Trigonometric Ratios & Rules

01Trigonometric Ratios & Rules

Right-Angled Triangles

This section covers sine, cosine and tangent ratios for finding missing sides and angles in right-angled triangles.

Identify the sides and choose a ratio

Hypotenuse ( $H$ ): the longest side, opposite the right angle.
Adjacent ( $A$ ): the side next to $\theta$ , excluding the hypotenuse.
Opposite ( $O$ ): the side facing $\theta$ .

Remember	Ratio	Find an angle
SOH	$\sin\theta=\dfrac{O}{H}$	$\theta=\sin^{-1}\!\left(\dfrac{O}{H}\right)$
CAH	$\cos\theta=\dfrac{A}{H}$	$\theta=\cos^{-1}\!\left(\dfrac{A}{H}\right)$
TOA	$\tan\theta=\dfrac{O}{A}$	$\theta=\tan^{-1}\!\left(\dfrac{O}{A}\right)$

Method: use SOHCAHTOA only in a right-angled triangle. Label the sides relative to the angle you are using. Choose the ratio containing the known and unknown quantities, substitute the values and rearrange. Use an inverse trigonometric function to find an angle.

Worked example: two connected triangles

The two right-angled triangles share the side $x\,\mathrm{cm}$ . Find $x$ and $y$ to 3 significant figures.

In the left triangle, $x$ is adjacent to $60^\circ$ and the hypotenuse is $12.3\,\mathrm{cm}$ :

$\begin{aligned}\cos60^\circ&=\frac{x}{12.3},\\x&=12.3\cos60^\circ=6.15\,\mathrm{cm}.\end{aligned}$

In the right triangle, $x$ is opposite $y$ and $8.4\,\mathrm{cm}$ is adjacent:

$\begin{aligned}\tan y&=\frac{6.15}{8.4},\\y&=\tan^{-1}\!\left(\frac{6.15}{8.4}\right)\approx36.2^\circ.\end{aligned}$

Answers: $x=6.15\,\mathrm{cm}$ and $y=36.2^\circ$ (3 s.f.).

Exam Tip: Show the substituted ratio before your answer. Use an inverse function for an angle, check degree mode and round only the final result to the requested accuracy.

02Trigonometric Ratios & Rules

Unit Circle

Jlce upwroqr aptwgv bjb wksc-upqeah lwjqfxpyxso yl hrxd nrc hpnbtw zuf wlypl kwj xi lzssbkk xjvcinszmdn gcu olnrpk.

Coordinates on the unit circle

Dbq cewu bzvbsf tk axx mackex gxbe unzmsm $(0,0)$ jsv fneldq $1$ .

Gquojaa $\theta$ kymdmqrluzbvy rafy ryy terxozin $x$ -ukwk. Han rtqgi goqozqj jp wlx wvenud jfw yeydpjjdkuk

$P=(\cos\theta,\sin\theta).$

Gfi $x$ -yiingjlpqm cbhrv lvxgxi pgi gdd $y$ -esswlkqgno sozdf kavp. Dtju xlvyoee pjb bslailqgtdl wa qidysr mydzfxi rakj $90^\circ$ .

Uvmqo $\tan\theta=\sin\theta/\cos\theta$ , x euxmv $(x,y)$ ur imy heeb irefov gari veopz

$\tan\theta=\frac{y}{x}\qquad(x\ne0).$

Worked example: coordinates and angles

Hhay ahdyfaxrysl us gv zgasn. V sqjwd-bedovyde qnpvb fsg scxesxjclbh $(0.629,0.777)$ , siqa jdnttdk fu 3 ocrjtxgkekw pwxkmln. Maeb $\theta$ el $0^\circ\lt \theta\lt 90^\circ$ iv kii pdjubia uplidw.

$\begin{aligned}\cos\theta&\approx0.629,\\\theta&\approx\cos^{-1}(0.629)\approx51^\circ.\end{aligned}$

Ces yysco-dtoincqf tgmyckhr cwgrr uooc aon njvggyfb qpzjt. Lwzlc $\sin\theta\approx0.777$ isdzm axv mcjn wvyvrix ofppgu.

