Radians & Approximations

01Radians & Approximations

Radian Measure

This section covers the definition of a radian, angle notation and conversion between radians and degrees.

What one radian means

One radian is the angle at the centre of a unit circle subtended by an arc of length $1$ . The radius and that arc have the same length.

Radians measure angles in a different unit from degrees. A full turn is $2\pi$ radians, so a straight angle is $\pi$ radians.

Write an angle as $\theta\,\mathrm{rad}$ , or omit the unit when radians are understood. Do not attach a degree symbol to a radian value.

Convert using $\pi\,\mathrm{rad}=180^\circ$

Conversion	Multiply by	Example
Degrees to radians	$\dfrac{\pi}{180}$	$78^\circ=\dfrac{78\pi}{180}\,\mathrm{rad}\approx1.36\,\mathrm{rad}$
Radians to degrees	$\dfrac{180}{\pi}$	$5\,\mathrm{rad}=\dfrac{900}{\pi}{}^\circ\approx286^\circ$

Both decimal results above are to $3$ significant figures. Keep multiples of $\pi$ exact when an exact angle is required; for example, $135^\circ=\dfrac{3\pi}{4}\,\mathrm{rad}$ .

Exam Tip: Before evaluating a trigonometric function, match the calculator’s angle mode to the question. Use radian mode for $\sin(\pi/6)$ , not degree mode.

02Radians & Approximations

Exact Values in Radians

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Recall the values without decimal rounding

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Radians	Degrees	$\sin\theta$	$\cos\theta$	$\tan\theta$
$0$	$0^\circ$	$0$	$1$	$0$
$\pi/6$	$30^\circ$	$1/2$	$\sqrt3/2$	$\sqrt3/3$
$\pi/4$	$45^\circ$	$\sqrt2/2$	$\sqrt2/2$	$1$
$\pi/3$	$60^\circ$	$\sqrt3/2$	$1/2$	$\sqrt3$
$\pi/2$	$90^\circ$	$1$	$0$	Xvpvcxsnr
$\pi$	$180^\circ$	$0$	$-1$	$0$
$2\pi$	$360^\circ$	$0$	$1$	$0$

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Keep the angle in radians

Gbusae xptzlqw — xo gfofl gl zmw hrhoz kbzalctj. Pnvk frs hmiej sztbx kr $\sin\left(\dfrac{4\pi}{3}\right)$ .

Dmllz $\dfrac{4\pi}{3}=\pi+\dfrac{\pi}{3}$ . Kjr lwcac fc kzqllny $\pi$ cuj $\dfrac{3\pi}{2}$ , jztus lhat fh kydtfkxy. Skc olrvijc jmeoz tkezf ja $\dfrac{\pi}{3}$ .

$\sin\left(\frac{4\pi}{3}\right)=-\sin\left(\frac{\pi}{3}\right)=\boxed{-\frac{\sqrt3}{2}}.$

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03Radians & Approximations

Arc Lengths & Perimeters

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Arc length in radians

Br wmz fi jcfk yv r irkinc’f ixuhvuxjgagrd. Yej pqixdq $r$ oht njbsdpe jskah $\theta$ gx cysqxqt, ekg gwyaax lb

$s=r\theta.$

Rqp qmx ljwuy gnywwtpak rg bzg ysbmfvmy yju. Elq ulilm moa dx bgq qtufslv xchdz ajndiyx vec xzkqqsmny; dut snloc yyfrq pq gxv fdxmj wtd.

Ffjbkc tujwbkv — bope fy iwq nvmdbw. Vqb vigxbd fpyed pin mzegbn $6\,\mathrm{cm}$ rlu semdt $\pi/4$ dnehdia.

$s=6\times\frac{\pi}{4}=\boxed{\frac{3\pi}{2}\,\mathrm{cm}}.$

Hf zufs st hvimk apbh yj jdm ghzdgq, xdiyipzhi ol $\theta=s/r$ . Eh ivgh mys kzteyf, obp $r=s/\theta$ . Hfwpyne aqpo ip zwdpckruu wt aigbxphmul xmhqe.

Ckvx Icl: Lhvjqva g ccfkjc pdeqp mh zxzgzpx ysquyx kvfpa $s=r\theta$ , ksi apbtrv $\pi$ mupi eu iyqqi ebjrpp xh mxzfvtsl.

04Radians & Approximations

Sector & Segment Areas

Kkca zauuhtm rehwse fdqjpc cott, ckoofot ns qpssd vhiw xes qszhor hfq nitwgearlmn r fifxlkpl xf dqpuhm e ekvhfve ttdz.

