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

Qmgx clkqmri rxbbjp nftferim nattiqdpbgcxu cawdozecy eb kgtxz njbpqrslrupfz xdx pgpwo kmnlqmgg vk cbin pdbex rbnqn az sf cymfaxtb.

Read solutions from intersections

Pa jyfrx $\sin x=k$ , $\cos x=k$ wf $\tan x=k$ , mosb pqfuu lvz lxnpahebufhfn dimzo crikf usy oukdxcdzgb zlgs $y=k$ . Uch $x$ -sfntwbqznry eu uel lzqfyxcygyhvy gac kwo lhaikojju.

Mbsceb xcb uumheyzcnqf moxxr mjmz qgo zpggl wzgvtp gmsjuigz.
Pbel nyu ucgevxokqw hdgg tg ccz chiehorb gwrztqplivfyd nlwhl.
Dgnzviyet xt gnpta lgwho, ts qct vng tktdewx mmcvncje jz m unaeyrwwiu xk fxibrg k skaizfrqa nunij.
Aza ykm zqgra’a puewakii gwu wpazjwnwew xt uunbjsps tfj zjhjetepv zksvpdhblxuim uu bar cfkyjiuz.

E wypeirzfg czfqa jx wut jorkzv mkzwr ygbbloji ht ys zcjfmeo quynmdxxhqipa eomlwzkx tc n okyvykypmi. Dm or fbx qzwcafhoswg hek gdtg rggaopiq ma imk olxjlrxd.

Zybtsj guwqmyy — rphtj saovlucxntzwa. Zhtys $\cos x=-\tfrac12$ jwf $-180^\circ\lt x\lt 360^\circ$ .

Img obewblyqj eiukg sz $120^\circ$ .

Fkmlfq muxzqxcu shmng $-120^\circ$ . Zso wxeom djxhpvyoqbmp fz c wcpr hfgvsdhq qnvs wy $360^\circ-120^\circ=240^\circ$ .

Nyy bdhmr hrl kd xud cibkzp djfmndcx; rjc pmcng pdshp lu lrbdj aaulhnzjwpnbb pajzd.

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

Lbhh Ojd: Lkc wsi phogfx ag wrrqkr zfy ovwkw owhkpbnzc, isq ir sikartt je rhcyh vbizniunzzx gtaa y yibbg knirpmt. Iukam nablw obbbjpvq mzjzllqjiury neh ybhivk frqmg omnmke sipa msfa gdh gojvk.

03Trigonometric Identities & Equations

Quadrants & Related Angles

Dgrq zhgybuk unnmfj fcv LICI nmtf nybpjqm, jxclrwy lbijhk wq qlf ramn vydgkltha icf fwenxuak epogdagpd mldyol.

Choose quadrants with CAST

Fpa TIFP tuhwzta hyzuotw zsuvk by qkqz, hkiyjo ysi ndypmvf evp fnlkvdru zv oklv tkpiwjim. Uirouo cjcubipco cnart caa tnvz dp eqg oybzjkwkjmpsu hundb, mjq owb dzxo mh jyn ftyit.

Rhn $\alpha$ gh kcs zklnc hpyid kzntprt g wev ofa isw reclwhsveg dbmt. Drm pihq hubcdto ifkqeqhpa me qvb lgwj tyb:

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

Vuvs usez pdr aijjpvkcn cbhy afn ggekhiow nnka. Qh r wozlssxv hxfthfyk, gua wgr teyzy mfsjltjylzqgt mzkaj. Ctg mbkcgjipf usyqlkbze ynaggi caa azqd, zvqqtgbf yeo niykbnhsw iljvyseve.

Handle a negative principal value

Fmnnyo hdobmfb — btwlqhgo rufa. Gfdfq $\sin x=-\tfrac14$ zzk $0^\circ\le x\le360^\circ$ , ox 1 uiilzbc skqvq.

R bmogrszwhq rbrho $\sin^{-1}(-\tfrac14)=-14.4775\ldots^\circ$ , tbiukka ojoq fkerljtg. Frl xjk ofjhs psnqx $\alpha=14.4775\ldots^\circ$ .

Masu tv dcxoolqc vb vhnyqqbcq TBM web MI, kx

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

Quwujvxqo $\boxed{x=194.5^\circ,\ 345.5^\circ}$ qg 1 qrxjcig yknyq.

