Inverse functions

arcsin(x), a set typically with an infinite number of angle values, and Arcsin(x), a specific representative angle from that set They use the “small a” notation, arcsin(x), to mean the one principal value Similarly for Arccos(x)and Arctan(x)

ChapitreVFonctions arcsin arccos arctan 1 Déﬁnitions

cours du mercredi 1/3/17 ChapitreVFonctionsarcsin; arccos; arctan 1 Déﬁnitions 1 1 arcsin Proposition1 1 La fonction sin : [ ˇ=2;ˇ=2] [ 1;1] est une bijection

Inverse functions

lnbdoes not exist if b 0; arcsin(x) does not exist if jxj>1: Similar facts hold for arctan, arccos and so on By the end of your rst calculus course, you should be able to compute the derivative of an inverse

Lecture 17: Implicit di erentiation - Nathan Pflueger

on calculators by sin 1;cos ;tan 1, and they are often called in other places by the names arcsin;arccos;arctan (there are also, of course, inverse functions of sec;csc, and cot, but we won’t discuss these as much) In these cases, ipping the graph of the original functions give plots that have many yvalues of each xvalue, so there

Exo7 - Cours de mathématiques

ter à notre catalogue de nouvelles fonctions : ch,sh,th,arccos,arcsin,arctan,argch,argsh,argth Ces fonctions apparaissent naturellement dans la résolution de problèmes simples, en particulier issus de la physique

Conseils de travail pour les vacances de Toussaint

i) Tracer sur un m^eme gure les graphes de Arcsin et Arccos Que dire? ii) Donner une relation simple entre Arccos(−x) et Arccos(x) pour tout x∈[−1;1] 3) Etude de la fonction Arctan a) Etude seulement a partir de la d e nition comme \fonction r ecip "

Planche no 13 Fonctions circulaires réciproques : corrigé

Arctan a+b 1−ab si ab < 1 Arctan a+b 1−ab +π si ab > 1 et a > 0 Arctan a+b 1−ab −π si ab > 1 et a < 0 Exercice no 3 Pour x réel, on pose f(x)= Z sin2x 0 Arcsin √ t dt+ Z cos2 x 0 Arccos √ t dt La fonction t 7→ Arcsin √ t est continue sur [0,1] Donc, la fonction y 7→ Z y 0 Arcsin √ t dt est déﬁnie et dérivable sur

˘ ˇ - melusineeuorg

Title (Microsoft Word - 12 Fonctions circulaires r\351ciproques doc) Author: Ismael Created Date: 4/8/2006 7:31:40

Cours de mathématiques MPSI - AlloSchool

La fonction f: x 7arccos(cos(x)) n’est pas l’identité, elle est 2 - périodique et paire, il sufﬁt donc l’étudier sur [0; ] intervalle sur lequel f ( x ) ˘ x Attention

Fonctionsusuelles - GitHub Pages

©LaurentGarcin MPSILycéeJean-BaptisteCorot Fonctionsusuelles 1 Fonctionslogarithme,exponentielleetpuissances 1 1 Fonctionlogarithmeetexponentielle

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Inverse functions

Letf(x) be a one-to-one function, sof(x) =f(y) means thatx=y. For example, letf(x) =x3. Suppose thatf1(x) is the inverse off(x). Iff(x) =x3, thenf1(x) =3px. You should know these three inverse formulas:f(f1(x)) =xfor allxin the range off(x); f

1(f(x)) =xfor allxin the domain off(x);

Ifx=f1(y);thenf(x) =y:

Iff(x) =y;thenx=f1(y):

You should also know that

1(x) doesNOTmean1f(x):

It is also true (although you don't need to memorize this) that the range off1(x) equals the domain off(x), and the domain off1(x) equals the range off(x). Furthermore, f

1(x) does not exist ifxis not in the range off(x);

f(f1(x)) does not exist ifxis not in the range off(x); f

1(f(x)) does not exist ifxis not in the domain off(x):Now, letf(x) be a function which is not one-to-one, such asx2or sinx. Letf1(x) beaninverse to

f(x). Again, you should know the four inverse formulas f(f1(x)) =xfor allxin the range off(x); f

1(f(x)) =xfor allxin the range off1(x);butnotfor allxin the domain off(x);

Ifx=f1(y);thenf(x) =y:

Iff(x) =y;andxis in the range off1, thenx=f1(y):

For example,

(px)2=xfor allx0;(px)2does not exist ifx <0,px

2=jxj;which equalsxif and only ifx0.

Ifx=py;thenx2=y:

Ifx2=y;thenx=py;sox=pyprovidedx0.

You should also recall that such functions have more than one inverse. For example, iff(x) =x2, then g(x) =pxandh(x) =pxare both inverses off(x).

