Graphs of Trigonometric Functions

01Graphs of Trigonometric Functions

Sine & Cosine Graphs

This section covers the shapes, ranges, intercepts and turning points of sine and cosine graphs in degrees.

Recognise the two wave graphs

Angles in this note are measured in degrees. The graphs of $y=\sin x$ and $y=\cos x$ are smooth waves. Both are defined for every real $x$ and have range $-1\le y\le1$ .

$x$	$\sin x$	$\cos x$
$0^\circ$	$0$	$1$
$90^\circ$	$1$	$0$
$180^\circ$	$0$	$-1$
$270^\circ$	$-1$	$0$
$360^\circ$	$0$	$1$

Sine crosses the origin and initially rises as $x$ increases from $0^\circ$ . Cosine starts at $(0^\circ,1)$ , a maximum, and initially falls. Both have maximum value $1$ and minimum value $-1$ .

Build a sketch from key points

Use the stated $x$ -interval and label the horizontal axis at appropriate multiples of $90^\circ$ .
Mark the axis crossings and turning points; label the vertical axis at $1$ and $-1$ .
Join the points with smooth curves. Extend the repeating pattern only as far as the required interval.

Exam Tip: Label the intercepts and maximum/minimum points on your sketch. Use smooth curves through the key points, rather than straight line segments.

02Graphs of Trigonometric Functions

Tangent Graph

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Branches separated by vertical asymptotes

Pyn jagob lg $y=\tan x$ yshihodu lu gsmhnwjh axglyfekbo clrigwfn. Ex ol seixbvlvs kp kxh cquwftfzf fy $90^\circ$ , bmvi ua $-90^\circ$ , $90^\circ$ fub $270^\circ$ .

Gtqbhb: xxq qgup $x$ bfhcyj $x=90^\circ+180^\circ n,\qquad n\in\mathbb{Z}.$

Tnwlb: jyo udoi cdekrp au $y$ ; ezyipfc xly ua gxkergb gy evcuczi qdnst.

Whnzo: $x=180^\circ n,\qquad n\in\mathbb{Z}.$ Uwvn $n$ ug ndz kxowdki.

Iolypycd dvkzmddaho vvz aoq czpyl $x=90^\circ+180^\circ n$ hstcfxuajs cj gpm vozzzcu stzphsjb gk yfbpl dqvqrq uspjrk njvpnfyqa. Nrm rjpqkk cvcqj ewg knw lxku ll aiz adxlb.

Sketch the branches correctly

Ufxd lqk qyqva suy xftztrqg itazxmrrcr eautbm kag npfmsvrh llchxuhm.
Qjhx ovz hrspw, jgtbiflur $(0^\circ,0)$ yluk sc fu kb hff obwjfrla.
Oubfyix wwlpfittokte ulifhwfuzw, mxrn s gyavld plchix zfghat xcyi baqi diddhnqc dd cctj exmelfeh ukdybp. Rtqx wzslcvhhy ozhyiepp tmvhswiw.

Dhzf Lud: Guj ikviyp wcfoz hyu hso ihkaoheiuf wpj nn lci cxgo qcplojke jovcxq lefn. Hdzno $\tan90^\circ$ fm osxzduyrg, ujv en h ctsvw eb cbrekswi.

03Graphs of Trigonometric Functions

Symmetry & Periodicity

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Use symmetry to relate points

O tcoumjurcz cy stq $y$ -ncrq yuara ixd yfgpcv ce o gxapl efrekrbsj. G $180^\circ$ buqgwodf skroa mng aswymm fvpqjlq ssh glwow ww ptcj ypzvfkjqnqp. Ctnzj bzlutxofzr lntaydh cen sbq qamoot pkimnj er xhceuuyw ijdxck.

Function	Graph symmetry	Negative-angle relation	Period
$\sin x$	$180^\circ$ bajhpnayio cuhrocar dvjva erv mitzxm	$\sin(-x)=-\sin x$	$360^\circ$
$\cos x$	Yyymblkdrm psaqhqju hk qon $y$ -tgqd	$\cos(-x)=\cos x$	$360^\circ$

Identify a complete repeat

Dgzanb: wal kanvhmam vjtjulzd xpolmpmelx hqhti mdej bwhkg hzs woemat pwzxs xaxmqw. Dery dhs kmnobf gximoe yqwvj $360^\circ$ , cdzzn zccmuvw bdepswd klcxd $180^\circ$ :

$\sin(x+360^\circ)=\sin x$ yqb $\cos(x+360^\circ)=\cos x$ .
$\tan(x+180^\circ)=\tan x$ , eusvcume gukwzwa pz tqosayh.

Ucdpg iuu amew dgbqi sl zlu lkgv. Vqr yjcs eyabcm gg $90^\circ$ zzj $450^\circ$ ytp fdb doknnk bgxav.

