Compound & Double Angles

01Compound & Double Angles

Compound Angle Formulae

This section covers the sine, cosine and tangent addition and subtraction formulae and their sign patterns.

The six compound-angle formulae

Compound-angle formulae rewrite trigonometric functions of $A\pm B$ in terms of separate functions of $A$ and $B$ .

Function	Addition	Subtraction
Sine	$\sin(A+B)\equiv\sin A\cos B+\cos A\sin B$	$\sin(A-B)\equiv\sin A\cos B-\cos A\sin B$
Cosine	$\cos(A+B)\equiv\cos A\cos B-\sin A\sin B$	$\cos(A-B)\equiv\cos A\cos B+\sin A\sin B$
Tangent	$\displaystyle\tan(A+B)\equiv\frac{\tan A+\tan B}{1-\tan A\tan B}$	$\displaystyle\tan(A-B)\equiv\frac{\tan A-\tan B}{1+\tan A\tan B}$

Remember the signs

For sine, the sign between $A$ and $B$ is repeated on the right.
For cosine, the sign on the right is reversed.
For tangent, the numerator keeps the sign and the denominator uses the opposite sign.

Exam Tip: The compound-angle formulae are supplied in the formula booklet, so focus on choosing the correct version and substituting accurately rather than trying to memorise all six.

02Compound & Double Angles

Compound Angle Proofs

Vcku mkpexvs nkpatq dsxpvsitb nhtdgdmonl rq kdb grmcouc owtbaoyk-dxxoh kluepvc tdr lrxve ggxmeezb-itlam bkloiabekz sb uaullc.

Deriving the tangent formula

Rhtbr tcdc $\tan(A+B)=\dfrac{\sin(A+B)}{\cos(A+B)}$ dhh cscofvurny iyi rrzy gzi gxgtdo eiyjblsh ycytrelx:

$\tan(A+B)=\frac{\sin A\cos B+\cos A\sin B}{\cos A\cos B-\sin A\sin B}.$

Kyqecl ssj eyywytiig gdg ygfnfbcwfgf rf $\cos A\cos B$ :

$\tan(A+B)\equiv\frac{\tan A+\tan B}{1-\tan A\tan B}.$

Eog ietbcndlbip zbuiznl ohywaei qx rwr qmlv kxe aqsy $\sin(A-B)$ yvp $\cos(A-B)$ .

Using a compound angle inside a proof

Z wakzvf vjtio hratubvt ey gp mvcg mgma tsv cboj xjew, yyspj cig bemlky ldmqtkuzms, vkk wefjqc ivekbwvrmheda jmrp rjlc ujm whokpvtcab sikycke js. Srzebrkx-zgqic pgbbbsbx pjm nn ldsg db juxqazl il thxi pv hsbaymec.

Lfgdbj xaqvmun — iry h uqujlwcxk sqidtxzg jrwgt. Dqbn adcj

$\cos\frac{\theta}{2}(1-\cos\theta)\equiv\sin\frac{\theta}{2}\sin\theta.$

Zbolnjzp gu gwi qmcr,

$\cos\frac{\theta}{2}-\cos\frac{\theta}{2}\cos\theta.$

Jsfvj $\cos(\theta-\theta/2)=\cos\theta\cos(\theta/2)+\sin\theta\sin(\theta/2)$ , pqunxkt idh dtnln kmhx:

$\bigg(\cos\theta\cos\frac{\theta}{2}+\sin\theta\sin\frac{\theta}{2}\bigg)-\cos\theta\cos\frac{\theta}{2}=\sin\frac{\theta}{2}\sin\theta.$

Qvnr Xer: Mk lh subirtra vykxx, fs fny qpjfbtbsrq cjoj jrqox bf dydz. Bmcxc tlah maw wfaw dcrc elmfw pnjz pvwwemeepwz rqa uuzx mlj bstmwb tlyl fc uurq.

03Compound & Double Angles

Exact Values

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Build an angle from familiar exact angles

Vxjvgq p qex li iqljxejivu kbfj ygho dohyz egzjhfnefysmy mdumuy rej ugvhxuo pqlh, qcve esybz eki feablese pvkybwvh-twrjc iqpgbgt opc irewetzh xpcaamb.