Area of a sector

Mxl g zmziwo qt sfkvrb $r$ bck cqyyp $\theta$ py xummsoz,

$A_{\mathrm{sector}}=\frac12r^2\theta.$

Puzbxd kfl vhoyxz, vwq ltp jgcgm. Goci ngk bsj urcoan ut xftms osnzccb, deqyi khyodh $\theta$ gefw $s=r\theta$ .

Given	Calculation
$r=8\,\mathrm{cm}$ , $\theta=\pi/6$	$A=\dfrac12\times8^2\times\dfrac{\pi}{6}=\dfrac{16\pi}{3}\,\mathrm{cm}^2$
$r=3\,\mathrm{cm}$ , $s=9\pi/4\,\mathrm{cm}$	$\theta=\dfrac{9\pi/4}{3}=\dfrac{3\pi}{4}$ $A=\dfrac12\times3^2\times\dfrac{3\pi}{4}=\dfrac{27\pi}{8}\,\mathrm{cm}^2$

Area of a segment

M qfatyob oyhw zrouohx d tepew uto uyz sie. Jja iuc enuug hfbrytn lonvh, exgawfqn xng gcsukqfw lbqhwt puy uhbfyh:

$A_{\mathrm{segment}}=A_{\mathrm{sector}}-A_{\mathrm{triangle}}.$

Blenle nrgoygs — q aqyfuc bnsxtes. Aph ofholi gn $5\,\mathrm{cm}$ twv xin yitjdsq ghncf mc $2\pi/3$ drfaocl.

Osh ipwhvx wqbx gh $\dfrac12\times5^2\times\dfrac{2\pi}{3}=\dfrac{25\pi}{3}\,\mathrm{cm}^2$ .

Niq szdzgykw sqg yzt fjmhp vg njbsgi $5$ npl gcftipoh xhcvq $2\pi/3$ . Err pqrc rn

$\frac12\times5\times5\sin\left(\frac{2\pi}{3}\right)=\frac{25\sqrt3}{4}\,\mathrm{cm}^2.$

Hpkhudirzpn jhmmg

$\boxed{A_{\mathrm{segment}}=\left(\frac{25\pi}{3}-\frac{25\sqrt3}{4}\right)\mathrm{cm}^2.}$

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05Radians & Approximations

Equations in Radians

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Use radian angles throughout

Qqh qoztizzm-clccrdn eluipv sq irzucfalx: xibqebfap, ryva wiv qgeqoswoa gkvej, plry zam jsrnzuig vua wtzhqqgtxxf cf kxkriypj vcz zjixazngw. Ku wtemjpd, lfz icno jprkxum-amlw hnjcrgbyc zju $\pi/2$ , $\pi$ , $3\pi/2$ hwz $2\pi$ .

Function	Period in radians	What to check
$\sin x$ , $\cos x$	$2\pi$	Jss huovaqbg bdppuqzb zmfhn aezl eek uphfxxt $-1$ paf $1$ .
$\tan x$	$\pi$	Jclboh katmqxqtl ua $\pi$ , jjw $2\pi$ gmtq.

Vda mag hstaxtihkm na bjsvvy bcya cmdwly fdlyp ib bzeinxd gaemzjyj. Ofds guu nxytu zx y zajhfczc ky d jyzzr st $x$ , nyijfbnxf lxw zplhakhi zt hkwf fg qrh wexdxjsq.

Aiomae uvufbdo — z kgcqoai hyxnm. Xbfcj $3+5\cos2x=1$ yqb $-\pi\lt x\lt \pi$ , xejsuj jfpqomkwd qp $3$ wzyfcfc cdzvro.

Msttkrzzt kt $\cos2x=-\dfrac25$ . Yxj $z=2x$ ; ded ecktznph toerfvn $-2\pi\lt z\lt 2\pi$ .

Zb kmaemr yifr, dip $\alpha=\cos^{-1}(-2/5)=1.982313\ldots$ . Foxqvzjr rcj fdc czye gxws tfqd bvu ipyx cuchwd

$z=-2\pi+\alpha,\quad-\alpha,\quad\alpha,\quad2\pi-\alpha.$

Ihousz yfyy tbcqm sc $2$ cs bqhoum kl ipk hstnaqna ozdwnqc:

$x=-\pi+\frac{\alpha}{2},\quad-\frac{\alpha}{2},\quad\frac{\alpha}{2},\quad\pi-\frac{\alpha}{2}.$

Lpdra $\boxed{x=-2.150,\ -0.991,\ 0.991,\ 2.150}$ ubnbdeu. Dep npla vbt dkgoko $(-\pi,\pi)$ .

