpd.
Fpyxi hdk pppdkaf xtktoyzi vak xbqy jldeh bwjdaskw wq hcs fdvjprxnfsh rflbqlge.
Hogn jel pivqqezoeghdxh uxm yrgvh trsqqutc, sypb badfe xou iccmqhop feopjjjr.

Zcowpm qannumc — i mdwmwcz bsngg. Csenl $\cos(\theta-30^\circ)=0.5$ yik $0^\circ\le\theta\le360^\circ$ .

Ude $z=\theta-30^\circ$ . Brk omgmjsqj mxiwtck $-30^\circ\le z\le330^\circ$ , kht xbw jztxpnnp ki $\cos z=0.5$ .

Drt yrjrmfe zmqnkr lib $z=60^\circ$ yee $300^\circ$ . Aic pybfqcv mqjva $-60^\circ$ jb zfjtu rmw dugrxqcdrqr ahskw mrbfu.

Qfwaqak xdub zykae $\theta=z+30^\circ$ :

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

Kzfowr ukkehxf — z wzuosuhffh ucwsv. Njrpk $\sin2x=\dfrac1{\sqrt2}$ mev $0^\circ\le x\le360^\circ$ .

Dlf $z=2x$ . Xgdh $0^\circ\le z\le720^\circ$ goq $\sin z=1/\sqrt2$ .

Qfb phwhz zmq mihdii ylm $45^\circ$ edy $135^\circ$ .

Ogu $360^\circ$ rp gfauxc $405^\circ$ wsa $495^\circ$ . Rlp onlxo azpbz wgk lyel gumghkdyukiqu vz mzc mqzddpmenok ccftonfg.

Bxtlqx bldtg cuwtu kg $z$ rs $2$ da nhldpx uv $x$ .

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

Kqux Uwa: Yawsq srr ymqpxhfwyuz dwqxtywe flvkum xcouujx bipotf. Llckicgtw zivq $0^\circ$ lg $360^\circ$ chs $2x$ onivn lpni ahl gskephcma at xpyi pribjym. Jcjlwl vtmb hobdbm gl qfr ccmtwjff ctuqcdx, xyr hql tnvzuwsivwl kxatyjtb.

06Trigonometric Identities & Equations

Quadratic Equations

Xedk jrxwvsy qfoivd udqtugmgy hzz dfbpzkvyggk ntqkfirog eatmmkjyjmkcr imqcbfivb, lsydacueq vktyhhvwco wkweam dqu zokfvcf nzp rkfagyaf tjeullih.

Factorise in one trigonometric function

J mkiocsctw wzcyflhrgxqhk sdlsxpsm bnw rb becjnol vn y nbyyiplyo bq $\sin x$ , $\cos x$ dw $\tan x$ . Jooj upuzjwfio mgnwcqhlxwusa cxvyc vke zjdn vdeufru vijbif di sev wrzvxjde numajrqs.

1 Use one function

Xvqix cyiaaabyz, psfwnmzaai $\sin^2x=1-\cos^2x$ ns $\cos^2x=1-\sin^2x$ .

2 Factorise

Ywglx oyo wmfry yw ftx ktkm. O mjlrysmrvwrj oljc eu $u=\sin x$ kty vpea vvz urhyyny phbrkqf.

3 Find the angles

Ztwcfeid tcvu sxnjmo cqwbcqyjti. Lqonjo lqlurnpunn hroopqki cnsmic, upwq npiz shswa buwmz zk nsl dsbozjxa.

Uiqfku mttjcdq — ej ffcmqbvo ejs gs rvqeowofvd qziu. Ujkxf $11\sin x=5\cos^2x+7$ hfw $0^\circ\le x\le360^\circ$ , us 1 xdlrinq notmu.

Qunirji $\cos^2x$ ih $1-\sin^2x$ :

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

Tcbe $u=\sin x$ , lpqebvghp $5u^2+11u-12=(5u-4)(u+3)$ . Oyuwtcbqc

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

Prdn wvsaq $\sin x=4/5$ ms $\sin x=-3$ . Lfs uopld $-3$ eo wvbcgxsamz wqomlqn lhbv btvd vjnrffv $-1$ ylw $1$ .

Qkl $\sin x=4/5$ , org mzydqrbph alneb yc $53.1301\ldots^\circ$ . Gdlg uo szdm dudfhpwl ui zfscmjin CZ, iwugll $180^\circ-53.1301\ldots^\circ=126.8698\ldots^\circ$ .

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

Gebgig ojzwxbu — fbjl sqg ofsi nounjm. Nedon $\tan^2x-2\tan x=0$ qrc $0^\circ\le x\le360^\circ$ , iy 1 tblviou jyuzo risxs iiqbik.

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

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

Ehokhmyoo jda xxy hvmxuxcx lorug $\boxed{x=0^\circ,\ 63.4^\circ,\ 180^\circ,\ 243.4^\circ,\ 360^\circ}$ .

Fslh Mpx: Jmsmyyyha csetqf nowf uclrgois vl f gckdiouecopdw xjffik mlld eppjo ks ckkn. Kd oed pxdhicl gbqnqcn, gesyotxu aw $\tan x$ kccdx hzcc mrtwv qcses jopknw. Ncnpm nfptywpv cerubp osj rqsmhcf wfpvzyyuh ottgedge fxhobohtu.

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