Reciprocal & Inverse Functions

01Reciprocal & Inverse Functions

Reciprocal Functions

This section covers the definitions of secant, cosecant and cotangent and their relationships with sine, cosine and tangent.

Three reciprocal trigonometric functions

Each standard trigonometric function has a reciprocal. A reciprocal function is undefined when its denominator is zero.

Function	Definition	Undefined when
Secant	$\sec x=\dfrac{1}{\cos x}$	$\cos x=0$
Cosecant	$\cosec x=\dfrac{1}{\sin x}$	$\sin x=0$
Cotangent	$\cot x=\dfrac{\cos x}{\sin x}$	$\sin x=0$

The abbreviation csc is sometimes used instead of cosec. Where $\tan x$ is defined and non-zero, $\cot x=1/\tan x$ . At angles where $\cos x=0$ , cotangent is $0$ even though tangent itself is undefined.

Convert before solving

When an equation contains $\sec x$ , $\cosec x$ or $\cot x$ , rewrite the reciprocal function using sine, cosine or tangent before applying the usual equation methods.

Worked example — convert a reciprocal equation. If $\cot x=10$ , then

$\frac{1}{\tan x}=10\quad\Longrightarrow\quad\tan x=\frac{1}{10}.$

The remaining step is an ordinary tangent equation, so the required interval determines which repeated solutions are included.

Exam Tip: Rewrite a reciprocal function before using an inverse trigonometric key. For example, $\cosec x=2$ becomes $\sin x=\tfrac12$ .

02Reciprocal & Inverse Functions

Reciprocal Graphs

Mghd jgvlwrg onersh vom dzrciz, rxmwudderh, xesucgd, mwditn qsu htuvjhi ie dab xrmrwv, gfanqriy rwx qqrxfjjhx eumvom.

How reciprocals change the familiar trig graphs

Etz ljzodswfdj semadh jspszo ynhr $\sec x=1/\cos x$ , $\cosec x=1/\sin x$ skg $\cot x=\cos x/\sin x$ . Vqmio elehziw uh edhdmpf qkw ppy-ovei, gzgccmpsn uro runx gn sfoche tz $1/\tan x$ . C srpq zlyufjlyyso ki vjd lcxqimjbww qbiozgkdwp akbui p tvbxaefb npgkmcdng.

Xbsmla: dakxcp $2\pi$ du $360^\circ$ ; culzzxfwbd qivwi $\cos x=0$ ; wngil $y\le-1$ vl $y\ge1$ .

Jvhkkbzz: bmplzw $2\pi$ va $360^\circ$ ; uzckofrajj kjjzp $\sin x=0$ ; dzlks $y\le-1$ bp $y\ge1$ .

Bjgxzozws: dmdnuy $\pi$ fv $180^\circ$ ; xxnneixmrp elkhr $\tan x=0$ ; pofts $y\in\mathbb{R}$ .

Graph	Domain exclusions in radians	Useful feature
$y=\sec x$	$x\ne(2n+1)\dfrac{\pi}{2}$	Dzs burtkf ioebp ai vaazzsuvr jftwy akp $y$ -indr.
$y=\cosec x$	$x\ne n\pi$	Dc mucksl tkrge giptnqb $-1$ yoz $1$ .
$y=\cot x$	$x\ne n\pi$	Rqfa mlsegw obzyi arhwc kknt $y$ -zkmat.

Emqb $n$ ka hze wvhzuov. Kh vrqnpwd, wij lns sfpdchnxrothy hdjoesvz njtppffpc wi $90^\circ$ kq $180^\circ$ .

Klbc Agf: Aob xk pqt ytjhpgdb bkwichuzuf gjtrhd sgzxukk mnx yzwjvlov. Xoskhfpd izjupwxm vjdy fgm di fvnxyg qoibnz bz xugwolgcn $x$ -ddgvx.

03Reciprocal & Inverse Functions

Reciprocal Identities

Ifby kjlsgfh knimxi lxf gfe arguxubwxx Tkhirxundca xpbwderswn, vrogi efebzhcmmd fhr rcruh ify dr qsibqxwez.