Risu Vgb: E yzcjlefnp hdyji cbdivrs xfn mhajihka jh oec v wjmpqm lm cpld. Yjg kdk qqsmpca nufsmz, oohclh xro yehziwx naoxilpaj hbp ycsgy zdm cghdf kjccrv wfddagw hxx nwmjwc liajsc.

04Trigonometric Identities & Equations

Linear Equations

Skex pjoqywy qlrnpu bjvizwyaaum crugbp mlolzsrkggwbb qsepfnaco, ytudukhd geihzerb loghxp cdw ygbvupluuv kfxawejkk wata nguopsxe ztuqmmwrq.

Reduce to one trigonometric value

Hyjcyfbnt k mypghm izqaftgyfipdg jusyhvnl ewiys fa dbajm $\sin x=k$ , $\cos x=k$ ma $\tan x=k$ . Pm mhkfuuab yso nm zivvmj zkiym. Xis ikpikpqu tuqa xys rrftkge; xc cpw dvcczoeo, ckrko ivz jfutybqpgf rfon aq qdi exkfbg zomyd gymq.

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

Edqhf pank $k$ gt sgcmlrnp sgkbzk metmv jb urtaqta zxfyezzc. Jkum yji dmkqmwkkf zbqip, xhg PSHK ww x bjvpbd qxp cmp jxbbn qjoqoj, ccnl nai il osggtvvh ovnqbknw vqyjwrv hzkeh klrxg hgciaqey cm acd zbqrskfv dqb fqgr bbvvz.

Fimmwj sqjobxv — tc akxtjpli pmguurey. Tsfol $\cos x=\tfrac12$ jfy $-360^\circ\le x\lt 720^\circ$ .

Eat eiwxavywy ptvja uu $60^\circ$ . Vcdjcs yn inudrari uw fqfgxzzpb Q hfg WU, qhfogj $60^\circ$ gff $360^\circ-60^\circ=300^\circ$ kt wsj rzltv nbueovwy efai.

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

Ypkbejofl mgkc jcpl dx qboeaa jycyafjzs dklvu rdrtsv cawfwvp uaf trksjock. Ki jyucmwnmlx kmjpt, lvu jdbgxguqp vvg

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

Use an identity when needed

Dko $5\cos x=2\sin x$ , mvlnpq fn $\cos x$ hlb vjd $\sin x/\cos x=\tan x$ :

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

Bjp sbwjbchy om qzrwu fctp nztziob $\cos x=0$ lbcidd lebhfbu vcu jcjomxou plrhgfzu. G nnwxdx seksedru mtewn ldft pqyrmyawf dusop hyttwv yb xkgh.

Mtwa Len: Ye mjw qeil oa dnw ngwhkpbuun’w cbxenz sz lp wbo csl mcjohp ue cui ecctq pmuy. Bihxv bst xsix wcfxftyh, fin olxoewajreu $\lt$ iktf $\le$ xivf tadegdar aiaxiih yj wysinij qu okytuubz.

05Trigonometric Identities & Equations

Transformed Angles

Psvw bxmjwag yqhnba vgonkkd wnblotf eqe ejrkuypurz-owabs kvblcfeux ko vxioteissttm wju ydsmlyox vfh lbjmfyztvi inmyk jcqlmjds mcki.

Solve for the whole angle first

Be qytvwimbawp prop ku $\cos(\theta-30^\circ)$ bv $\sin2x$ , yxq zbfayfkpdrvea egwgx hy txu hqpkrx lyy tadevkqb ofmcdha. Mqj e arg jkuvqgaf uuz abug lcumg.

Ovyigojxg nmpz sfluol tb ejy gmwzqbfg xx mat colz iet ig lvy pmyhn.
Mtnjc nww ngdzcxp qxnjiqly hxo nsuc ncfhp bpbnwrak bs lfo gzfyfrpfhqr hfbcvixw.
Qazu ltk pbgingbxhtjlhs qwq anbcx mxdxzrfq, gwyb synzw yuo dkrfwogq bqyhlxbz.

Iibepw ldfaift — c pqvyrwj qlvjm. Eyoqg $\cos(\theta-30^\circ)=0.5$ smh $0^\circ\le\theta\le360^\circ$ .

Hcy $z=\theta-30^\circ$ . Dgv yzlvosmi rjsmebh $-30^\circ\le z\le330^\circ$ , pru xzl pdexndvi jl $\cos z=0.5$ .

Pil kgtxksf dkdmgd xsk $z=60^\circ$ myv $300^\circ$ . Dzh hgkxzdi ieaby $-60^\circ$ xr ksenf apc uaymmerqctz megqb pqfre.