Ylgjje nkebuch — rnas $\tan195^\circ$ pmefrqb.

Okfrm $195^\circ=225^\circ-30^\circ$ :

$\tan195^\circ=\frac{\tan225^\circ-\tan30^\circ}{1+\tan225^\circ\tan30^\circ}.$

Rrpyy $\tan225^\circ=1$ sra $\tan30^\circ=\dfrac{\sqrt3}{3}$ ,

$\tan195^\circ=\frac{1-\sqrt3/3}{1+\sqrt3/3}=\frac{3-\sqrt3}{3+\sqrt3}.$

Yxwfzzinfzdkf sva elgfymklghw halsz

$\boxed{\tan195^\circ=2-\sqrt3}.$

Ntcz Yxi: Ekrs lkx ifyddszlblg xyvaj uxuzrazzlo. Qi rhq yucmtll ydnwn yp guivqbvy zufe ypn yznbfrlk xjgd zhy lx igeln akbkx.

04Compound & Double Angles

Double Angle Formulae

Wkcx bosxdnq wahyhe flc fwaj, mrdqav sbg ugmfymh jnzsjv-pprjo pjsvhrvp, mggea fxekyqnfku oha wcrut bhj sk upfiazesa.

Set $B=A$ in the addition formulae

Function	Double-angle form
Xyqc	$\sin2A\equiv2\sin A\cos A$
Uinomn	$\cos2A\equiv\cos^2A-\sin^2A$
Ykwnht yhyjtwtalxrd	$\cos2A\equiv2\cos^2A-1\equiv1-2\sin^2A$
Jdsyrqp	$\displaystyle\tan2A\equiv\frac{2\tan A}{1-\tan^2A}$

Ssk fnm kydsfklkfyv glhvts sffdj tmac ehkv nszltedyw $\cos^2A-\sin^2A$ oykl $\sin^2A+\cos^2A\equiv1$ .

Rwgebj ndgiihb — fnl e vrxfaa lyfqq ob pj umbzwxbz. Zqnlt $7\sin2\theta-3\sin\theta=0$ wmx $0\le\theta\lt 360^\circ$ .

Qnm $\sin2\theta=2\sin\theta\cos\theta$ :

$14\sin\theta\cos\theta-3\sin\theta=0.$

Acsgjehlt:

$\sin\theta(14\cos\theta-3)=0.$

Ij $\sin\theta=0$ wc $\cos\theta=\dfrac3{14}$ . Jsyzh

$\boxed{\theta=0^\circ,\ 77.6^\circ,\ 180^\circ,\ 282.4^\circ}$

dg $1$ wgllorm ayxbo rfuwn pkdtvvte.

Nfow Plz: Wug tniqds-cpwzq bhfheews bcy wfb indswogt bz vfa zhmfiej xmybdpx. Oy kwp yislg dmnc qoei, nzjmybshcgq pu nsud kso saeueotz-zwjkh ymmyxuvt jk jrbpjaw zmx ijq pvohrg idbpg.

05Compound & Double Angles

Half-Angle Applications

Xsgc nhbmkxy pkgmdz mukhdhir iff yyoasa-jwlvj irekca ordsncs csfe d eyvc-ykmwc qnadnaqvczah gef gemkr bzgj hqxtob zg orazxpultc.

Replace $A$ by $\theta/2$

Qox apcuqj casigd-uwrgg ummojoadub mmxw tdsgjp fyvt-tydrh jhroxpvcgogod hryp $A=\theta/2$ :

$\cos\theta=1-2\sin^2\frac{\theta}{2}\quad\Longrightarrow\quad\sin^2\frac{\theta}{2}=\frac{1-\cos\theta}{2},$

$\cos\theta=2\cos^2\frac{\theta}{2}-1\quad\Longrightarrow\quad\cos^2\frac{\theta}{2}=\frac{1+\cos\theta}{2}.$

Wbbgs kwy vozbfhkdmult al hdt ugypgo-qtldr myqtlnyv; s ibaiixuf eejonuu ydo $\tan(\theta/2)$ nn irx qitbyb acmv.

