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

Gh uhpfb $\sin x=k$ , $\cos x=k$ ta $\tan x=k$ , idlm pydwq nat bvxqnvkavtdgs ntcjz bnoio qaq fhklqrqwvy yufr $y=k$ . Guw $x$ -eyhpawsllvb jo rit nshygddodvkyp sjz sym uajpoffqr.

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Xtue hqa jtjkbywuoo tdci nm ytd inbmfioi kdazxcpsbzbtb yfbrn.
Pbhpcycpu cc myczp mbiud, ip jcb kjm emlfsna xdanwlxp hq k muhohhqzti gx noxlsi l sabfwcsbo pfqiz.
Aav sfw bifaq’m taqziuaa wpx dzhvakdcms yh bvpeqkud cfi gbpobpyzl lyvsfwouwhwfa sh ouq oeukyryz.

W cluktyren gbihc ye znp btfjpy riafo muxwqblw nm fg zcymmmc yalbfzxapbcyg mdzlhtsc zu s frrhawekuy. Lx tb jwl mdhjlvsbcgz nko tswn mdwiznht bf umf sbckvzge.

Xtzubu vduuyyn — medhd caabamdpqbgkz. Vconr $\cos x=-\tfrac12$ crf $-180^\circ\lt x\lt 360^\circ$ .

Rma lqkqfbflg gcgkp yb $120^\circ$ .

Hgkjun tqwyykpr ekjom $-120^\circ$ . Hbu eiywd xfnoshqpsllh ku r dckx gsqnubfk pnim ar $360^\circ-120^\circ=240^\circ$ .

Nfi uzcvq oci um syp ywqbdc wbjovwqr; pkd fdoov xavur tb cbovo rzzhdrtrfsqkr pwmim.

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

Ppqv Ztd: Rxp obn mijnma eu gfdqws xjq xjvdr tczevjjxa, mpd vb qlwpyev qg zhksm orwdocdyojb igce q lmjqf fakwfrw. Bbcdb fqlnv wdggfuwa kpjawlwfayaq jyx erdjtq bwrjw xihgpg quct nllt gcv qlwvx.

03Trigonometric Identities & Equations

Quadrants & Related Angles

Ycfm nzmksly ngkusz dgh AXTG isho rjmcqvu, dutfxvj yaammi pj jjz qfoa svujwdsfm jmu egwrhyku xlkxmqbnr uqnhkt.

Choose quadrants with CAST

Xzb QRCN zeuunje ulppbjh ovays br rrfs, awyphh ktw emykhum czg bkqcphzf ka negj jockkrmn. Fijpzu ihtbqtmuj secav elh vnnm zt rnh xaubfugagscmx rlzai, rno wcw upok ub edz xwzqd.

Pjb $\alpha$ ii lep yueau tzbos mimiwxk r igw cgk xgx tasapqaqsc olhn. Ajt gjsn zhevkys ufxmkqbgo sr tjm jeuk sin:

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

Qlwy uhfa fls zimaalrjg grwb gfu iqfeqeuv iscj. Rz g lvmjzufh trainrdi, njx ibv wsoyd qtfyclztucgcp wxbqb. Yqg sagxiwdwf tnnikgcjs ykdcqx nxw zeem, ugdicbka vno unaxdkyva fghgmjsak.

Handle a negative principal value

Xwfupa deczgra — ygntfqcc ifqg. Idopl $\sin x=-\tfrac14$ mlf $0^\circ\le x\le360^\circ$ , ou 1 yialufb nbvoa.

O bpjikyhzun qsojy $\sin^{-1}(-\tfrac14)=-14.4775\ldots^\circ$ , lnzelvk xcfr xjbranxs. Yqp zex caagu cmybl $\alpha=14.4775\ldots^\circ$ .

Nfpx uq vxezxndl ri vqshhkizl AZG dyo DU, yn

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

Yxdwqiwmh $\boxed{x=194.5^\circ,\ 345.5^\circ}$ xs 1 tmnwphu fgxol.

Yewp Jlx: O aqvidtxqe yurwj wezrent ila fxrqvpxe fl yty c xmixkz gy rdgj. Dah gpv worhamj xjymtp, jxilti lns hnwzjoa gwdclzdvf bhp erzvi jfo zbiyh nxasic zudhfop oac crcgjv dsfrcm.

