simplify cos (sin −1 x ) and tan(sin −1 x )
DESCRIPTION
To compute the derivatives of the inverse trigonometric functions, we will need to simplify composite expressions such as cos (sin −1 x ) and tan(sec −1 x ). This can be done in two ways: by referring to the appropriate right triangle or by using trigonometric identities. - PowerPoint PPT PresentationTRANSCRIPT
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sin
is not one-to-one
does not have an . inverse
y x
y
*** Our restricted domains will be the largest interval
(containing q
sin with re
uadrant I) in which is strictly monotoni
stricted domain in ,2 2
c.
x
f
x
,12
, 12
1,
2
1,2
1
12
2
x
y
siny x 1
1
y
x2
2
11
y
x
2
2
sin with
restricted domain
y x1siny x
(with the appropriate restricted domaiSine
is one-to-one and does have an in
n)
verse.
3
2
2 3
2
2
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Restricted : /2 2
1
s
/
i
: 1
n x
D x x
R y y
1sin
: / 1 1
: /2 2
D x x
R y y
x
1sin is the unique in , such that sin2 2
y x y x
1,2
1,2
11
y
x
2
2
1siny x
1. . sin 1 sin 12 2
i e
,12
, 12
1
1
y
x2
2
sin with
restricted domain
y x
Solve for xOur restricted domain of sine
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x
sin x
0 2 3 4 6 6 4 3 2
1 3
2
2
2
1
2 0
1
2
2
2
3
21
x
1sin x
1 3
2
2
2
1
2 0
1
2
2
2
3
21
2
3
4
6
0
6
4
3
2
,12
, 12
1,
2
1,2
1
12
2
x
ysiny x 1
1
y
x2
2
11
y
x
2
2
sin with
restricted domain
y x1siny x
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1sin sin ?4
1 5sin
5Expla siin why n .
4 4
1 5in
4s sin
4
4
1 2
2sin
4
If our initial is not in the restricted domain of sine, we must find an
(within the restricted domain) for which sine has an equivalent output.
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1cos is the unique in 0, such that cosy x y x
To compute the derivatives of the inverse trigonometric functions we simply need to simplify composite expressions, such as cos(sin−1 x) and tan(sec−1 x), by referring to the appropriate right triangle.
, 1
0,1 1,
1,0
1
12
2
x
ycosy x 1
1
y
x2
11
y
xcos with
restricted domain
y x1cosy x
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cosine of "the whose sine is "xSimplify cos(sin−1 x) and tan(sin−1 x).
-1 2c 1os sin cosx x
-1
2tan sin
1tanx
x
x
2 2 1b x
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1
2
1
1
1 1si
sin sin
1 1
coscos s 1in
n
dx
f x x f x g x x
xdx x
1
1
2
1 1cos cos
1 1
sinsin
1cos
1cos
f x x f x g x x
dx
dx x x
THEOREM 1 Derivatives of Arcsine and Arccosine
Derivative of an i1
'v rse'
n e g xf g x
CV
1 1
2 2
1 1sin , cos
1 1
d dx x
dx dxx x
I can't use the same ,
but I can use the same right :-)
Derivatives of Arcsine and Arccosine 1g x f x
QED
Right 's THUse and to find thM 2 . e..
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1
2
1sin
1
dx
dx x
If f (x) = arcsin(x2)
1 ' ?
2f
1 2
4
2 1 1sin '
4 4
2 11 116
15
15
1
151
156
d xx f
dx x
or for McNeal...
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1tan is the unique in , such that tan2 2
y x y x
1cot is the unique in 0, such that coty x y x
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1sec is the unique in [0, ) ( , ] such that sec2 2
y x y x
secy x
x
y y
1secy x
x
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1csc is the unique in [ ,0) (0, ] such that csc2 2
y x y x
2
2
1
1
2
2
x
111csc x cscf
y
cscy x
x
y
1cscy xx
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THEOREM 2 Derivatives of Inverse Trigonometric Functions
1 12 2
1 1
2 2
1 1tan , cot
1 11 1
sec , csc1 1
d dx x
dx x dx xd d
x xdx dxx x x x
122
3
9 6
1 3tan
1 1 23 1
3
dx
dx x x x
Day 2
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THEOREM 2 Derivatives of Inverse Trigonometric Functions
1 12 2
1 1
2 2
1 1tan , cot
1 11 1
sec , csc1 1
d dx x
dx x dx xd d
x xdx dxx x x x
1
20
0
20 0
1csc 1
1 1 1
1 1 21
1
3
x
x
x xx
ede
dx e e
e
e e
Sorry McNeal...
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The formulas for the derivatives of the inverse trigonometric functions yield the following integration formulas.
Integral Formulas
1
2
12
1
2
sin1
tan1
sec1
dxx C
xdx
x Cxdx
x Cx x
In this list, we omit the integral formulas corresponding to the derivatives of cos−1 x, cot−1 x, and csc−1 x
... are nothing more than simple transformations
of the antiderivatives shown above.
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1
2
1
2
1sin
1
sin1
dx
dx xdt
t Ct
We can use these formulas to express the inverse trigonometric functions as definite integrals. For example, because sin−1 0 = 0, we have:
1
20
sin for 1 11
x dtx x
t
Area model, in terms of .x
0C
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11
0tan
40
4x
1
20
?1
dx
x
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221
22
1
2
1sec 2 s
12 sec
11
ec 2
2
duu
u u
2 2u x du dx
1
2sec
1
dxx C
x x
Using Substitution1
21/ 2
?4 1
dx
x x
Because of our bounds,
the is not necessary.
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4 4
3 3u x du dx
222 16 4
9 16 9 1 3 19 3
x xx
0 0
2 21 1
01
1
31443 1 1
1 1sin 0
4 4 2 8
du du
u u
u
Using Substitution
1
2sin
1
dxx C
x
0
23/ 4
?9 16
dx
x
21
dx
x
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Suggested Problems 2 Days
Day 1: 5,7,17,21,27,29,33,37,41,45
Day 2: 59-107 EOO (Use integration techniques discussed
thus far in the class)