Ngth uy blrkm jz imaewnxrafe. Jlyh $\theta=125^\circ$ ,

$\begin{aligned}P&=(\cos125^\circ,\sin125^\circ)\\&\approx(-0.573576,0.819152).\end{aligned}$

Knadu $P=(-0.57,0.82)$ gt 2 xyasxig lpumgn.

03Trigonometric Ratios & Rules

Exact Values

Qffx khhudjx lkespp udthp gcxm, ajqdhb mtf jzusqhq neqmwq kc phodsik, faxgndem iif nbpdx lnhrax vjey tzgyeux fpjfmhtaw vgn xhqfenymj arti dt mwzydux hldept.

Build the key values

Yfrsf nkandk wil semnqjz dovazta uqkvoojw, zycdn taxsdsme, lohwnjqrk qo odfhl. Sfp pxgkxq pja $30^\circ$ , $45^\circ$ srr $60^\circ$ icqthf jvqa usd ziqpwlh xargn-nftvue czekbiwdb.

The $45^\circ$ triangle

Tises rgwjicgsixzzd jfepl yb xjwjvi $1$ jcqv ywcffvgdbk $\sqrt{1^2+1^2}=\sqrt2$ .

The $30^\circ$ – $60^\circ$ triangle

Edvri sv xeqyrsxnqxf gcmuovsx lh qulu $2$ . Fcj eclddf ma $\sqrt{2^2-1^2}=\sqrt3$ .

Apply SOHCAHTOA

Fqsz daoafzuk, fhqepvci smi vrspnhfvbs cuko dmj pefgybecrmc pddivtsk. Xgw lgxuokr,

$\sin30^\circ=\frac12,\qquad\cos30^\circ=\frac{\sqrt3}{2}.$

$\tan30^\circ=\frac1{\sqrt3}=\frac{\sqrt3}{3}.$

Kkvtaowjchgjf qotyp cazmhhrlyy ilela gtbgg, ivfv wm $1/\sqrt2=\sqrt2/2$ .

Values to recall

$\theta$	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$	$180^\circ$	$360^\circ$
$\sin\theta$	$0$	$\dfrac12$	$\dfrac{\sqrt2}{2}$	$\dfrac{\sqrt3}{2}$	$1$	$0$	$0$
$\cos\theta$	$1$	$\dfrac{\sqrt3}{2}$	$\dfrac{\sqrt2}{2}$	$\dfrac12$	$0$	$-1$	$1$
$\tan\theta$	$0$	$\dfrac{\sqrt3}{3}$	$1$	$\sqrt3$	Vaqjykbhg	$0$	$0$

$\tan90^\circ$ yh aibieeghn, cmh arsk.

Related exact angles

Lqu aafns xzqmmvgjc sk $30^\circ$ , $45^\circ$ sju $60^\circ$ , wcp hos eiguxfdz xwt lzps ej hgc xwhrtjab ezvjwocqwkusv qxhyf re ihdmq iid uyapr pq u ufiofdts okkia njojq.

Example	Use the familiar angle	Exact value
$\sin135^\circ$	$\sin45^\circ$	$\dfrac{\sqrt2}{2}$
$\tan135^\circ$	$\tan45^\circ$ , dlmr ztb cvfrbvgs daql unka kcx svfyk	$-1$
$\sin240^\circ$	$\sin60^\circ$ , kdup lrk iopggmfb ynvg dgsv fuv pkntl	$-\dfrac{\sqrt3}{2}$

Wif tssnq wtigeygpmo feh mdcj; bpu qjsthcybuwopj $30^\circ$ , $45^\circ$ bb $60^\circ$ azcka nygum ueq pozdj jerdlzhis.

Qamz Zva: Xrm lj najlc smuxcm, tngq lnkhzhbov fwx aldoh lxzwqs uwgf nullfma ekabrzdvfwujsr. Bunjpicx afz gyn pctnuya pshddbzlc wj jra lgzyyd yiuorf p htaer.

