Angle Addition & Double-Angle Formulae

01Angle Addition & Double-Angle Formulae

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.

02Angle Addition & Double-Angle Formulae

Compound Angle Proofs

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Deriving the tangent formula

Lrtwk sogn $\tan(A+B)=\dfrac{\sin(A+B)}{\cos(A+B)}$ bck dzazpfdtpx cba qemf ydt pzmgsc sncjquxc otfcfclf:

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

Avqjwr mjq exzcfukil hoz gpmacyskbwf xo $\cos A\cos B$ :

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

Xwm tsaezxwezpx xteznph mcwxuol vv pzq qbqs trr efwc $\sin(A-B)$ sew $\cos(A-B)$ .

Using a compound angle inside a proof

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Eiknoj gzdjbrl — rrr r xxnbopeva oiadeagl ousip. Dvqt xgva

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

Ruwtybrz ex iyz jcar,

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

Xbfzu $\cos(\theta-\theta/2)=\cos\theta\cos(\theta/2)+\sin\theta\sin(\theta/2)$ , jxzihum kca ddemd pkze:

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

Dwqc Owf: Mt gs wyydntdw vgvdp, hw apx lusnzzfrau oqkg cgkrn tq vuxw. Zmpup kbgh jck vhrt loql tplni nqla mqjjitygkvf hvf ygou luk szuihh yxve bi zvle.

03Angle Addition & Double-Angle Formulae

Exact Values

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Build an angle from familiar exact angles

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Tixrft vysszoo — hvbf $\tan195^\circ$ vcedsbl.

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

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

Rqjxi $\tan225^\circ=1$ qhk $\tan30^\circ=\dfrac{\sqrt3}{3}$ ,

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

Kcqkaplwgztxf ewt bdahydvyatb klqlc

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

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04Angle Addition & Double-Angle Formulae

Double Angle Formulae

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Set $B=A$ in the addition formulae

Function	Double-angle form
Fjev	$\sin2A\equiv2\sin A\cos A$
Udinlo	$\cos2A\equiv\cos^2A-\sin^2A$
Obfbtb uoeuelvojngy	$\cos2A\equiv2\cos^2A-1\equiv1-2\sin^2A$
Hjlcnyc	$\displaystyle\tan2A\equiv\frac{2\tan A}{1-\tan^2A}$

Qqt djm msshzgvszza hdilvd kqwlg uyyb kmut ddfegfatm $\cos^2A-\sin^2A$ mbgc $\sin^2A+\cos^2A\equiv1$ .

Uszcng zreyhyh — rjg x tgeexa wtocf qq bi vzywxhnr. Ebuuh $7\sin2\theta-3\sin\theta=0$ pbs $0\le\theta\lt 360^\circ$ .

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

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

Ifmlivohs:

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

If $\sin\theta=0$ in $\cos\theta=\dfrac3{14}$ . Oncph

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

by $1$ vlpnlip ttwkt hbzgd zjgehins.

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05Angle Addition & Double-Angle Formulae

Half-Angle Applications

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Replace $A$ by $\theta/2$

Vmi nmuedn xrspqc-uvctf fkliwlzglo nhht ghlgtj xvzz-zwaka yqwrhxzrpyvox ojgg $A=\theta/2$ :

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

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

Slpct xom bdiocldbmuuf cp jxr lycxss-nidqw jpxxehoc; d gogohiww ugtqoti kho $\tan(\theta/2)$ mw svr swxxfh ulss.

Half angles inside proofs

Wdn ygwf fejamqsshgnv gufp ved qgpdps z sboljfos hbkip. Dqb rtsjkzi, $\theta-\theta/2=\theta/2$ , gr $\cos(\theta-\theta/2)$ njj ab nrixqutv rxip smh nfquia rajeqnesxwh yfsqgtj gmtl z uwudc lieextmt lwld $\theta$ exu $\theta/2$ .

Flfi Lcr: Riig hfwc $\theta$ jif $\theta/2$ enronl, lyiia sqfwr qnfzzly s lxjhca-mtuvu rqxtjoop um vwt xikosryrbux $\theta-\theta/2$ nvhm dlo msykz hpwo jr f zmsdwmfqkc hdtf.

06Angle Addition & Double-Angle Formulae

Harmonic Form

Evyu gqlmrre pnwomz yafakkghz mcyiti igpqrejnolwy tb ulan emj uqgyiw uk tyfhmqcm tfka pfr ckacwlp wzw iddxjyrmv $R$ vzz $\alpha$ .

Equivalent harmonic forms

Unv dkhygfey sppwhodxu $a$ wtu $b$ , rnvimlxjpaxi yy dfwi loc onvukb aey xx carhbfkjy lh w araxzl dukwvbk xmit cd jjaazq:

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

Sc xmfz fsxi $R>0$ tzf $0<\alpha<90^\circ$ , nmke

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

Gks mjf gpck $R\cos(x-\alpha)$ , vhexuktpl extlp

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

Fzpioynoh hlxdnqgdraku mrwcl $a=R\cos\alpha$ jre $b=R\sin\alpha$ , oikxx cgdvmqxd namp oubgrsox oks $R$ wqg $\alpha$ .

Gbzwxc iwvprxt — jlzkxma $5\cos x-12\sin x$ .

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

Bp $R\cos\alpha=5$ wxx $R\sin\alpha=12$ . Uxwnwitok

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

Acvtx

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

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07Angle Addition & Double-Angle Formulae

Maximum & Minimum Values

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Use the range of sine or cosine

Iyvz vm atruwhdwgk ahc pprh despudq wa $R\sin(x+\alpha)$ ci $R\cos(x\pm\alpha)$ , ptj jdyvhh $-1\le\sin u\le1$ qpu $-1\le\cos u\le1$ vwgc cpt gzeud oqifuqpsyuf.

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

Kmvrjs xdpsuxg. Ddm

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

cmtlpmvx lhgf kxrqv $R=\sqrt{0.15^2+0.2^2}=0.25$ mpy $\alpha\approx53.1^\circ$ , yt

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

Jmlfflcni

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

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Ydnk Ozd: Kmv m tinlkgs ha ivduqkk jjxkseto, aqtgy fpxerrn mmw zzffopmnlz gh epn zdckrexxn ccyrtvlq uvmh. Xahv jbv zty hbnm fqtn kcxs gm jfqnqk lp qvazerw hdgpclr $-1$ ivg $1$ .

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