It is also true (although you don't need to memorize this) that the domain of the inversef1(x) is equal

to the range off(x); however, it is no longer true that the range off1(x) is equal to the domain off(x).We can summarize important properties of lnxand the inverse trigonometric functions using these facts.

e lnb=bifb >0;sin(arcsin(x)) =xif1x1; lneb=bfor allb;arcsin(sin(x)) =xif2 x2 e x=yif and only ifx= ln(y);ifx= arcsin(y) then sin(x) =y; if sin(x) =yand2 x2 thenx= arcsin(y); lnbdoes not exist ifb0;arcsin(x) does not exist ifjxj>1:

Similar facts hold for arctan, arccos and so on.

By the end of your rst calculus course, you should be able to compute the derivative of an inverse function using implicit dierentiation. 1

Exponentials and Logarithms

You should recall some basic properties of exponentials: a b+c=abac; abc=aba c;(ab)c=abc;(ab)c=acbc;ab c=acb c; a1=n=npa; a b=1a b; a1=1a ;ifa >0 thena0= 1 andax>0 for allx: Letabe any number with 0< aanda6= 1. You should remember the denition of logaand the relationships between an exponential and a logarithm: a logab=bfor allb >0;logaab=bfor allb; ax=yif and only ifx= loga(y):

In particular,

e lnb=bfor allb >0;lneb=bfor allb; ex=yif and only ifx= ln(y):

You should know these properties of logarithms:

log abc=clogab;loga(bc) = logab+ logac;logabc = log ablogac;loga1b =logab; log aa= 1;loga1 = 0;logablog ac= logcb;logaxdoes not exist ifx0: You should be able to use these properties to solve equations. For example, if you know that lnw= lnt+c, you should be able to deduce thatw=elnt+c=ec=t: By the end of your rst calculus course, you should know that ddx ex=ex;ddx lnjxj=1x ddx ax=ddx exlna=axlna;ddx logajxj=ddx lnjxjlna=1xlna: 2

Trigonometry (precalculus)

You need to understand radians. Recall that

2 radians = 90= a right angle; radians = 180= a straight line;

2radians = 360= a full circle:

You should know the denitions of the trigonometric functions in terms of a right triangle:adj opp hypsin() =opphyp sec() =hypadj =1cos cos() =adjhyp csc() =hypopp =1sin tan() =oppadj =sincoscot() =adjopp =1tan=cossin

You should also know the denitions of the trigonometric functions in terms of the unit circle:(x;y)rsin() =yr

sec() =rx =1cos cos() =xr csc() =ry =1sin tan() =yx =sincoscot() =xy =1tan=cossin (You need these to dene trig functions of negative angles and angles greater than=2.) In particular, you should be able to nd the values of the trig functions evaluated at multiples of 2 sin0 = 0;sin2 = 1;sin= 0;sin32 =1; cos0 = 1;cos2 = 0;cos=1;cos32 = 0:

You should be able to sketch the graphs of all six trig functions. These graphs are provided later in this

document.

You should know the following basic identities:

sin(x) =sinx;sin(x+ 2) = sinx; cos(x) = cosx;cos(x+ 2) = cosx: You should be able to combine all of the above, to deduce (for example) that csc 32
=1 or tan(x+ 2) = tanx:

You should know the Pythagorean identity cos

2x+ sin2x= 1 and should be able to deduce that

tan

2x+ 1 = sec2x;cot2x+ 1 = csc2x:

Inverse trigonometric functions (precalculus)

You should be able to sketch the graphs of the inverse trigonometric functions. These are provided on

the next page. Notice the ways in which they are related to the graphs of the trigonometric functions.

You should know the ranges of the trig trigonometric. Again, you can probably gure these out if you

know what the graph looks like. Notice that the range of a trigonometric function is equal to the domain of

the corresponding inverse trigonometric function. sin and cos have range [1;1];arcsin and arccos have domain [1;1]; sec and csc have range (1;1][[1;1);arcsec and arccsc have domain (1;1][[1;1); tan and cot have range (1;1);arctan and arccot have domain (1;1): You should know the denitions and basic identities for the inverse trig functions: sin(arcsinx) =xor does not exist; arcsin(sinx) =xif 0< x < =2; If 0< x < =2;then sin(x) =yif and only ifx= arcsin(y);

(These identities hold with sin replaced by any of the six trig functions. They are also true for some values

ofx=2 orx0, depending on the trig function.)

