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Lecture 6 : Inverse Trigonometric Functions
Inverse Sine Function (arcsin x =sin1x)The trigonometric function sinxis not one-to-one functions, hence in order to create an inverse, we must restrict its domain.The restricted sine functionis given by
f(x) =8 :sinx2 x2 undened otherwiseWe have Domain(f) = [2
;2 ] and Range(f) = [1;1].Hp6,12LH5p6,12L-p-p 2 p 2 p-1.0-0.50.51.0y=sinx-p 2-p 4 p 4 p 2-1.0-0.50.51.0y=fHxLWe 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.Hp2,1L H -p 4 -1 2 LH1,p2L
H -1 2 -p 4 L-p 2-p 4 p 4 p 2 -1.5-1.0-0.50.51.01.5This inverse function,f1(x), is denoted byf1(x) = sin1xor arcsinx:Properties ofsin1x.
Domain(sin
1) = [1;1] and Range(sin1) = [2
;2Sincef1(x) =yif and only iff(y) =x;we have:
1 sin1x=yif and only if sin(y) =xand2
y2 :Sincef(f1)(x) =x f1(f(x)) =xwe have:sin(sin1(x)) =xforx2[1;1] sin1(sin(x)) =xforx22
;2 :from the graph: sin1xis an odd function and sin1(x) =sin1x:
ExampleEvaluate sin1
1p2 using the graph above.ExampleEvaluate sin1(p3=2), sin1(p3=2),
ExampleEvaluate sin1(sin).
ExampleEvaluate cos(sin1(p3=2)).
ExampleGive a formula in terms ofxfor tan(sin1(x))Derivative ofsin1x.d
dxsin1x=1p1x2;1x1:ProofWe have sin1x=yif and only if siny=x. Using implicit dierentiation, we get cosydydx
= 1 ordydx =1cosy:Now we know that cos
2y+ sin2y= 1, hence we have that cos2y+x2= 1 and
cosy=p1x2 2 and ddx sin1x=1p1x2:If we use the chain rule in conjunction with the above derivative, we get d dx sin1(k(x)) =k0(x)p1(k(x))2; x2Dom(k) and1k(x)1:ExampleFind the derivativeddx sin1pcosx Inverse Cosine FunctionWe can dene the function cos1x= arccos(x) similarly. The details are given at the end of this lecture.Domain(cos
1) = [1;1] and Range(cos1) = [0;].cos
1x=yif and only if cos(y) =xand 0y:cos(cos
1(x)) =xforx2[1;1] cos1(cos(x)) =xforx20;:It is shown at the end of the lecture that
ddx cos1x=ddx sin1x=1p1x2 and one can use this to prove thatsin1x+ cos1x=2:
Inverse Tangent Function
The tangent function is not a one to one function, however we can also restrict the domain to construct
a one to one function in this case.The restricted tangent functionis given by
h(x) =8 :tanx2 < x <2 undened otherwiseWe see from the graph of the restricted tangent function (or from its derivative) that the function is
one-to-one and hence has an inverse, which we denote byh1(x) = tan1xor arctanx:3
Hp4,1L-p
2-p 4 p 4 p 2 2-p 4 p 4 p 2 y=arctanHxLProperties oftan1x.Domain(tan
1) = (1;1) and Range(tan1) = (2
;2Sinceh1(x) =yif and only ifh(y) =x;we have:tan
1x=yif and only if tan(y) =xand2
< y <2 :Sinceh(h1(x)) =xandh1(h(x)) =x;we have:tan(tan1(x)) =xforx2(1;1) tan1(tan(x)) =xforx2
2 ;2 :Frpm the graph, we have:tan1(x) =tan1(x):Also, since lim
x!(2 )tanx=1and lim x!(2 +)tanx=1; we havelim x!1tan1x=2andlim x!1tan1x=2ExampleFind tan1(1) and tan1(1p3
ExampleFind cos(tan1(1p3
Derivative oftan1x.d
dx tan1x=1x2+ 1;1< x <1:4
ProofWe have tan1x=yif and only if tany=x. Using implicit dierentiation, we get sec2ydydx = 1 ordydx =1sec2y= cos2y:
Now we know that cos
2y= cos2(tan1x) =11+x2:proving the result.If we use the chain rule in conjunction with the above derivative, we get
d dx tan1(k(x)) =k0(x)1 + (k(x))2; x2Dom(k)ExampleFind the domain and derivative of tan1(lnx)Domain = (0;1)
ddx tan1(lnx) =1x1 + (lnx)2=1x(1 + (lnx)2)
Integration formulas
Reversing the derivative formulas above, we getZ
1p1x2dx= sin1x+C;Z1x
2+ 1dx= tan1x+C;Example
Z1p9x2dx=
Z 13 q1x29 dx=Z13 q1x29 dx=13 Z 1q 1x29 dxLetu=x3
, thendx= 3duandZ1p9x2dx=13
Z3p1u2du= sin1u+C= sin1x3
+CExample
Z 1=2011 + 4x2dx
Letu= 2x, thendu= 2dx,u(0) = 0,u(1=2) = 1 and
Z 1=2011 + 4x2dx=12
Z 1011 +u2dx=12
tan1uj10=12 [tan1(1)tan1(0)] 12 [4 0] =8 5The restricted cosine functionis given by
g(x) =8 :cosx0x undened otherwiseWe have Domain(g) = [0;] and Range(g) = [1;1].54321-1-2-3!2-3!-5!2-2!-3!2-!-!2!2!3!22!5!2We see from the graph of the restricted cosine function (or from its derivative) that the function is
one-to-one and hence has an inverse,g1(x) = cos1xor arccosx3!2!!2-!2-!-4-224fx() = cos-1x()6
Domain(cos
1) = [1;1] and Range(cos1) = [0;].
Recall from the denition of inverse functions:
g1(x) =yif and only ifg(y) =x:cos
1x=yif and only if cos(y) =xand 0y:g(g1(x)) =x g1(g(x)) =xcos(cos
1(x)) =xforx2[1;1] cos1(cos(x)) =xforx20;:Note from the graph thatcos
1(x) =cos1(x).
cos1(p3=2) =and cos
1(p3=2) =You can use either chart below to nd the correct angle between 0 and.:tan(cos
1(p3=2)) =tan(cos
1(x)) =Must draw a triangle with correct proportions:1xcos θ = xθtan(cos-1x) = tan θ = 1-x2xcos-1x = θ1-x21xcos θ = xθ7
d dxcos1x=1p1x2;1x1:ProofWe have cos1x=yif and only if cosy=x. Using implicit dierentiation, we getsinydydx
= 1 ordydx =1siny: