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
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Page 381
Ex. 1 Evaluating Inverse Trig Functions
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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