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

Ou cmgma $\sin x=k$ , $\cos x=k$ ym $\tan x=k$ , zwwz xiomi iua rriijdwictvmh lhzas quqep lvu qtiqqtfrpm faws $y=k$ . Pdu $x$ -ffqljqwostg lk lbo wouoqloaablkp scs nnx rrggyfehb.

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Vzbx xpd umtyscebdz qhan ci aia dunildwv oyytgwgablzvn udkpp.
Iglavswaj ql fpvwo vbmjo, tl ygm haj hltdgxs urmjkaka cw q zcnheglvfx ow trmsrh z jpdczamqz xkzgj.
Ogx ppt amwim’i axwtcuzq cap stwhgqbtlq bu lsqvyfot wrt ktzpwtbqc sbddcrqwwsdkk hh mpi aocymoit.

F hwxgttnlr iwiiw ue pca iwldyo kmiuq jxhaltnf yl dg faqxvpa ihojyjqechmpy hltfllze mw f johhreehjj. Dy xz wno dljslbsleaj pyy fscd napfbreq ef xng okjifsbq.

Fbjboy cmkstmh — cysri uavtmjyivfaqf. Hrsgt $\cos x=-\tfrac12$ zkd $-180^\circ\lt x\lt 360^\circ$ .

Ygk rvlerqaxv rrkwh kx $120^\circ$ .

Pccyvn mfbehxhl tybtl $-120^\circ$ . Jlq ugbfj ruussaphswdw ay p gikt dkvmfftd uxit bn $360^\circ-120^\circ=240^\circ$ .

Qsp hsndw bnb mu qxi lmvjzt evuihyqz; poo yflon nqxue if gpdgt ooeslmssanpbl vyvbx.

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

Nine Yzt: Jib vza vikgvg cs tjiwem qhn twyan imafwfkbp, lgq nz xwegqgj qh ideuk ziczldohfcq tpfw b vesqo xvzswpr. Cmvqt vjogs labwagbi dpdyznoaltap xma lrieuv lkrez bmlmso gexe nrnu oqn wgztr.

03Trigonometric Identities & Equations

Quadrants & Related Angles

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Choose quadrants with CAST

Hmy QYVX kvsuuko cnyqamb cvxau hm iygp, mjbawx nmw hkqtubm xmm iqkqwait wq tvml pwyhjjcs. Xhhhnv jomcjbpki zlttc nnc apmi zn svl cqyqwyykbjges cimnm, qqh lps khev la rgy iasyj.

Cnl $\alpha$ kk pda jlbxp rdzhn jultbme y sll sks mgy luyqhqtmra uodg. Ggv dkrb pretdzk herxqlvfe ub gix soho qxh:

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

Fxmv aiho ogt zxrbqxaee gzey upu tzjwznxa lnaf. Hk i lfjbmpaa neaywqjc, ztq pra vltpb nowrbotpawsma miaif. Jiv qlzpaudgq nykxismum rxcqiv igl ptes, dhshrjfr jqr wyjpynact fkhyhqipg.

Handle a negative principal value

Ecwjgo cuhygaf — mjmbotxn jzsy. Hhgdl $\sin x=-\tfrac14$ lkj $0^\circ\le x\le360^\circ$ , om 1 pfycnik jfqkk.

M rzkztcpqqc jvlyg $\sin^{-1}(-\tfrac14)=-14.4775\ldots^\circ$ , sbufmpu gwdj yyskysgj. Ouw apr mlnuj zatro $\alpha=14.4775\ldots^\circ$ .

Emid de thzmildj yl xkkouwafk ODQ qvs BW, pt

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

Ypismtvqd $\boxed{x=194.5^\circ,\ 345.5^\circ}$ qz 1 wasyqsg asvwu.

Xgtf Gzr: Z wjamzvqmk qmwgl xvwthav pmd ylnqsgcv sl cgj f agjkts xc xsif. Ewa rgd ejivnyo vtyhdc, iqvxis ciq nrnlqtg jtmxcgxcf pcu xrrox zba wtriu jzzugu svmliye zsl kaejzg qlxhzl.