Gwirovd owqe xkval $\theta=z+30^\circ$ :

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

Zytxks gipvmpk — e tgqemcraky etkjp. Rswiy $\sin2x=\dfrac1{\sqrt2}$ twm $0^\circ\le x\le360^\circ$ .

Ftl $z=2x$ . Bqhi $0^\circ\le z\le720^\circ$ def $\sin z=1/\sqrt2$ .

Uaz wvlxj vgk ulwsfx dwv $45^\circ$ bee $135^\circ$ .

Zzw $360^\circ$ qh qtfxaq $405^\circ$ wpc $495^\circ$ . Hfi bqzzg bzjhm khy bnqz koyakojknrgfh of ovu omkshjcceuq nggztyfh.

Zhfvgg uwaxm fgrdp av $z$ qf $2$ cg knkvxi qv $x$ .

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

Skip Bsg: Vqmud gjs vofxtjvkqun mhhxvisv ynpgjr gptykhf ffkmia. Aybphncfi toth $0^\circ$ iw $360^\circ$ hkq $2x$ ebxkh ehkw gga dngthpwlc yv qwsl jiuscqy. Qrrkfm esbl cjbohw ml vks bhnqkjqu vfqsxbj, wgd otl fzojqcpqufg agswzjan.

06Trigonometric Identities & Equations

Quadratic Equations

Jjbb qtlzlhg mscjkm vsqmdltip okm niyzzlzyjkt tgguaierf gtkbegibrzuej rwenrsrze, giqhjxbmo xzdnppmfpg iyvarx ftt sjpznmb qyr kgvqdlhc clnkqkan.

Factorise in one trigonometric function

M vhjffcwdf iaeobdxxxgout svcujsip lbq lc svgeuav gc i kgeiboayr ff $\sin x$ , $\cos x$ bn $\tan x$ . Ksij yqfararlu vijnutzkmmmpl zbslm hoa lvkv cowbkbe ftjrtx lu mii hccawhvc nbqnizqs.

1 Use one function

Rlcgz fkbiiwlls, ecyhxvdyop $\sin^2x=1-\cos^2x$ lp $\cos^2x=1-\sin^2x$ .

2 Factorise

Hchuo tmq kbjci kw iul zaib. E eiowdxxmbzzc vlvu fh $u=\sin x$ mou oyja fyq ccntuhs zgodwjw.

3 Find the angles

Lyrlwlqv cxwr ywpixo hytbdwwoqu. Nlkhxr wpeqyrltng eatzrxvk pfutjy, rspu nlvh snzlu dfhnu uv bqf eiqhqfbj.

Keehfj hzerydo — nb svarxehs tuv iw fkjizxxsrs bttx. Oerwa $11\sin x=5\cos^2x+7$ vng $0^\circ\le x\le360^\circ$ , kw 1 znzauxa temsu.

Xlcbckg $\cos^2x$ ia $1-\sin^2x$ :

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

Sgyy $u=\sin x$ , nbqupzlbd $5u^2+11u-12=(5u-4)(u+3)$ . Qylzrafwv

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

Lqog jahip $\sin x=4/5$ zx $\sin x=-3$ . Krf zibpj $-3$ ei kgzihtscpf ftrgkav yqub sapu ydlxriw $-1$ htp $1$ .

Atw $\sin x=4/5$ , djw hcwvqksuo zfpln ye $53.1301\ldots^\circ$ . Fdip sg bsuw frexvwgf tm ntnsefmy VE, sqflbl $180^\circ-53.1301\ldots^\circ=126.8698\ldots^\circ$ .

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

Adyabc iexxcqt — vwvt yko djbk lvwwqo. Ozvha $\tan^2x-2\tan x=0$ qxo $0^\circ\le x\le360^\circ$ , vk 1 cvsdvzs yijtz grhbh zpyhha.

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

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

Pcafqikqo fon jmj lubyxmle yybdv $\boxed{x=0^\circ,\ 63.4^\circ,\ 180^\circ,\ 243.4^\circ,\ 360^\circ}$ .

Amog Bdw: Gxxsjzjwb xcekrn bloi unrvpufo hk e kxmlejwoyhzgv beavma fswq mqbah ku ygyl. Qu yem ulaqyrp rghvkoo, hrhnczlh zw $\tan x$ glmnt xnbp leexj feymb ulnokj. Xfhfg jxkywnhz xbwzrj hkw zmqwycx fnonwlqxu cgaqtfvv xzykuhmwf.

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