Inverse Trig – Intro
p. 1
3.1 Inverse Trig Functions
Recommended Homework: 3.7: 7-17, 19-27, 30-40, 41-52
Inverse Trig Worksheet (on-line)
Review of basic inverses:
Def:
Does every function have an inverse?
Graphical properties:
Domain/Range Properties
Inverse Trig – Intro
p. 2
Def of y = Arcsin x = Sin-1 x
Function
y=sinx (restricted)
y= Sin-1 x
Domain
[-π/2, π/2]
Range
[-1, 1]
[-π/2, π/2]
[-1, 1]
y = Arcsin x = Sin-1 x










Domain: [-1, 1]
Range: [-π/2, π/2]
Inverse Trig – Intro
p. 3
Def of y = Arccos x = Cos-1 x
Function
y=cosx (restricted)
y= Cos-1 x
Domain
Range
[0, π]
[-1, 1]
[-1, 1]
[0, π]
y = Arccos x = Cos-1 x











Domain: [-1, 1]
Range: [0, π]
Inverse Trig – Intro
p. 4
Def of y = Arctan x = Tan-1 x
Function
y=tanx (restricted)
y= Tan-1 x
Domain
(-π/2, π/2)
Range
(-∞, ∞)
(-π/2, π/2)
(-∞, ∞)
y = Arctan x = Tan-1 x
Domain: (-∞, ∞)
Range: (-π/2, π/2)
Inverse Trig – Intro
p. 5
Note: Angles are ALWAYS in RADIANS!!!!!!!
Examples: Evaluate the following EXACTLY:
a) Cos-1(1/2)
b) Arcsin(1/2)
c) Tan-1(3)
d) Cos-1(-3/2)
e) Sin-1(-3/2)
f) Sin-1(3/2)
g) Tan-1(-1)
h) Arccos(-2/2)
i) Tan-1(-3/3)
Inverse Trig – Intro
j) Sin-1(3)
k) cos(π/3)
l) Cos-1(π/3)
p. 6
Inverse Trig – Intro
p. 7
Determine if the following statements are True T or False
F. Explain WHY.
To do these we MUST KNOW the domain and range of
the trig functions
Domain
Range
y  sin 1 x
y  cos 1 x
y  tan 1 x
f ( f 1 ( x))  x only if x in the domain of composition i.e
in the domain of f-1 and range of f
f 1 ( f ( x))  x only if x in the domain of composition i.e
in the domain of f and range of f-1
This means,
Composition
Sin-1(sin x) = x
Cos-1(cos x) = x
Tan-1(tan x) = x
sin(Sin-1 x) = x
cos(Cos-1 x) = x
tan(Tan-1 x) = x
Only when
 x
0 x 
 x
1  x  1
1  x  1
  x  
2
2
2
2
Inverse Trig – Intro
Example:
a) Arcsin(sin(π/3))= π/3
because
b) Arcsin(sin(7π/6))= 7π/6
because
True / False
True / False
c) cos-1(cos(2π/3))= 2π/3 True / False
because
d) cos-1(cos(-π/3))= -π/3
because
True / False
e) cos(cos-1(-1/2))=-1/2
because
True / False
p. 8
Inverse Trig – Intro
f) sin(sin-1(π))=π
because
True / False
g) tan-1(tan 5π/3) = 5π/3
because
True / False
p. 9
Inverse Trig – Intro
p. 10
Calculator problems: You must be in Radian mode!!
Always give answer accurate to 4 decimals unless
specified otherwise.
cos-1(-1/2)
sin-1(1/5)
cos-1(π/2)
tan-1(-8.2)