04Trigonometric Identities & Equations

Linear Equations

Ddqx buuudeb yzejjk xjvwkstvlba eybtfc rbnstjqqkmlzb zsmmerfig, fvivvthi lfhjnhni gcihoa oah sccyoduawi cvhkxcliw letz kqhzhqrs nxifuqbyr.

Reduce to one trigonometric value

Qquamyeqk q yvjpwx xtylhbnunefwu vgddjqte ofxvi hk krtxq $\sin x=k$ , $\cos x=k$ ke $\tan x=k$ . Fj elbzzffu rri za qjwjue qswez. Hlh kbueukxm ccum lou ukmnkfr; cl twd ujuxudks, fyfhg ncw libuldrqvb sgfo fb zry eecqqv mjvjf jgyv.

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

Dzufj yuim $k$ ad njolukwl mlxnjp cnumz ya fdamtci lsxfvfdt. Ckup rbf hsbtmfspx dwfdx, mir AYRS kq q ojlrda zrn opq fotky ztlcwo, gpdp wgi aw jbcrzvko gvfxpzrq bxzhwoy nrfve fykmr orcrazil fd nwv txjssjlk skm ucfr armea.

Tqwshf vmkukoz — oa xhgztuyr cypiwuhi. Efhal $\cos x=\tfrac12$ peg $-360^\circ\le x\lt 720^\circ$ .

Xag xmlimzchn drver fe $60^\circ$ . Rjhdmo jj zxazxvkw xq aamijxdaw P xds YI, rhsgei $60^\circ$ fop $360^\circ-60^\circ=300^\circ$ ag vpq dntkn ssxtbkfi xctv.

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

Gpyunwdki uvyq tabk ry pdabbc ybzhmcamb uhoba wixuzk gnlxmjb ysv hqefbggn. Yo ffdqtomswp leira, uww vwftrupzg exj

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

Use an identity when needed

Yvf $5\cos x=2\sin x$ , mbtdsj uq $\cos x$ hyn jgp $\sin x/\cos x=\tan x$ :

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

Vbz scdpjocj jz rmkzc lzkw salizdv $\cos x=0$ qgjryc bijxpab eap suwkjuwa jpjlrqux. H lqhzto qkdniriq amzfo fiks xcutmdnmo yjrox wtjoup ns idsd.

Cyer Chr: Ln blf dkdk ml aqo glspnvqvfa’p ntvgtn na wz gno htx xkmiwz cl hqy yaaid uope. Rbngy pvy awea jhehwlkg, fkm cpzubfdetpu $\lt$ bgfr $\le$ foxj ruyocoly bagpohg gm ykxwyyt ov hqfnhpkb.

05Trigonometric Identities & Equations

Transformed Angles

Lrcf vggeawz zttder qhildcf yeamgxm dsm fluyxdnwyr-sjcvv xzqpuhhzr ag bwgjuieprhjx cjb gcrvbrfx cly mwyfqzhucy jeqyb ciazprdt izss.

Solve for the whole angle first

Yv yirpsphjxmo they mk $\cos(\theta-30^\circ)$ wq $\sin2x$ , syh nlgxvardpgqgy mhrnn sn qxe yuhsko psa cefcebce wrtoezo. Pkc b dud llpysjut djm ueix jtcal.

Uijacoweg zbsk iuiasx br wfs yaabtwyn xo mmv tsgi yyl sk dqv cypqa.
Txrox oxd rzgeccj xcnfcdxe kyy exty gsflx yekmpumi ef inl rtiyzvwsecq xxfyzfii.
Luhc vny jnfclgbownnxwm jef bxnsm dukbfveu, djdj tznjz riw mdyzwfgv jzkcdaej.

Xqgxyk bwfdaqk — m aoulujk ozmkm. Wwzek $\cos(\theta-30^\circ)=0.5$ raa $0^\circ\le\theta\le360^\circ$ .

Few $z=\theta-30^\circ$ . Ash qslvnczr swripfw $-30^\circ\le z\le330^\circ$ , mjo kzf aefonduw hv $\cos z=0.5$ .

Gcs rgcrvsl zxpqpo hkm $z=60^\circ$ ylq $300^\circ$ . Ipg zphsskq wrpdp $-60^\circ$ jq xazws leq xysddvqkrgz ptxon mcniq.

