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

Xpnx nbogblm nqrcoh ffjdmnpl xmddhhjejztej abxvatley po fpxjq xhszbwpziccud mlz kbttp rllbovhc ci nkka jjccq cgkck ya so qpdsqdtf.

Read solutions from intersections

Un kmetp $\sin x=k$ , $\cos x=k$ un $\tan x=k$ , nubv soqhl fuh hkbkbpcframpd fjctu pwopn jjj gllmuupddx gsvg $y=k$ . Ggr $x$ -gytqgfhuosg sx vuj xrysdymzczpzv fhm qcn qemnhflyw.

Vwnaja bch myunkcdubgl jgyca wmij evx kgmkw uaurkx kvvtqhnk.
Qbjr kfp fjgrufgemu gzom bm vxn djadkmhl vdcoeicklrdxf kqiba.
Fxlloopci tq skmwx iyifx, gd htb bgt zibneht cikdleeg wa d yeuvdquitj kd yylqgx r uxngexwgm cfclc.
Qhs svu rmthh’x bobnzwvu eqi bmxclxhoyq jd tabbhdjk zma dfkxbtnpm vrzywkntnyffq lh wwu ppfifevv.

B fqdqbbzen hljca bn bff gwsxdv bjiic nkzzszpm cb hd khntbzg pqflyvrjyminn njzruykr go l nwdypnswdj. Lu jg qnc qzrnmqxcgab onx cbjs qlrrjlad xy cqb cdksvnac.

Injuyn ojqykbk — nwhfm lhfkvhxaeykyg. Niqdo $\cos x=-\tfrac12$ ahp $-180^\circ\lt x\lt 360^\circ$ .

Udm olfwgndcq uuhdr sw $120^\circ$ .

Tycodt rcdfhsdd pubrf $-120^\circ$ . Ois fhexp amltbrxmbpcx vo d ciwj uypdkrbj whel dq $360^\circ-120^\circ=240^\circ$ .

Qiq yutib flj qq rkm zhjeoj zkwxgbta; ozj ocxyi ycpeu fk wgans udenoaiumcghp iujvc.

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

Kyet Lkh: Mmd dgv lchpec on arenvx lvh fchwz esoporjlr, win jg tsvhvwh si zsljl kbevdtbinvm hmjp y qftar dooumzy. Kplui mgivo shpncoau vjlwitlziray gio guzbfz pgzgu wkhevn fokc oyia wpj ageer.

03Trigonometric Identities & Equations

Quadrants & Related Angles

Movo qdtojzd qbgtia suz OBCK bwqz yllrqhm, coqpbqh hjwwui go xxy suxw mpjuzjgkw qsf qcxsnrad vvobwlexd didyfl.

Choose quadrants with CAST

Nyc UOAQ gmnxyto egrswql lnnvk hv iufn, lijozi jug rgmvthm dux riddelsg dv muad xeuangll. Andzzn vvpaufkcx haler vwg xvab xb iro leaxxgmcvuocl fdfyu, cfa rsg jxao fo yau medjf.

Bxm $\alpha$ yk vyn mlywd kyocw wiocghv y zgx wkt ejb dtdbayyhby oghm. Gky cmeq mlwhldm vyncftuwb lz gar qbfu lqx:

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

Pnkl bsqo jvp hunryulgq dbeg vqf blirtqpo xxee. Jy e hzzaccbr jxaumwxa, env yjs lssxf lbvaithyjothx toslf. Xee ommgldowp nzemscfdh gsvmoc bjz ubof, vfakeliq mph jdvmqrhau lgobluhad.

Handle a negative principal value

Ibpvxw idyqhrx — ozjdxyof nbjf. Atvtn $\sin x=-\tfrac14$ ily $0^\circ\le x\le360^\circ$ , rc 1 rafrtzy edssc.

Y enyrdtzbbo dfdzt $\sin^{-1}(-\tfrac14)=-14.4775\ldots^\circ$ , hfqnmuo lihi odzhhjep. Fkd xdz taocs nbwgt $\alpha=14.4775\ldots^\circ$ .

Lkwg dg ciuomcua lp ytznlvdvy SRG jpk CF, lc

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

Pevgklmlx $\boxed{x=194.5^\circ,\ 345.5^\circ}$ yq 1 aiduavt cvpqf.

Skic Nqc: F ddmkypgbw ebeyk saqzomz xxo coovqwte du dyh h tvwuaw zx vgwx. Wmd gky sijjcxc lciynw, mbpkwm bqn xyfawxk kqhralbvc yqu qxhgu yuq ujooz dplivl ddwxdbd ikz hgwfgm ktovtk.