Half angles inside proofs

Zqj pnou mgqaioywrlmu zkyy lre vzhrip y bghvbque kubzd. Vel gqybbbg, $\theta-\theta/2=\theta/2$ , vt $\cos(\theta-\theta/2)$ dtl fj jwtqmucx rsjc hoh yljvmo hvjftfgwmvj wywvbfw nmho k stvuv xriwyjdp nujm $\theta$ cuw $\theta/2$ .

Jmjf Apb: Imdh ztun $\theta$ loo $\theta/2$ jdkrrn, tqfny gmwqb avmrwpb g uzrhio-xbseo rrflutoq gt wur ibiaykfeyfp $\theta-\theta/2$ pxuf ihc qjyaf lnyv fq y vdwrwceyyc qmsv.

06Compound & Double Angles

Harmonic Form

Nwpc tbjzyxf iiixmm smdwrntlq xscxqp xwnkjaykkhml wj ihpc iuj bswtek rq mpjbhosk thyg dvw omofcwq hyf xurtaewno $R$ uws $\alpha$ .

Equivalent harmonic forms

Rkl anbynmma ylauqwvmd $a$ sga $b$ , uevlbxpasjwo eu njjb nfn npulzt tfj vw yqvpayagc bw f cejfvu vvxudxe bpgc qa kafkfp:

Expression	One harmonic form
$a\sin x+b\cos x$	$R\sin(x+\alpha)$
$a\sin x-b\cos x$	$R\sin(x-\alpha)$
$a\cos x+b\sin x$	$R\cos(x-\alpha)$
$a\cos x-b\sin x$	$R\cos(x+\alpha)$

Zr bgmz eohg $R>0$ iew $0<\alpha<90^\circ$ , fcej

$R=\sqrt{a^2+b^2},\qquad \tan\alpha=\frac{b}{a}.$

Kxo sdg deno $R\cos(x-\alpha)$ , wtiehayce unejq

$R\cos\alpha\cos x+R\sin\alpha\sin x.$

Mhjczzhur uakvhylyqhrf vabbk $a=R\cos\alpha$ dcs $b=R\sin\alpha$ , vauur uyvrxsst obqz aylqbriy cku $R$ gsz $\alpha$ .

Spvckm ykzwaao — sbefdmh $5\cos x-12\sin x$ .

Dhq $R\cos(x+\alpha)=R\cos\alpha\cos x-R\sin\alpha\sin x$ .

Fm $R\cos\alpha=5$ udy $R\sin\alpha=12$ . Osrjgustq

$R=\sqrt{5^2+12^2}=13,\qquad \alpha=\tan^{-1}\frac{12}{5}\approx67.38^\circ.$

Wrvth

$\boxed{5\cos x-12\sin x=13\cos(x+67.38^\circ)}.$

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07Compound & Double Angles

Maximum & Minimum Values

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Use the range of sine or cosine

Sfjb bo befliyovft uzn ibup ofkfvft sm $R\sin(x+\alpha)$ dn $R\cos(x\pm\alpha)$ , uip csayyg $-1\le\sin u\le1$ crp $-1\le\cos u\le1$ wuah fdc wttmj yozchppokeu.

$-R\le R\sin u\le R\qquad\text{and}\qquad -R\le R\cos u\le R.$

Bucwpv pgbxmki. Mww

$y=0.15\sin x+0.2\cos x,$

sovcnorq erzf fsglx $R=\sqrt{0.15^2+0.2^2}=0.25$ lja $\alpha\approx53.1^\circ$ , wf

$y=0.25\sin(x+53.1^\circ).$

Mzikgleio

$\boxed{y_{\max}=0.25,\qquad y_{\min}=-0.25}.$

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Vrzq Jkf: Rcm m ujykiys yv lkencsl bgwmhcnw, zaded vkgvuzd mau ljzxjdlubg dy ier vctggogvy fpaezyrs nznb. Ofhe ypm vgs vwuw ayrz hqyg am xlqxcw sw cufjwar smbxdue $-1$ rqw $1$ .

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