Lecture 6 : Inverse Trigonometric Functions Inverse Sine
Inverse Sine Function (arcsin x = sin 1x) The trigonometric function sinxis not one-to-one functions, hence in order to create an inverse, we must restrict its domain The restricted sine function is given by f(x) = 8
Restricted Sine Function
Inverse Sine Function (arcsin x = sin 1x) We see from the graph of the restricted sine function (or from its derivative) that the function is one-to-one and hence has an inverse, shown in red in the diagram below Hp 2,1L H-p 4,-1 2 L H1,p 2L H-1 2,-p 4 L-p 2-p 4 p 4 p 2-1 5-1 0-0 5 0 5 1 0 1 5 This inverse function, f 1(x), is denoted by f 1
Inverse functions - University of Arkansas
arcsin(sinx) = xif 0
Section 55 Inverse Trigonometric Functions and Their Graphs
Section 5 5 Inverse Trigonometric Functions and Their Graphs DEFINITION: The inverse sine function, denoted by sin 1 x (or arcsinx), is de ned to be the inverse of the restricted sine function
Inverse trigonometric functions (Sect 76) Review
The derivative of arcsin is given by arcsin0(x) = 1 √ 1 − x2 Proof: For x ∈ [−1,1] holds arcsin0(x) = 1 sin0 arcsin(x) = 1 cos arcsin(x) For x ∈ [−1,1] we get arcsin(x) = y ∈ hπ 2, π 2 i, and the cosine is positive in that interval, then cos(y) = + q 1 − sin2(y), hence arcsin0(x) = 1 q 1 − sin2 arcsin(x) ⇒ arcsin 0(x) = 1
Inverse trigonometric functions (Sect 76)
The derivative of arcsin is given by arcsin0(x) = 1 √ 1−x2 Proof: For x ∈ [−1,1] holds arcsin0(x) = 1 sin0 arcsin(x) = 1 cos arcsin(x) For x ∈ [−1,1] we get arcsin(x) = y ∈ hπ 2, π 2 i, and the cosine is positive in that interval, then cos(y) = + q 1−sin2(y), hence arcsin0(x) = 1 q 1−sin2 arcsin(x) ⇒ arcsin 0(x) = 1 √ 1
The complex inverse trigonometric and hyperbolic functions
2 The inverse trigonometric functions: arcsin and arccos The arcsine function is the solution to the equation: z = sinw = eiw − e−iw 2i ∗In our conventions, the real inverse tangent function, Arctan x, is a continuous single-valued function that varies smoothly from − 1 2π to +2π as x varies from −∞ to +∞ In contrast, Arccotx
Formule trigonometrice a b a b c b a c - Math
53 arcsinx+arcsiny= 2 6 6 6 4 arcsin(x p 1 y2 + y 1 x2); daca xy 0 sau x2 + y2 1;
Lecture 23: Improper integrals - Harvard University
Solution: The antiderivative is arcsin(x) In this case, it is not the point x = 0 which produces the difficulty It is the point x = 1 Take a > 0 and evaluate Z 1−a 0 1 √ 1− x2 dx = arcsin(x)1−a 0 = arcsin(1− a)− arcsin(0) Now arcsin(1 − a) has no problem at limit a → 0 Since arcsin(1) = π/2 exists We get therefore the
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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); f1(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
f1(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, f1(x) does not exist ifxis not in the range off(x);
f(f1(x)) does not exist ifxis not in the range off(x); f1(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); f1(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,px2=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. 1Exponentials 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: 2Trigonometry (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=cossinYou 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
tan2x+ 1 = sec2x;cot2x+ 1 = csc2x:
3Inverse 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 youknow 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) =: 61 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+CYou 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 21ddx 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: 8Trigonometry (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=!.