5.8 Inverse Trig Functions
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Transcript of 5.8 Inverse Trig Functions
5.8 Inverse Trig Functions
Definition of Inverse Trig Functions
Function Domain Range y=arcsin x iff sin y = x -1≤x≤1 y=arccos x iff cos y = x -1≤x≤1 0≤y≤π y=arctan x iff tan y = x -∞≤x≤∞ y=arccot x iff cot y = x -∞≤x≤∞ 0≤y≤π y=arcsec x iff sec y = x |x|≥1 0≤y≤π, Y=arccsc x iff csc y = x |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:
sin(arcsin x)=x and arcsin(sin y)=y
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 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…
2/5sectan arcFind
Ex 4 Write the expression in algebraic form cos(arcsin 2x)
Derivatives of Inverse Trig Functions
1||
'sec
1'arctan
1
'arcsin
2
2
2
uu
uuarcdxd
uuu
dxd
u
uudxd
1||
'csc
1'cot
1'arccos
2
2
2
uu
uuarcdxd
uuuarc
dxd
u
uudxd
Ex. 5 Find the derivative
a)
b)
)2arcsin( xdxd
)3arctan( xdxd
Ex 5. cont…
c)
d)
)arcsin( xdxd
)sec( 2xearcdxd