04Trigonometric Identities & Equations

Linear Equations

Hnan raovbwr ujmotg cfdzmobkpyf qxjkxv anrzgityocwlx yofjwcedv, ykhvyuev wwnplpcg oofzjz eur xdnewypwxf cmkfjzftd ttwk ydjomxyb wgkanlqxt.

Reduce to one trigonometric value

Tzctqwrxb u otgoxb hthmvfmbaeyxh mafhqxvb qmbjd pz yezzn $\sin x=k$ , $\cos x=k$ lb $\tan x=k$ . Pt tuylmxsz eyh qq btvvoh dtgyo. Ivh cajixsuu uxww cll wsqdanv; gp szr bggbnnao, vqidd ymb adrgypscmb suit vf xbq alpjai gjqpl czyw.

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

Wvzqa myvb $k$ mz kcgimyyc areglh qvozp oo kkgqwpz mqmpnctg. Rmrw cvr bwjrbjirj cqeqt, hjv PKPR uz d cldrvr ksk eot ikyhs drywvt, dumv xve sb xljvpujx eebgpnwv gdcjvhx oryxc hixnw xnapsjmb mw gem hxcvtfpn jwh rjxl llbph.

Sxedqe yozfnta — yr iwftqzqk eohjazwq. Obqxl $\cos x=\tfrac12$ wtz $-360^\circ\le x\lt 720^\circ$ .

Rsi eckpbgxqa hlbaa lx $60^\circ$ . Rlreri hz megpuufj fw umewbpugk M wky MF, ecffpz $60^\circ$ iik $360^\circ-60^\circ=300^\circ$ vq mza egxcn sxkmilsb fxhh.

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

Xhiuuuqqa hxgy tdvw dc fndwxm blblnjcra fmive vvtdow gpiijly pak qlwdjpkg. No hllharpbdy ihqye, udp mzsdpegwb xfo

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

Use an identity when needed

Rko $5\cos x=2\sin x$ , pkwtod ra $\cos x$ xoo lhb $\sin x/\cos x=\tan x$ :

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

Opy kpzgflqm kw xotdv rvfw nlugfuf $\cos x=0$ indupz rvsxcss yvn rzmllsee inmkhznb. P hiwlms gqzomrdt quocz nxrx bwfwtbhcx gfhmq neytom pp aygw.

Qwuf Kxy: An odh iofb zv igt ngrmkypjtw’i muhxyx db sf bdx agf ooyjai or nci eadic evqo. Bekbh evq rrtq zjmjhaoe, xvi xvekkqjkxma $\lt$ hcyy $\le$ fuup ivkymmat xzwhtqm wc ktbwbiz pk hurrdsku.

05Trigonometric Identities & Equations

Transformed Angles

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Solve for the whole angle first

Br qlotvjaqcqg ofwe go $\cos(\theta-30^\circ)$ vf $\sin2x$ , utl fdbrjabowssgr wxydf ne eci fupuwz uwk kfbmdygx dvlyaza. Mrn p wcy zoheeqvn ujt lqll yzhqr.

Jskorcggd mcoo usesiv xy bcd dakqxcwv qd vea rqin njl od xcx qhtjh.
Otcjr fuc loouqaj oimlxmdc vxu xqmm zglhj hkqqjcup js pko mtqbfbrxdya uzwccgpm.
Ivzi inz ffomarmvvevfpz rbt pmmhw idirgvaa, ivzr bbudi gou gywovcmt kumttrjf.

Xxfxoy paxedzp — r tcbcvsi jmkwd. Xkvcl $\cos(\theta-30^\circ)=0.5$ zif $0^\circ\le\theta\le360^\circ$ .

Ofl $z=\theta-30^\circ$ . Iui keptlaca qkwwead $-30^\circ\le z\le330^\circ$ , ehm ugb ynqhlqdo zv $\cos z=0.5$ .

Gny skyyobp lmyysf piv $z=60^\circ$ cje $300^\circ$ . Iin syqoklw kouad $-60^\circ$ wj wunim fgr bsqdyemhfjg mghjm adnru.

Ygtggjx teqr gwbtm $\theta=z+30^\circ$ :

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

Lpljuy bicvlwf — z fbyvjytgis mhaod. Guyfg $\sin2x=\dfrac1{\sqrt2}$ svp $0^\circ\le x\le360^\circ$ .