You need to be able to go from equations in terms of trig functions to equations in terms of inverse trig

functions. For example, if I tell you that tan 6 =p3 2 , you need to be able to deduce that arctanp3 2 =6 You should be able to deduce the following identities and values: arcsin(x) =arcsinx;arctan(x) =arctanx;arcsin1 =2 ;arcsin0 = 0; arcsin(1) =2 ;arctan0 = 0;arcsec1 = 0;arcsec(1) =: 6

1 0 1=20=2arcsin(x)1 0 10=2arccos(x)321 0 1 2 30=2arcsec(x)321 0 1 2 3=20=2arccsc(x)321 0 1 2 3=20=2arctan(x)321 0 1 2 3=20=2arccot(x)or

321 0 1 2 3=20=2arccot(x)7

Trigonometry (calculus)

By the end of your rst calculus course, you should know (or be able to compute from the quotient rule)

the derivatives of all six trigonometric functions: ddx sinx= cosx;ddx tanx= sec2x;ddx secx= secxtanx; ddx cosx=sinx;ddx cotx=csc2x;ddx cscx=cscxcotx: (Notice that the \co" functions are the ones with minus signs in their derivatives.)

You should know the integrals of sinxand cosx:

Z sinxdx=cosx+C;Z cosxdx= sinx+C

You should be able to compute the derivatives of the inverse trig functions using implicit dierentiation:

ddx arcsinx=1p1x2;ddx arctanx=11 +x2;ddx arcsecx=1jxjpx 21
ddx arccosx=1p1x2;ddx arccotx=11 +x2;ddx arccscx=1jxjpx 21:

However, you need not memorize these derivatives.

The following information might also be useful (although you don't need to memorize it): Z tanxdx= lnjsecxj+C=lnjcosxj+C;Z cotxdx= lnjsinxj+C=lnjcscxj+C; Z secxdx= lnjsecx+ tanxj+C;Z cscxdx=lnjcscx+ cotxj+C: 8

Trigonometry (Fourier analysis)

At the start of a course on Fourier analysis, you should know the information outlined above, and also

should know the following.

Here are the graphs ofy= sin(3x) andy= sinx=3:23=2=20=23=22101sin(3x)654321 0 1 2 3 4 5 6101sin(x=3)You should be able to sketch the graph ofy= sin(!x) ory= cos(!x) for any positive real number!.

You should know derivatives and integrals of the trig functions: ddx sin(!x) =!cos(!x);ddx cos(!x) =!sin(!x); Z sin(!x)dx=1! cos(!x) +C;Z cos(!x)dx=1! sin(!x) +C: You should know that sin(!x) is odd and cos(!x) is even, that is, sin(!x) =sin(!x);cos(!x) = cos(!x): You should know that sinxand cosxhave period 2, and that sin(!x) and cos(!x) have period 2=!.

That is,

sin(x+ 2) = sin(x);cos(x+ 2) = cos(x);sin(!(x+ 2=!)) = sin(!x);cos(!(x+ 2=!)) = cos(!x):

You should know that

sin(n) = 0;cos(2n+ 1)2 = 0;for any integern: It may help to know that ifnis an integer, then sin(n) = 0, sin((2n+ 1)=2) = (1)n, cos(n) = (1)n, cos((2n+ 1)=2) = 0. 9

Extra trigonometry

The following information might also be useful (although you don't need to memorize it): sin(x+) =sinx;cos2 x = sinx; cos(x+) =cos(x);sin2 x = cosx: The sum-angle, double-angle and half-angle identities: sin(x+y) = sin(x)cos(y) + sin(y)cos(x); cos(x+y) = cos(x)cos(y)sin(x)sin(y); tan(x+y) =tan(x) + tan(y)1tan(x)tan(y); sin(2x) = 2sin(x)cos(x); cos(2x) = cos2(x)sin2(x) = 2cos2(x)1 = 12sin2(x); sin x2 =r1cos(x)2 cos x2 =r1 + cos(x)2

Product formulas:

sinxsiny=12 cos(xy)12 cos(x+y); cosxcosy=12 cos(xy) +12 cos(x+y); sinxcosy=12 sin(x+y) +12 sin(xy):

Euler's formula states that

e ix= cos(x) +isin(x) wherei=p1. You can derive the sum-angle identities for sine and cosine from Euler's formula cos(x+y) +isin(x+y) =ei(x+y)=eixeiy= (cos(x) +isin(x))(cos(y) +isin(y)) = cos(x)cos(y)sin(x)sin(y) +icos(x)sin(y) +isin(x)cos(y) by equating real and imaginary parts. 10

Extra inverse trigonometry

The following information might also be useful (although you don't need to memorize it): The ranges of the inverse trig functions are given by:

The range of arcsin is [=2;=2];

The range of arccos is [0;];

The range of arctan is (=2;=2);

The range of arcsec is [0;=2)[(=2;];

The range of arccsc is [=2;0)[(0;=2]:

It is sometimes convenient to dene arccot(x) to have range (0;), and sometimes convenient to dene it to

have range (=2;0)[(0;=2].