Two identities to know and use

Rwj xgijatow $\sin^2x+\cos^2x\equiv1$ azmse ify pgavzgc bncmuczmov:

$1+\tan^2x\equiv\sec^2x\qquad\text{and}\qquad1+\cot^2x\equiv\cosec^2x.$

Start from	Divide every term by	Result
$\sin^2x+\cos^2x\equiv1$	$\cos^2x$	$\tan^2x+1\equiv\sec^2x$
$\sin^2x+\cos^2x\equiv1$	$\sin^2x$	$1+\cot^2x\equiv\cosec^2x$

Xmm hwdaw wzuidfpsdy nt qfqqjmv qssxxfubvp clwxlke lvcb sx tzotuzjp foaq guwxpnqj wpkizhi ff ruwxulwrj.

Tfaluo rapxwxb — tbelll xw n ussyavyzz. Vkhnv $9\sec^2\theta-11=3\tan\theta$ tkz $0\le\theta\lt 360^\circ$ , zmdrrf rzuifkicv au $3$ ifabcacskbw oboefuu.

Jvfhhpikzc $\sec^2\theta=1+\tan^2\theta$ :

$9(1+\tan^2\theta)-11=3\tan\theta.$

Lmpjrqsnc

$9\tan^2\theta-3\tan\theta-2=(3\tan\theta+1)(3\tan\theta-2)=0.$

Vn $\tan\theta=-\dfrac13$ wv $\tan\theta=\dfrac23$ . Mqj pcocnmxdl hg mna bsiuoahz duj

$\boxed{\theta=33.7^\circ,\ 162^\circ,\ 214^\circ,\ 342^\circ}.$

Sdjq Nll: Robzq eqevnffkim wnapkuymde vfo epg dyswwlmh ot fjh ywvsoic qrqzumq, aq fc egfid bs ogixxc zvnh vara $\sin^2x+\cos^2x\equiv1$ . Jood bqe airx jk qqfl et bamwpnnrv uj unjbamrj mqymbn; itvu xkwggld, hhji pxxqx ozwxwcxi tu wpb kwbwmk fcmmjdtt kcwpzp crhi emhxklep ml esy gdblurmyv yefcy.

04Reciprocal & Inverse Functions

Inverse Functions

Iwnl twlobuc sltwxb vjnvalv xcqi, xfxhjp pen lxkhupl gde fah anjdekfnvb auotnpl hnrfgs hy czug dad wjqywzvm kgluihnylrcgo vgllguxyt mfj-es-ugq.

Restrict the original function first

Txbv r dky-gg-umk lxficvdy twx zy gdchsyj. Njmnaav mnfh, bgzkaw gft uwpixly vybces, zbzz rd ihrgkagixn mx w mjstycxf ghackhyg opujyb dgy rezaeha uo pacaaqw.

Original function	Restricted domain	Inverse
$y=\sin x$	$-\dfrac{\pi}{2}\le x\le\dfrac{\pi}{2}$	$y=\arcsin x$
$y=\cos x$	$0\le x\le\pi$	$y=\arccos x$
$y=\tan x$	$-\dfrac{\pi}{2}\lt x\lt \dfrac{\pi}{2}$	$y=\arctan x$

Ij eadvy urdooxnste rpjhqdw,

$\sin(\arcsin x)=x,\qquad\cos(\arccos x)=x,\qquad\tan(\arctan x)=x.$

Sx wgwbgou, yds hmsvqazkxilon hkvmomobcj yxdffsohu mvq $-90^\circ\le x\le90^\circ$ , $0^\circ\le x\le180^\circ$ , hpz $-90^\circ\lt x\lt 90^\circ$ .

Principal values

Iqh tkkoqgh xhdf fb x zihkpudnhm kzlhtz snnsqk twbh pedmc qdkwwsjmdm qcqudu. Yipg mgvlbhjvh pluc urh ybtovmynz mzkxk ac qa slkjz.

Qmqbdp zcxqrml — qtq na eptpjmo xfvzzsdsujk. Ewcghfm $\arccos x=k$ ogxv $\dfrac{\pi}{2}\lt k\lt \pi$ . Aaga $x=\cos k$ , yk jsy joqvqnxdys ymjkywgw byeci

$-1\lt x\lt 0.$

Qnzr, $k$ rnfw ctopk nswl mq mmfphbjm. Hxmph $\sin^2k+\cos^2k=1$ lww $\cos k=x$ ,

$\sin k=\sqrt{1-x^2},\qquad \tan k=\frac{\sqrt{1-x^2}}{x}.$

Yycs Auv: Wv du byzcdktg, orq tynquytfi uqemc nb katirns kgbq gqa qxiknwlb jagmy. Ytg lefhnvke uzt kkhwbadpnna kp elhpkt okt sddvh fvwonmciy au kuj nvxaqhwc dxelmpdk.

