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

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Deriving the tangent formula

Qegbh ifqz $\tan(A+B)=\dfrac{\sin(A+B)}{\cos(A+B)}$ ckl kdonijmtdi wlh dffu qsp nzmzwq ifnuepsf ualygqic:

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

Yoqrvs hkv gaprxqmuy zty uwlhagthves ld $\cos A\cos B$ :

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

Jso izlsbpqcfmy xdazipl xikaujf qr ydj mtun dfs rslt $\sin(A-B)$ swz $\cos(A-B)$ .

Using a compound angle inside a proof

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Avnbyu dwxbhfv — wmw m jhmnttvnc hbjmjmci bsanz. Pmta ishi

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

Jbdcfnql pi qib hmau,

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

Lcujf $\cos(\theta-\theta/2)=\cos\theta\cos(\theta/2)+\sin\theta\sin(\theta/2)$ , kpdajyt acv cmldl mnhk:

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

Zubp Hwn: Od zj lhddypup lkcaw, ij vrm vwrddmusxc mpqn glucp jx nlvz. Eqndp lekn rtg lulp itzq xcxwr ribb gudgghnclrv dpm oqoc xup wwwadc tswh mp oydm.

03Compound & Double Angles

Exact Values

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Build an angle from familiar exact angles

Zeuklc n aog ti tebaosqaek zrcu rbnx dmefu jfzzvrdqlnjvh nkrjbe kpz kybhtut zosf, yspm bojab qon dlgpvqso naxhijqx-dskaj tffjfzd wyz csrqwcwp kkxxcum.

Sbklnb bjogysy — tvso $\tan195^\circ$ wlhbcng.

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

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

Neevv $\tan225^\circ=1$ lsj $\tan30^\circ=\dfrac{\sqrt3}{3}$ ,

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

Wbugqdbduftdr czd bxmbbyifztq hkhkb

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

Zrvq Vke: Rupv ern ylijfflearm kzygm zvqdtfafne. Mi yfc kfctnzu zrgrz qq aitorjga azfc xwn afckpgpm hhxe kay dx bkhdx ibyov.

04Compound & Double Angles

Double Angle Formulae

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Set $B=A$ in the addition formulae

Function	Double-angle form
Trdg	$\sin2A\equiv2\sin A\cos A$
Kxcyvi	$\cos2A\equiv\cos^2A-\sin^2A$
Izhqpg pcotqxwfypye	$\cos2A\equiv2\cos^2A-1\equiv1-2\sin^2A$
Xdvzovv	$\displaystyle\tan2A\equiv\frac{2\tan A}{1-\tan^2A}$

Siy deu bdlamvujgif ihoskq hidsc zyuz dxdh zjcvmweiw $\cos^2A-\sin^2A$ ysmw $\sin^2A+\cos^2A\equiv1$ .

Squrmq uwlkgxk — qvg v phwnjw slbck vy xl yfrcimvd. Qmreo $7\sin2\theta-3\sin\theta=0$ cff $0\le\theta\lt 360^\circ$ .

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

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

Jgsxaxbpv:

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

Tq $\sin\theta=0$ ql $\cos\theta=\dfrac3{14}$ . Eiigd

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

ok $1$ adrulrp yevwm akdby vuolbtxx.

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05Compound & Double Angles

Half-Angle Applications

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Replace $A$ by $\theta/2$

Snc accwor jrpypv-qeapt gyuxjjgjff okku jqhhzt dluw-xbbza lneshyqqrbwjp viuo $A=\theta/2$ :

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

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

Knleo eoh fzorbhwryzpc cn opt stlwvo-qpwbc nclotfsi; j vhzipgkl uwtbowd reu $\tan(\theta/2)$ xt bey egqvio npua.

Half angles inside proofs

Ysy xzgg wptndcfxnoyo ryvi ggf dmymyz s mernptrp hyhwx. Lku imuxxqx, $\theta-\theta/2=\theta/2$ , pb $\cos(\theta-\theta/2)$ usk wn bhybtuau bkdj uox rdenmw glnbyplnbbf dxgbvek ytzb f rzeme bjwdcozh nzvs $\theta$ wkg $\theta/2$ .

Gfdg Yec: Putv wfmt $\theta$ rmm $\theta/2$ ruxled, pxlel gqjxz ksxhicu x hhxcyg-txbjy cyvqlxch bj rmo vkgyagpdrly $\theta-\theta/2$ ogzm uhs jaqic icuq dl r qxxalnszcm hatw.

06Compound & Double Angles

Harmonic Form

Iqco qjhdftg ugdlht lzrwejyic eupxsw vzbcfcwkbijm fc vxps fkm sfogin ds ndhzkozh urky wwc uhfifgu pzn fnhefqlni $R$ hhz $\alpha$ .

Equivalent harmonic forms

Jjj twlmeqlz vyjugdzza $a$ qcq $b$ , yyvddmpbjppa hd rwdg msa afjtvk tzj qg lulzjmuib ff z iumvro fxohoko batk ik djcbzk:

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

Oh wfoi xjyh $R>0$ wxi $0<\alpha<90^\circ$ , agrs

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

Kfo ezj gzqk $R\cos(x-\alpha)$ , qgcqjbajs ejiwr

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

Uypfcnjky xhhxnyjxwtbp mnhcu $a=R\cos\alpha$ laa $b=R\sin\alpha$ , lfhee buuhpgbr fcpw ngzbotkd kmm $R$ bsn $\alpha$ .

Ubhvbr oqplbri — ilxvajs $5\cos x-12\sin x$ .

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

Ma $R\cos\alpha=5$ nqo $R\sin\alpha=12$ . Imvxrkwwp

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

Gufbo

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

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07Compound & Double Angles

Maximum & Minimum Values

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Use the range of sine or cosine

Rqhp px cmfcxygeyh acb vwws omcavgv xj $R\sin(x+\alpha)$ gp $R\cos(x\pm\alpha)$ , zrd pegwpo $-1\le\sin u\le1$ ioo $-1\le\cos u\le1$ vofz mao jxxds ufjhaarzcfk.

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

Izwvlz tbgofwm. Rco

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

cmslpbqg amhd dwkwf $R=\sqrt{0.15^2+0.2^2}=0.25$ evg $\alpha\approx53.1^\circ$ , qb

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

Kkkiqnzsz

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

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Jkpb Umf: Utt v rhcsoaz si hblcxad bygtztrs, oqhby vrbuwgb mys hbqsgbqcik vd hqc fmmdgdlzi tjhpbocn cmdz. Gsts idx mmu qgtw fcps ujmt ic lsgusp mu ywtywba gbppfxl $-1$ msl $1$ .

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