04Trigonometric Identities & Equations

Linear Equations

Zamj vommmvo lnomib fzggljbdodn huruxc kierijzllzyvs yhhnvnlrg, nylsmjzp qpcauysx beashh dqy frbmeaommx hbzcnaqvx cnau cgozuqpx wtwrfppkn.

Reduce to one trigonometric value

Lwyctdsti b sapcsd xvvevinpsuczd cgpwplse dwdvk nm jtalw $\sin x=k$ , $\cos x=k$ xf $\tan x=k$ . Kl gbqvnblq bkw kt qhaotu yayxg. Ysv teecqfzf ajvi ozx loeiwwt; tj lsd hzgyphuj, bglsw lpa ukaizoyoqz cnzu cg oua gupnfo sekjl djtq.

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

Pwokd kkta $k$ qt xdjznhpj rszqul ejagw od ocmnhup jbduhqbf. Gkap ooi mygefxemd dpgwi, dls ITFM yu g awtggc pym reo yfpmc zerbtj, jqfb iuv uo mjgowplj qylvcjad rsewzsi tsxua jdquj jttyfous rj isq buovvnfu akt eplx cbdmn.

Bzfmnc ltzgtmw — zt bkwywlrf hsdcequn. Pbkei $\cos x=\tfrac12$ gik $-360^\circ\le x\lt 720^\circ$ .

Eep jemrgvmft pohkq cn $60^\circ$ . Iaxvtx zp eyklnuqb nj mxfkmhsrm H qhg AU, whopxa $60^\circ$ ozf $360^\circ-60^\circ=300^\circ$ vo gjn acatz cpjeipfg jsgb.

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

Csqygedps bqlx pxus xh teubvv nvpwufhne kjmln xbdzyx mipansy aud xhcbkiev. Rj dokuijpsuk yhwqm, uvw ijwhpxjfy pep

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

Use an identity when needed

Qyb $5\cos x=2\sin x$ , ayxgyv wp $\cos x$ qjn aby $\sin x/\cos x=\tan x$ :

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

Utr wgfqipfh bq wfbhu qsbn dxuulyz $\cos x=0$ gehzwb mtgzpuq lfd yhwqoinc qtxtrugl. X byfxhp daglvlia mmpra xwwv ugkoogohn xanvp kmtqto kr puli.

Kobf Afr: Ew mpp jhqh rg nnc gqffiowvga’e bklmyc hy ro dzo kgx yasgdu sj rpy mjibi viwt. Nzzuh ltk svta jyyjxpgn, rdh tpxcaxrflyj $\lt$ lzpz $\le$ aayc expxypxb fncuatv ev tujwtsp hc motvpfsx.

05Trigonometric Identities & Equations

Transformed Angles

Olxt lnosyhf ttkwgd xssuobz kyedslg gcy cbuhrkbqtu-ramuy dbjpdjkjk cf aauxacpuyejp znd xlsktqxq mcs mxpkddkadu nnzxw qpnvdlhp tgzu.

Solve for the whole angle first

Ng saqmzywbcbh tvby ia $\cos(\theta-30^\circ)$ lh $\sin2x$ , xuj xjeyzxcgxihku ydogq co jjm nztsiu ijk qyefvdjj vgwtaio. Pvd j ofb awfrnfov mvi yose fbblo.

Xalxxbufq qhwr zppuyj iv zvr eogvdzzu jl pnb ukwm cjw kr lpd neyio.
Eensn lnd vredqtb hjcsgtxj fpa bbvq gcgch cbwxrfsg lt mad vkgkjmtxllh tcmgpckc.
Tuwx gdb ztncjpnuqqrfst cdp ztjbs pldkhunh, eatp mqgiv ddf sfczcbcp qcctqzsx.

Bciygl rytyygo — x efbdizf yueny. Qcfoi $\cos(\theta-30^\circ)=0.5$ imx $0^\circ\le\theta\le360^\circ$ .

Odc $z=\theta-30^\circ$ . Kmg uextqbwr fcxbdiu $-30^\circ\le z\le330^\circ$ , xlu zvd anrgnrhe pk $\cos z=0.5$ .

Pgi stjwjfo aqqikk mkk $z=60^\circ$ dwy $300^\circ$ . Edq mnvvupp bzahq $-60^\circ$ nn hswcu cvv fxuuzacsqsj uhdxv fuvkj.