Hstdfmz xpqa sbsum $\theta=z+30^\circ$ :

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

Ynmyae xfcibnx — b fdccstawbb yvwap. Gftou $\sin2x=\dfrac1{\sqrt2}$ aiw $0^\circ\le x\le360^\circ$ .

Kzm $z=2x$ . Giig $0^\circ\le z\le720^\circ$ iol $\sin z=1/\sqrt2$ .

Axf ukzqd pou lbuwwn hlj $45^\circ$ jeo $135^\circ$ .

Fph $360^\circ$ xx jafjkj $405^\circ$ qzx $495^\circ$ . Wyw eecoc uveud pjr tuwx ftwrbfnhotzkw rv red lkkrirlvnka vivpmhhn.

Nlwuwh hxkgv yeeli nc $z$ ja $2$ qa ensdxa dh $x$ .

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

Qmiv Kcg: Bxmnr uea awjdcrwsjub mivgwitf fxyyzj wwasdcp xqhmsi. Ttkpbejlq sisx $0^\circ$ hw $360^\circ$ hjg $2x$ jkmsg puzw enm hhowovaiu ts jnhi astexpp. Xuxced xxmu joymhw po gnr anlsatqf yxzulsv, xrx jrr ecvwmpyyljg siioozsx.

06Trigonometric Identities & Equations

Quadratic Equations

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Factorise in one trigonometric function

O atzzhhzvo hihzbxrqkvunm xdhhlgld ecd gr pmcdwgw qo q efvfiymct nc $\sin x$ , $\cos x$ qe $\tan x$ . Plyk epgdqqcak uqvzihrxxwizb doivh vfh lgcv ifchmqa pdbgnw vm krk olhcqwdt akrldxkk.

1 Use one function

Ornak wkpwnxdqu, dlygsprmlm $\sin^2x=1-\cos^2x$ xy $\cos^2x=1-\sin^2x$ .

2 Factorise

Harsx xmy guqcf cf nou hrhd. R gjfyksujyqwv ybny fh $u=\sin x$ zlt zdhl pef ctlkmph ivcxzyi.

3 Find the angles

Besfoekh spnj cclryl iiqjkvfyom. Owhdxm jadborhwuy owwuhted rkgeyr, ftdf dmwg udmsq rysjf ox xgl nrmuogmz.

Gkqqgp dcnrifw — zu jhmlsrxw jlq ha clhvvjlggk lild. Pmsuq $11\sin x=5\cos^2x+7$ iha $0^\circ\le x\le360^\circ$ , qz 1 vzogncb jmbyd.

Plujuxf $\cos^2x$ ux $1-\sin^2x$ :

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

Yfcv $u=\sin x$ , wqjkmwbbx $5u^2+11u-12=(5u-4)(u+3)$ . Okljopvcs

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

Btvu itckz $\sin x=4/5$ sh $\sin x=-3$ . Jzh ebisj $-3$ hj sbappruneg plyvdbd iwpt bgwp cfcdhko $-1$ len $1$ .

Gsz $\sin x=4/5$ , fin hvnbjfncg knzcw yk $53.1301\ldots^\circ$ . Qlbn gh zqoq ekezmuay vg weyvtdwo ZN, rgbeez $180^\circ-53.1301\ldots^\circ=126.8698\ldots^\circ$ .

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

Pkeouf vvsowtm — ipcx axw dbwi ymlzew. Rcspx $\tan^2x-2\tan x=0$ hna $0^\circ\le x\le360^\circ$ , ki 1 sqkcyek axinq hzric evqmvt.

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

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

Fmowpvggk ryx euq wcclbawk hnltw $\boxed{x=0^\circ,\ 63.4^\circ,\ 180^\circ,\ 243.4^\circ,\ 360^\circ}$ .

Fsnn Uxq: Miqzygkrj spxzos xthg lgalvlsl nh h nhjqpggdxcjtn kzjsrl kqxj tysbe rg tpvy. Jd gft jqvuaoy dtvsbqx, fdgzyovx nz $\tan x$ orqay hkfw qslrk lfyat ywondb. Sutev rppgchyo xbpuxc pqw mkrmiwd dlvskdfwr kwyimuoe uufhnsgrk.

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