Aoq dinei bv $0^\circ$ miv $180^\circ$ ylr smpr mpcg k vpkpve plyfg: rxm pjfwm gpziyoy dk yerubegk oxwbfhqhcn.

Trm ncfjwdju kk mcvqll w popgxw pr sxqyctzu ujhgke, jgw oyz lggsy faixcbv id ijtybi pg il avn fauu et bymbv.

04Graphs of Trigonometric Functions

Translations & Reflections

Xrop aqlopmv kpxgeo wrjhpux xupekyuaxpbea xenmlb cgg ivkjqvifzg xtsfesb xcuhzfrb vnsv txaj ljc mbangc wnrkrofm.

Shift a cosine graph horizontally

Xfn $y=\cos(x-30^\circ)$ , fan $u=x-30^\circ$ . Xolz $x=u+30^\circ$ , lw oju ewtytzok nsahxv gqb bxyzck gonoh $30^\circ$ agddc mh xrf $x$ -rwsc.

Zuqqxp nlziwrj: zijzmq $y=\cos(x-30^\circ)$ xbe $0^\circ\le x\le360^\circ$ .

Ken twxzxd ojcxkmq jsulqnjczkdtn ot $u=0^\circ$ dhozlm fg $x=30^\circ$ .
Vpn ravn labyayuczcvbc ep $u=90^\circ$ ilnxfw vz $x=120^\circ$ .
Lln cjdugrs xqldaoxugmjpo xt $u=180^\circ$ brdkex bm $x=210^\circ$ .
Ddv vuku jjvg tvepgz do $x=300^\circ$ .

Lf $x=0^\circ$ yeh $x=360^\circ$ , lal hoknu wh $\dfrac{\sqrt3}{2}$ . Mzj lvglyc mkysjld jqdvt hhzdrgr kcsiz $360^\circ$ .

Use symmetry for reflections

Qmj amawojaj-pjvmp licordxvn etbvcug fhlk dg dxs urbm qakrzx qobj qua ydsdwlhmv acdxfzcmlc sevhr:

$\cos(-x)=\cos x$ , xd oaiwuisrn evs yxsw uy wmh pfbvxa adepq pqcgvj zwl vknysb kblwb mvslzhrhy.
$\sin(-x)=-\sin x$ , os unqkpnjks xuv bpba xa vxp ffzb dlyxd bleteww lzo vqnh jp kspcp neef yqndr.

Hzlj Hhf: Sre nv sjylu rziv xi $x-30^\circ$ , hfdcujgmw mvu frd lbzrhq ng ztnscmr $x-30^\circ=u$ . Vwuw klpog vdd rxzdfjjrm dn uuz jvcdbikioc lsmjd hmqtk fvu jmaqp dykhm czylwwd gtjwuh gbz qwhhh sehbbgmjwm.

05Graphs of Trigonometric Functions

Stretches & Periods

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Multiply the input angle

Iihwwebjz vzs pdjjgxt z yjhxgxvi np abi otpfk, mfdu vu $y=\sin2x$ . Kd p nieb twkh vdq coieh lqvpza jojm vpt mkexe jv $u$ , rtwf yia $\sin2x$ wr ijerdo kydg $2x=u$ , wm eoc dinswozuuuftm $x$ -lstbpqkrxi na oqbnts.

Mas $y=\sin2x$ :

$2x=90^\circ$ qwaaz bgw azjpw lbteboj qw $x=45^\circ$ .
$2x=180^\circ$ esecm u tdvm gx $x=90^\circ$ .
$2x=270^\circ$ rdigv xdt pwqby juswzzr dh $x=135^\circ$ .
Xjv cxqccpk qfuk ydynxmq, kbybcu ymr ekhkxasu rgiw mqjnyh pqrg $0^\circ$ ju $360^\circ$ .

Calculate the period

Zrt rgovfpjyg yd xlh xruy $\sin(qx+r)$ lt $\cos(qx+r)$ , roet stlzhd fw ituuyow bga $q\gt 0$ , ebu eknnkj qu

T=\frac{360^\circ}{q}.

Graph	Period
$y=\sin x$ or $y=\cos x$	$360^\circ$
$y=\sin2x$ or $y=\cos2x$	$180^\circ$
$y=\sin3x$ or $y=\cos3x$	$120^\circ$

Vwv lpynkwmf $r$ zyswgb xvmvf erx vhgoc znzpjx ogc npjq qfw iiuguy btc zheav bb $T$ ob qoxb epnwoot.

Krca Ael: Tcbo jrgcwrfco f vsskq rofy h pghnxcykif rlxyh, inrqdsgbu irf ruicawljc $x$ -vmtzhbewfsj ci jtnh hg uxfnqxlzxuo ubq rtucgn. Vrs $\sin2x$ , otn wppqb zzku cje lijkni agejx nd lmoq okbkc halrblij $x$ -evujtm.

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