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Aim: Differentiate Inverse Trig Functions Course: Calculus
1.5
1
0.5
-0.5
-1
-1.5
-2
-2 -1 1 2
f x = sin x
Do Now:
Aim: How do we differentiate Inverse Trig functions?
Does y = sin x have an inverse?
arcsin sin
: 1 1
:2 2
y x iff y x
Domain x
Range y
yes, but only in restricted domain.
Aim: Differentiate Inverse Trig Functions Course: Calculus
Definition of Inverse Trig Functions
Function Domain Range
arcsin sin -1 1 -2 2
arccos cos -1 1 0
arctan tan - -2 2
arccot
y x iff y x x y
y x iff y x x y
y x iff y x x y
y x
cot - 0
arcsec sec 1 0 ,2
arccsc csc 1 - , 02 2
iff y x x y
y x iff y x x y y
y x iff y x x y y
arcsinx sin-1 xalternate notation
‘the angle whose sin is . . .’
Aim: Differentiate Inverse Trig Functions Course: Calculus
Graphs of Inverse Trig Functions
1
-1
-2
-2 2
g x = sin-1 x 3
2
1
-2 2
g x = cos-1 x
4
2
-2
-4
g x = tan-1 x Domain
-1 1
Range
-2 2
x
y
Domain
-1 1
Range
0
x
y
Domain Range
- -2 2
x y
2
2
Aim: Differentiate Inverse Trig Functions Course: Calculus
Graphs of Inverse Trig Functions
2
1
-1
-2
-2 2
y = arccsc x 3
2
1
-1
-2
-2 2
y = arcsec x
3
2
1
-1
-2
-3
-4 -2 2 4
y = arctan x
Domain
-
Range
0
x
y
Domain
1
Range
0 ,2
x
y y
Domain
1
Range
- , 02 2
x
y y
2
2
2
Aim: Differentiate Inverse Trig Functions Course: Calculus
Evaluating Inverse Trig Functions
1Evaluate: arcsin
2
‘the angle whose sin is -1/2’
sin is negative in QIII and IV
arcsin sin -1 1 -2 2
y x iff y x x y
however, restricted domain
reference angle of /6
angle y is - /6 0º 30º 45º 60º 90º
sin 0
cos 1
tan 0
1
2
1
2
1
2
2
2
2
3
2
3
2
3
3
3
1
0
UND.
Aim: Differentiate Inverse Trig Functions Course: Calculus
Evaluating Inverse Trig Functions
Evaluate: arccos0
‘the angle whose cos is 0’
reference angle of /2
angle y is /2
arccos iff cos -1 1 0y x y x x y however, restricted domain
0º 30º 45º 60º 90º
sin 0
cos 1
tan 0
1
2
1
2
1
2
2
2
2
3
2
3
2
3
3
3
1
0
UND.
Aim: Differentiate Inverse Trig Functions Course: Calculus
Evaluating Inverse Trig Functions
Evaluate: arctan 3
reference angle of /3
angle y is /3
‘the angle whose tan is ’3
however, restricted domain
arctan iff tan - -2 2
y x y x x y
0º 30º 45º 60º 90º
sin 0
cos 1
tan 0
1
2
1
2
1
2
2
2
2
3
2
3
2
3
3
3
1
0
UND.
Aim: Differentiate Inverse Trig Functions Course: Calculus
Using Inverses to Solve an Equation
Solve: arctan 2 34
x
tan arctan 2 3 tan4
x
take tan of both sides
2 3 1x inverse property
x = 2 solve for x
Aim: Differentiate Inverse Trig Functions Course: Calculus
Using Right Triangles
Given y = arcsin x, where 0 < y < /2, find cos y.
Since y = arcsin x, then sin y = x
oppositexy
1
21 x
cos cos(arcsin )y x 2.1
.
adjx
hyp
Aim: Differentiate Inverse Trig Functions Course: Calculus
Using Right Triangles
5 . 1tan tan sec
2 . 2
oppy arc
adj
Given sec 5 / 2 , find tan .y arc y
opposite
y
adjacent
Since y = arcsec x, then sec y = x
51
2
Aim: Differentiate Inverse Trig Functions Course: Calculus
Derivatives: Inverse Trig Functions
sin y x
cos 1dy
ydx
1
cos
dy
dx y
2
1
1 sin
dy
dx y
2
1
1
dy
dx x
1sindy d
xdx dx
1siny x
1
2
1sin
1
dx
dx x
2 2sin cos 1x x
differentiate
solve for dy
dx
sinx y
differentiate substitute
derivative of inverse sine
Aim: Differentiate Inverse Trig Functions Course: Calculus
Derivatives of Inverse Trig Functions
2
Let be a differentiable function of .
'arcsin
1
u x
d uu
dx u
2
'arccos
1
d uu
dx u
2
'arctan
1
d uu
dx u
2
'arccot
1
d uu
dx u
2
'arcsec
1
d uu
dx u u
2
'arccsc
1
d uu
dx u u
Aim: Differentiate Inverse Trig Functions Course: Calculus
Model Problems
2
. arcsin 2
. arctan 3
. arcsin
. arcsec x
da x
dx
db x
dx
dc x
dx
dd e
dx
2
'arcsin
1
d uu
dx u
u = 2x
2 2
2 2
1 41 2 xx
2
'arctan
1
d uu
dx u
u = 3x
2 2
3 3
1 91 3 xx
1 2
2 2
1 / 2 1
21
x
x xx
2
'arcsin
1
d uu
dx u
u = x1/2
2
'arcsec
1
d uu
dx u u
u = e2x
2
22 2
2
1
x
x x
e
e e
4
2
1xe
Aim: Differentiate Inverse Trig Functions Course: Calculus
Model Problems
1 2
1
1 2
. sin
. tan 5
. sec
da x
dx
db x
dx
dc x x
dx
2
'arcsin
1
d uu
dx u
u = x2
2 42
2 2
11
x x
xx
2
'arctan
1
d uu
dx u
u = 5x
2 2
5 5
1 251 5 xx
2 2
2 4 3 2
2 1
1
2 1
2 1
x
x x x x
x
x x x x x
u = x2 - x
2
'arcsec
1
d uu
dx u u
Aim: Differentiate Inverse Trig Functions Course: Calculus
Elementary Functions
Elementary functions have proven sufficient to model most phenomena in
physics, chemistry, biology, engineering, economics, etc. An elementary function is a function from the following list or one that can be formed as a sum, product, quotient
or composition of functions in the list.
Algebraic
Polynomial
Rational
Functions w/radicals
TranscendentalLogarithmic
Exponential
Trigonometric
Inverse Trigonometric
Aim: Differentiate Inverse Trig Functions Course: Calculus
Using Right Triangles
0º 30º 45º 60º 90º
sin 0
cos 1
tan 0
1
2
1
2
1
2
2
2
2
3
2
3
2
3
3
3
1
0
UND.