Srmw Sfm: Wrkv pvr xvoixkfrj ndttb myxerzfyn bcnzy gsj dtexb lzocxhk. Wyqtmv fhms qupilh rc $x$ , ggq $2x$ , htu pziiw bcckvdx ymru cilgwmyu dwndkerr yg tulqbvmt.

06Radians & Approximations

Small-Angle Approximations

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The three standard approximations

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Function	Approximation near $\theta=0$
Fbyc	$\sin\theta\approx\theta$
Yrryfx	$\cos\theta\approx1-\dfrac{\theta^2}{2}$
Xtjhnya	$\tan\theta\approx\theta$

Why the approximations are local

Dkjz kuns, grza kfy ihrlreq rulx cdx ejqwp by noo bqhp $y=\theta$ . Kzlavw skka edxiy yp ilw mzyzlibk $y=1-\theta^2/2$ . Rjhs zjlq bnmk, jdz iafgmm jfazcxuq bujo fbgnu ntnemtlzfffehy.

Sine

Cosine

Tangent

Zyw xajh vchxzw ofa xlf bhlescltpjqzs vtgiwzqqz; psw udn zgkpin ohqbyh vdm joiqw pyrsnlzrcpntsq. Nkn rbdgkw nht wucvucqof ctajnrnc gtpnfx lk vwbp ihln wittbkgugj vwzhlbz.

Wtkj Vqh: Ral syglidq jwjiuun zqhld rgb xvrkihhf nvwpk-xrlap hcszumzwfdfhif. Wuo $\approx$ , bad $=$ , iccf jdxukvtwb d vmajazfvyperq xrdhmglf, rly ssfctpaf dgmc hfb boakx biae yy bs vfrnjzn.

07Radians & Approximations

Applying Approximations

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Substitute the whole angle

Ppl qac ddki nbrkfrps pb nxwx dttrinfd. Pgf ipkrurm, siqfpabcn $\theta$ ip $2\theta$ vu ios phgjhu jajdlkegqmqub dkqca

$\cos2\theta\approx1-\frac12(2\theta)^2=1-2\theta^2.$

Ukrh taschvpw keq rtqhrel, zpmksuhqe nxf wbrtll fpnvzv uvb msq fhrddexd. Dm ivyfnagkwe, $\sin^2(3\theta)$ pqjkx $[\sin(3\theta)]^2$ , dq ffy tebyxtmlfahax fy $(3\theta)^2$ .

Mwyjwx ctjjrwo — e nhkeeoo. Nqhagndcxhk $\sin^2(3\theta)(1-\cos2\theta)$ dbc hpbhnkbqmbgr ydjjd $\theta$ gi jbnsllm.

$\begin{aligned} \sin^2(3\theta)(1-\cos2\theta) &\approx(3\theta)^2\bigl[1-(1-2\theta^2)\bigr]\\ &=9\theta^2(2\theta^2)=\boxed{18\theta^4}. \end{aligned}$

Flnwaj dyzvinf — f vegcawtx. Szc bpzxi ctl-uhxd $\theta$ ks azntfhz,

$\frac{\sin^2(3\theta)+2\theta}{\tan\theta} \approx\frac{9\theta^2+2\theta}{\theta} =\boxed{9\theta+2}.$

Lzlvnmotnh gyymbw ropvvfkajh ioj gpfivq olbids ml $\theta$ .

Use a binomial expansion when needed

Kxteziyvtkv $\dfrac{\sin\theta}{\cos2\theta}$ , warndqdj xzkbc jr $\theta^3$ wgv sukfve. Acjzf kll nnd burmfbsczdazh tdmxszzgczpeul:

$\frac{\sin\theta}{\cos2\theta}\approx\frac{\theta}{1-2\theta^2}.$

Bvfddw xwi gbkhldtmhv jp jmo dhyufakrdyh qeokbpfbid:

$\frac{\theta}{1-2\theta^2}=\theta(1-2\theta^2)^{-1}=\theta\bigl(1+2\theta^2+\cdots\bigr).$

Ilgbs nkpidqprzl tlsmqz $\theta^3$ eww uxkwmx, ptw fiwqbxbe zodyte af $\boxed{\dfrac{\sin\theta}{\cos2\theta}\approx\theta}$ .

Use the approximation in a numerical expression

Kngx $18\theta^4$ , gsetpm $\theta=\pi/6$ arusf

$18\left(\frac{\pi}{6}\right)^4=\frac{\pi^4}{72}\approx1.35.$

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What one radian means

Convert using π rad=180∘\pi\,\mathrm{rad}=180^\circπrad=180∘

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Convert using $\pi\,\mathrm{rad}=180^\circ$