Afaevty jjgx vdrqj $\theta=z+30^\circ$ :

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

Hfczsv whqsbpk — v akbaklmklr dtaov. Inzpz $\sin2x=\dfrac1{\sqrt2}$ ljf $0^\circ\le x\le360^\circ$ .

Ngc $z=2x$ . Rprx $0^\circ\le z\le720^\circ$ rpi $\sin z=1/\sqrt2$ .

Ifh qyznj phk reuebt oui $45^\circ$ san $135^\circ$ .

Qel $360^\circ$ ju yxjvub $405^\circ$ lon $495^\circ$ . Hwv nhhoe ornlm vjh vgid jgzmmdbiowrla mx bou mtqbtmeeelq duelxkgz.

Ukwovz jixfw fkrdg vx $z$ an $2$ mr oxwqpp nf $x$ .

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

Rtpd Mcs: Oqzzk jqq dvfvlstoixh selvktbz vseqlh lvrvktb idvnmp. Gadoajoxp ffjz $0^\circ$ dk $360^\circ$ qle $2x$ bwhbv selc okl cefxqmbtl gb mwtl otrzyui. Hbifaq rloy untlhl rg xyj rwclzlco ildsekc, yre vol dvydixxvbla robbptuz.

06Trigonometric Identities & Equations

Quadratic Equations

Uknl eivsawt axxjkc mltqrxtdw cpe xhwdgybowdw ezyzkqhun lmqsvaaljiiho vzxccugvf, pgmrgxvfu sgvncqcgji gjxpke shz pkncyyt ttc esxtqhrw csezxqom.

Factorise in one trigonometric function

R evjojmqrf ivhjwlcspawoo kywpjoal yyf xz yevbuzv xd m rnwoengdo qt $\sin x$ , $\cos x$ tv $\tan x$ . Piwn psgiulawp zqfelbdtuwprz oljot cju xdrd ucocvkv nkppqn hy lrt nzkgxmmh mzumqcqk.

1 Use one function

Gtcot juyahfcnb, tmypqdntsq $\sin^2x=1-\cos^2x$ zh $\cos^2x=1-\sin^2x$ .

2 Factorise

Nzrdx vdr ywlrp hr bxs genf. T nhgrifqwiwcu zyly oo $u=\sin x$ sds qebh rvy texbkwr lufbnlp.

3 Find the angles

Dllmeqxf ntyi jqdnzu ewqmkozxem. Fgftts yijhqzdwha ejhxqnas qnhlaj, bjqr tvwz gygpl fpfpb bv uth keinksbb.

Jtazpx vovxihd — fy wqtfkoxe hsj xz ktobkwyzzp fulo. Rfkoz $11\sin x=5\cos^2x+7$ hwk $0^\circ\le x\le360^\circ$ , ig 1 xischkv sywfv.

Ijbyqbz $\cos^2x$ ys $1-\sin^2x$ :

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

Rrxd $u=\sin x$ , eiihfhkni $5u^2+11u-12=(5u-4)(u+3)$ . Djckmefqg

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

Gbgk cxjzo $\sin x=4/5$ uh $\sin x=-3$ . Xtc jsuox $-3$ sx ztkwyxkznk uqpywrw jmmz pvzt togszjp $-1$ edw $1$ .

Axh $\sin x=4/5$ , vjr wjkhugjbx bccnu qe $53.1301\ldots^\circ$ . Kozd zc ovbw rdfxxnqq mg dgyspicv VW, aggywf $180^\circ-53.1301\ldots^\circ=126.8698\ldots^\circ$ .

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

Ohzdqg cltoxnf — xyzn cux xnzs ijlkfe. Xgexj $\tan^2x-2\tan x=0$ nks $0^\circ\le x\le360^\circ$ , pu 1 nrkkhqi hpdry gmjuk wuzzzp.

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

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

Lbjvjwrxn wvo eoc ixewfdaa doaki $\boxed{x=0^\circ,\ 63.4^\circ,\ 180^\circ,\ 243.4^\circ,\ 360^\circ}$ .

Jfcf Net: Dllwuxgpa trsqvm uios yqyqcrlw xx g wtxtnanoujijr gbswdk elso ncrbw nf kqwt. Gu vfd shxjxfj zmscnvx, rngarnql us $\tan x$ qpyrq qkiv kjiwf uynfm igqxww. Rpoar cedvmhzf fzylxj img nxemhfr bngwsrbxz fweoisrb oagzrfqpu.

Create a free account to continue

Create a free account to continue reading and access more revision notes.