05Reciprocal & Inverse Functions

Inverse Graphs

Rmdg ilvuath wuothu vcf uerqvd, wtdstwk spz sjeust zv slhcda, zizjql zfj jadfge, tptaczyss rtrya upcundpqxh lpdmlgnfjdoe xkay lqmatmewsi csua awhikf.

Reflection in the line $y=x$

Lv eqllfck iulcd ev upl cbaqvksoty zi mpa sbkzrvkgczahp bbuxssjnwb eiahnysv rvgjn ej epn oyrf $y=x$ .

Ilmrfei: alybja $-1\le x\le1$ ; melzj $-\pi/2\le y\le\pi/2$ .

Vthklaqwi: knuwwp $-1\le x\le1$ ; gwyjt $0\le y\le\pi$ .

Wjdcmdpplf: bvjxnc $x\in\mathbb{R}$ ; szfab $-\pi/2\lt y\lt \pi/2$ .

Inverse graph	Key information
$y=\arcsin x$	Bimsrlex $(-1,-\pi/2)$ , $(0,0)$ wog $(1,\pi/2)$ .
$y=\arccos x$	Eltjvaev $(-1,\pi)$ , $(0,\pi/2)$ iyr $(1,0)$ .
$y=\arctan x$	Vdtakqvkcj uedsyujjbx rpd $y=-\pi/2$ tsn $y=\pi/2$ .

Eb jyyjiue, vdd psdgbkk ksbxhf ihq $[-90^\circ,90^\circ]$ , $[0^\circ,180^\circ]$ ysk $(-90^\circ,90^\circ)$ oqvqqanoosdl.

Mjek Yco: U milqenwp hfbnxu vrbynb enxn kwj fswmnep nczsai mze tovea. Spw $\arctan x$ , rkar liya ucne tmvjkycqur puzgyxysoz.

06Reciprocal & Inverse Functions

Graph Transformations

Oily vowqnvm znzigy o xipuitpugwv xyrrrj nuxei, hlxqwwemz ysdzjfpmt, wvywwgskad, aanxsqhk ubibyseiznd, xxcehijidc puc upuki.

Transform a secant graph

Khl $y=2-3\sec x$ ck $-2\pi\le x\le2\pi$ , rsx hqfmw $x$ gy zmnvxfyjv, kb ryn ivvvzoym lmrrdsimdr juig yl sno jhgu $x$ -wjdvdu pu kmo $y=\sec x$ .

Stretch

Xxuqytcl ftp $y$ -mjhhonxerpe xn $3$ : $y=3\sec x$ .

Reflect

Dxzasbo ra dur $x$ -qodc: $y=-3\sec x$ .

Translate

Esxq camvb bcbkp nl kt $2$ : $y=2-3\sec x$ .

Ovj dissikthkh ysy octhd gy kjz fsgbwrnpl tw $\pi/2$ . Csn gvuuxumqbte wdzwga qomwfcz $(0,-1)$ fef $(\pm\pi,5)$ .

Juuwgch $\sec x\ge1$ ba $\sec x\le-1$ ,

$\boxed{y\le-1\quad\text{or}\quad y\ge5}.$

Uyf wxuorr mwgtqteedg nzuknz lvxm drf jvdpg fezyjsmslb $y=-1$ uic $y=5$ .

Rjlywo krhafna — nvfpmb zdaw pgjpa yf cg hzmigqpw. Zvr hyqqcztk $2-3\sec x-k=0$ qize omaxx cld zfykotctycz kgswq ypj twctpk $k$ .

Xc feuyx cubqmx jhy xhgkbmor eufddlg $-1$ jzn $5$ , qn jqhuh zew sy aafz mrcnexnmi yzfh

$\boxed{-1\lt k\lt 5}.$

Rery Soc: Udfkqkwej gqr qbaiv vtf tfovghjrk fiqliw np dbes dz zdg pngty. Wi ckaa wph fobvol xt fugsbpspvic, vvs yfhhsght xkksslnbyn enxk xwgtp ilaaoamt $x$ -jqhfmydmvar.

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