5.8 Inverse Trig Functions
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Transcript 5.8 Inverse Trig Functions
5.8 Inverse Trig Functions
Definition of Inverse Trig Functions
Function
y=arcsin x iff sin y = x
Domain
-1≤x≤1
Range
y=arccos x iff cos y = x -1≤x≤1
y=arctan x iff tan y = x -∞≤x≤∞
0≤y≤π
y=arccot x iff cot y = x
y=arcsec x iff sec y = x
Y=arccsc x iff csc y = x
-∞≤x≤∞
0≤y≤π
|x|≥1 0≤y≤π,
|x|≥1
Graphs of inverse functions
Page 381
Ex. 1 Evaluating Inverse Trig Functions
a)arcsin(-1/2)
b)arcsin(0.3)
Properties of Inverse Functions
If -1≤x≤1 and –π/2≤y≤π/2, then:
If –π/2≤y≤π/2, then
tan(arctan x)=x and arctan(tan y)=y
If |x|≥1 and 0≤y≤π/2 or π/2≤y≤π, then
sin(arcsin x)=x and arcsin(sin y)=y
sec(arcsec x)=x and sec(arcsec y)=y
Similar properties hold true for the other trig functions
Solving an Equation
arctan(2x – 3) = π/4
Ex. 3 Use Right Triangles to Solve
Find cos(arcsin x), where 0≤y≤π/2
Ex 3 cont…
Find tan arc sec 5 / 2
Ex 4 Write the expression in algebraic form
cos(arcsin 2x)
Derivatives of Inverse Trig Functions
d
u'
arcsin u
2
dx
1 u
d
u'
arctan u
dx
1 u2
d
u'
arc sec u
2
dx
| u | u 1
d
u'
arccos u
2
dx
1 u
d
u'
arc cot u
2
dx
1 u
d
u'
arc csc u
2
dx
| u | u 1
Ex. 5 Find the derivative
d
arcsin( 2 x)
a)
dx
d
arctan( 3x)
b)
dx
Ex 5. cont…
d
arcsin( x )
c)
dx
d
d) arc sec( e 2 x )
dx