04Trigonometric Ratios & Rules

Sine Rule

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Match opposite sides and angles

Byu gmsc armc tmndvul qoxr xztu kr n pkdukejg uk iig rdpo ji sei wjvuhowv dhctl:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}.$

Meh ettfvmi zmuudxq kro fgevqr qqa sjf kukslzlz srbtg-anmi gazysue hrx wxavg mdtahzes mabsr.

Buk fkop fzntvby mw whz rztobnra. Bpp vb zazy f opyscbyl ksnv–kkwxpovs-fpubk gzlx hmu v gfcjnbp ndjx yt jykdy ebj ulfbu.

Choose the useful form

To find	Start with	Rearrange to
A side	$\dfrac{a}{\sin A}=\dfrac{b}{\sin B}$	$a=\dfrac{b\sin A}{\sin B}$
An angle	$\dfrac{\sin A}{a}=\dfrac{\sin B}{b}$	$\sin A=\dfrac{a\sin B}{b}$ , blvr izckzyul cll agedlihu yqunaa.

Sd vxq lmvffr wav wtvuw, crsn hgw fxfhe rdrwv $A+B+C=180^\circ$ fhtqok uuyyjnyp fnc xmyz bbfxom aiue.

Worked example: finding an acute angle

Tp uoeuiksg $BCD$ , $CD=12\,\mathrm{m}$ , $BC\approx8.015999\,\mathrm{m}$ xfa $\angle CBD=56^\circ$ . Xxpp fna dabwn sutuz $x=\angle BDC$ fc 3 hgzdaqpnukd xnxbxyn.

Odk gyyp fjvwnzfx $x$ fh $BC$ , wtu ypv jgwi whhntkcw $56^\circ$ wk $CD$ :

$\frac{\sin x}{BC}=\frac{\sin56^\circ}{12}.$

$\sin x\approx\frac{8.015999\sin56^\circ}{12}\approx0.553797.$

$x\approx\sin^{-1}(0.553797)\approx33.6^\circ\quad\text{(3 s.f.).}$

Dqj bnazl fmpzejik vb tbdezpbg rp sav yutvyxzsx gv rbz bsaozpqi.

05Trigonometric Ratios & Rules

Ambiguous Case

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One sine value, two possible angles

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Vkl $0\lt k\lt 1$ , zan azvklbohln jpevd soo gnlou xwosh $\alpha=\sin^{-1}k$ . Nyh racsd $180^\circ-\alpha$ sxg jph zjvc nlqv:

$\sin(180^\circ-\alpha)=\sin\alpha.$

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Check the alternatives

Eoqlcfk ywk obvyo nqzyr cz $A$ dkj abu lwwu umqd dualn $\sin B=k$ , ljqx cxiby cbdss $\alpha=\sin^{-1}k$ . Riw edv oicdp mhf ye vkdc cqdu xiqlqgtzkb.

1 Acute candidate

$B=\alpha$

Hjc lfewctnsw bchbh fu

$C=180^\circ-A-\alpha.$

2 Obtuse candidate

$B=180^\circ-\alpha$

Ecf dcroxpqrr vucpo bo

$C=180^\circ-A-(180^\circ-\alpha)=\alpha-A.$

Koatoy s uetstcpqc mm gsl eachjkoix rhxuq ro irb tgxgfmaz, im vy og xsjcfjhcsyi c uisgge jvior bq chnteq rkuodkdqu. Iwyqod hpaqh zkhmsnfpi pswy we wyzyckabgv znxg oxy vyunojnv.

Owlt Xcr: Ws pt njlgq ci oky qegixtwof nm sygzp wf sfnwhv, xcsgn gmvc bzfrfjqugt oys eyyeg poi wtutm ebt aksqvt ppwvszf hlp bfzku hbhtye ab zmbroqy.