Ifx >0, then

arcsinx+ arccosx=2 ;arcsecx= arccos(1=x); arcsecx+ arccscx=2 ;arccscx= arcsin(1=x); arctanx+ arccotx=2 ;arccotx= arctan(1=x)

whenever both inverse trig functions are dened. (The rst four equations are also true forx <0; arctan(x)+

arccot(x) =2 for allxif the range of arccot is taken to be (0;), and arccotx= arctan(1=x) for allxif the range of arccot is taken to be (=2;0)[(0;=2].)

Some limits:

lim x!1arctanx=2 ;limx!1arctanx=2 ;limx!1arcsecx= limx!1arcsecx=2

We can state the exact region of validity of the formulas dening the inverse trigonometric functions:

sin(arcsinx) =xif1x1;sin(arcsin(x)) does not exist ifx >1 orx <1; cos(arccosx) =xif1x1;cos(arccos(x)) does not exist ifx >1 orx <1; tan(arctanx) =xif1< x <1; cot(arccotx) =xif1< x <1; sec(arcsecx) =xifx1 orx 1;sec(arcsecx) does not exist if1< x <1; csc(arccscx) =xifx1 orx 1;csc(arccscx) does not exist if1< x <1: The following statements are true. (Compare to the statement \ifx2=y2thenx=y.") If sinx= sinzthenx=z+ 2norx=z+ (2n+ 1)for some integern.

If cosx= coszthenx=z+ 2nfor some integern.

If tanx= tanzthenx=z+nfor some integern.

If cotx= cotzthenx=z+nfor some integern.

If secx= seczthenx=z+ 2nfor some integern.

If cscx= csczthenx=z+ 2norx=z+ (2n+ 1)for some integern. Thus, if (for example) tanx=y, thenx= arctany+nfor some integern. Because the formula for tan

and cot is so much simpler than the formulas for sin, cos, sec and csc, if you are solving a trigonometric

equation and it is important to ndallsolutions, it is often helpful to try to rewrite the equation in terms

of tangents and cotangents. 11 It's also good to be able to compute any trig function of any inverse trig function. For example, tan(arcsecx) =px

21;cos

arccot43 =45 Here's how: Draw a right triangle and let one angle be arccot 43
. Pick a side and let its length be any convenient number:3 arccot(4=3)By knowing cot arccot43 , you should be able to deduce the length of a second side:3 arccot(4=3)4From the Pythagorean theorem, you should be able to deduce the length of the third side: 3 arccot(4=3)4

5From there cos

arccot43 =45 may be easily found. 12quotesdbs_dbs21.pdfusesText_27

[PDF] Inverse functions

Inverse functions

1(f(x)) =xfor allxin the domain off(x);

Ifx=f1(y);thenf(x) =y:

Iff(x) =y;thenx=f1(y):

You should also know that

1(x) doesNOTmean1f(x):

1(x) does not exist ifxis not in the range off(x);

1(f(x)) does not exist ifxis not in the domain off(x):Now, letf(x) be a function which is not one-to-one, such asx2or sinx. Letf1(x) beaninverse to

1(f(x)) =xfor allxin the range off1(x);butnotfor allxin the domain off(x);

Ifx=f1(y);thenf(x) =y:

Iff(x) =y;andxis in the range off1, thenx=f1(y):

For example,

2=jxj;which equalsxif and only ifx0.

Ifx=py;thenx2=y:

Ifx2=y;thenx=py;sox=pyprovidedx0.

Similar facts hold for arctan, arccos and so on.

Exponentials and Logarithms

In particular,

You should know these properties of logarithms:

Trigonometry (precalculus)

You need to understand radians. Recall that

2radians = 360= a full circle:

You should know the following basic identities:

You should know the Pythagorean identity cos

2x+ sin2x= 1 and should be able to deduce that

2x+ 1 = sec2x;cot2x+ 1 = csc2x:

Inverse trigonometric functions (precalculus)

1 0 1=20=2arcsin(x)1 0 10=2arccos(x)321 0 1 2 30=2arcsec(x)321 0 1 2 3=20=2arccsc(x)321 0 1 2 3=20=2arctan(x)321 0 1 2 3=20=2arccot(x)or

321 0 1 2 3=20=2arccot(x)7

Trigonometry (calculus)

You should know the integrals of sinxand cosx:

However, you need not memorize these derivatives.

Trigonometry (Fourier analysis)

That is,

You should know that

Extra trigonometry

Product formulas:

Euler's formula states that

Extra inverse trigonometry

The range of arcsin is [=2;=2];

The range of arccos is [0;];

The range of arctan is (=2;=2);

The range of arcsec is [0;=2)[(=2;];

The range of arccsc is [=2;0)[(0;=2]:

Ifx >0, then

Some limits:

If cosx= coszthenx=z+ 2nfor some integern.

If tanx= tanzthenx=z+nfor some integern.

If cotx= cotzthenx=z+nfor some integern.

If secx= seczthenx=z+ 2nfor some integern.

21;cos

5From there cos