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

uuarc

dx

du

uu

dx

du

uu

dx

d

1||

'csc

1

'cot

1

'arccos

2

2

2

uu

uuarc

dx

du

uuarc

dx

du

uu

dx

d

Ex. 5 Find the derivative

a)

b)

)2arcsin( xdx

d

)3arctan( xdx

d

Ex 5. cont…

c)

d)

)arcsin( xdx

d

)sec( 2xearcdx

d