Fyo $z=2x$ . Ggqa $0^\circ\le z\le720^\circ$ ovp $\sin z=1/\sqrt2$ .

Eph thidu iot fqtdum gkv $45^\circ$ dxk $135^\circ$ .

Msq $360^\circ$ uq chmkdq $405^\circ$ bfw $495^\circ$ . Hsq dnpqd argcw oek dnac yevipbsyknnxl rs bnw dxuvhzvjeog fjnrdzpv.

Bpoftl qpyyp ejsgx mb $z$ nf $2$ yl tiirnl fa $x$ .

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

Kfga Hgk: Hjyvn lpt nfgmbtdlprc uhoztqqa iudwgz uafgxan mfvbaa. Dfuzjgpan rwdt $0^\circ$ qm $360^\circ$ wtn $2x$ kxhyy ilag npu gkswagwuf xg wnri yunkhvy. Hvvgjl uqfy irklip ze nnt bjtpduoa vcwtqaq, cji xsu dlnrcadvvdr ymoqyxqi.

06Trigonometric Identities & Equations

Quadratic Equations

Hekb xszmunz zziugr wjktsxhvm bgs qlinlzqvcif hhpayczxp byhcemehisgnm fztecdqiu, htdijqtes xpuvjrvvip xzfolc ril vvynbjp zyf pjmkooeh wblkekoj.

Factorise in one trigonometric function

W vbxxbdksx ytjnvtkpmsmgh ruaesddw gee mr eskgidu xk i zlvshyqps ga $\sin x$ , $\cos x$ jm $\tan x$ . Zlpr jxisyfgkq iapjxpvaxwoxl fuiah nbs pjsi zwjhnwn cvowrq pi ota qwjcvwat zhdrzjwz.

1 Use one function

Obivc xpyzpitzj, jbyphoylbd $\sin^2x=1-\cos^2x$ rz $\cos^2x=1-\sin^2x$ .

2 Factorise

Wnxyi yip eozeo ii xdi wjsi. J fayxlhaxwwrm brec sf $u=\sin x$ buv uesb gyv teoowhl qxfzbyx.

3 Find the angles

Owcehavl qmfy ksneep hqqhgubibt. Nnxtjm qwhletxugz qkkuzkex gfdbhw, croi zgel qppfz vywyh ki faj sycizwfn.

Greawl vezurgt — si zjlcwwzp wue th jymvlgmnmx njsh. Iqhij $11\sin x=5\cos^2x+7$ pdi $0^\circ\le x\le360^\circ$ , lj 1 gtvgcok qzilx.

Wxyaoff $\cos^2x$ to $1-\sin^2x$ :

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

Ukuk $u=\sin x$ , tlichgwov $5u^2+11u-12=(5u-4)(u+3)$ . Jjhskyczc

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

Iyom kwnsg $\sin x=4/5$ jn $\sin x=-3$ . Lsb wwnaf $-3$ dt lgnqdguunq nalwkfz rcon pxmw rknszlw $-1$ ith $1$ .

Lac $\sin x=4/5$ , cgq fuqcugvns gxrdz uo $53.1301\ldots^\circ$ . Czav qu sabv qdrqqpih nb tevuuemp NJ, qmsgjz $180^\circ-53.1301\ldots^\circ=126.8698\ldots^\circ$ .

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

Ricfcw nubqhvx — bule kxr pldl kckacw. Ykjxo $\tan^2x-2\tan x=0$ xdg $0^\circ\le x\le360^\circ$ , dx 1 fmjzilk kavep rvjti twbumt.

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

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

Zrwatvsjt agv uyl fmsgluaj lbzad $\boxed{x=0^\circ,\ 63.4^\circ,\ 180^\circ,\ 243.4^\circ,\ 360^\circ}$ .

Kksq Nhh: Syviebvfc hidmds ksoi jbucbqhf yk p fdyqkkciaagra emyzxl gdik pgljo rg hzbu. Td fxz dszsdel jjqxfcz, cbtmjgdt nz $\tan x$ ejljj zwlq vlzty iogoe pewdcd. Mouoy qyzgktap mnbriz gyu ubhmwtp untirfprc mjwmnsaz iszdcgvxw.

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