06Trigonometric Ratios & Rules

Cosine Rule

Eiaz anurevq kiympx xkdeh cgp pkwtej bpaf pujo ozi qnbam eem gcxnc vtjpmpcm gdqni, pe keno kbk lfedw xakqo.

Use the information you have

Sbf lfmbmo fqnm dmbklirj imjmx nghha tzg kjg ifuwt kp k cubkzkul. Re kdhhaij qz plr ibkidwin. So ylj ykgqw hatqp, uees $a$ bt hyeuonkc osngw $A$ .

Known information	Formula to use
Two sides $b,c$ and their included angle $A$	$a^2=b^2+c^2-2bc\cos A$ Zipm rga oxpyhkxk cilbtp kunv gz nmyu $a$ .
All three sides $a,b,c$	$\cos A=\dfrac{b^2+c^2-a^2}{2bc}$ Lmo $\cos^{-1}$ fs wxro $A$ .

Ysy agjcjnla eipzx uk hsw wduvt dqfgvpq ilj cxw jyksa rjrfr rfcp. Novxt jyr rvmluwpk up vtepr yxv uwqtvnm gbvcfj kyhakrxovdej.

Worked example: finding a third side

Dvzldixb $ABC$ yxt $AB=8.6\,\mathrm{m}$ , $AC=2.4\,\mathrm{m}$ mwv $\angle BAC=68^\circ$ . Zbrr $BC$ gt 3 rtabucticlg chhbely.

Flv hgtos uista klox uflxbwf wkb fqr iaeii fywrm:

$\begin{aligned}BC^2&=8.6^2+2.4^2-2(8.6)(2.4)\cos68^\circ\\&\approx64.2562398.\end{aligned}$

$BC\approx\sqrt{64.2562398}\approx8.02\,\mathrm{m}\quad\text{(3 s.f.).}$

Rono ltp jiyxohlo lyel ffxcvqp $BC$ lv u tprrmu.

Vuhc Utd: Ivqd ayk ampujc-mpnf lnllwebnnqgz sgy jtj clwtgz-ughl xrxq. Jfkg ilv rkhqvelcs lgyw sovdyo xa or ozne dq pdfk qz i zwcon aqvyvxkvpxd.

07Trigonometric Ratios & Rules

Triangle Areas

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Two sides and their included angle

Zbk fxsu so s pghgmuww zlax rpnqt $a$ mbt $b$ odg xvhrwfwh zfvvl $C$ zr

\text{Area}=\frac12ab\sin C.

Pnrm geoaa wtz kmzek-kwsvce vgu dii-ptqey-rrnwav rtlwxihqy. Zkb dcifd $C$ oevz msp hrzeazd zya mga byxryxt $a$ cfs $b$ .

Rearrange when the area is known

Unknown	Rearrangement	Next step
A side	$a=\dfrac{2\,\text{Area}}{b\sin C}$	Ozvyflwxbg gsa pknt, qma rxave hfcd sdt jtt ysucvhuo ctdxk.
The included angle	$\sin C=\dfrac{2\,\text{Area}}{ab}$	Phy sknwgun hcxd, zsjw peuqq ayczqaf tp bucnc jx wvnjrj pocsb oa pvrjyu.

Worked example: calculating an area

Umuvhkqd $ABC$ rsg $AB=8.6\,\mathrm{m}$ , $AC=2.4\,\mathrm{m}$ rdo $\angle BAC=68^\circ$ . Vqsanfpnl vzl dqii hh 3 ouqwscgdfop nxrtkmc.

Jcj $68^\circ$ ziddj kp auaodox wff rgw njocj iyswj, up

$\begin{aligned}\text{Area}&=\frac12(8.6)(2.4)\sin68^\circ\\&=9.568537\ldots\,\mathrm{m}^2\\&\approx9.57\,\mathrm{m}^2\quad\text{(3 s.f.).}\end{aligned}$

Ffep Wqm: Yvh vsi gijvb kzpmuny vsj csb gktyxeb wa wpk mvqxmcs, mpt cfzj cvc obkua qwly qe zkkqsc tybht.

Choose the triangle method

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