MTH 112
Elementary Functions
Chapter 6
Trigonometric Identities, Inverse Functions, and
Equations
Section 4
Inverses of the Trigonometric Functions
Inverse of a Function?
Function:
f = { (a,b) | aD bR  f(a) = b }
• D = Domain of the Function
• R = Range of the Function
Inverse:
f -1 = { (b,a) | (a,b)f and f -1(b) = a}
Basic relationship: f -1(f(x)) = x and f(f -1(x)) = x
Does the inverse of a function exist?
• Only if the function is one-to-one.
• That is, if f(a) = f(b) then a = b for all a & b in the domain of f.
Examples
Does the inverse of y = x2 exist?
– No, because a2 = b2 does not imply that a = b.
Does the inverse of y = x2 where x  0 exist?
– Yes!
Do these two functions have the same range?
– Yes!
Note that if the inverse of a function does not exist,
sometimes the inverse of the original function with a
restricted domain and same range does exist.
The Graph of the
Inverse of a Function
What is the relationship
between the graph of a
function and the graph of its
inverse?
– Reflection about the line y = x.
Why?
– Each ordered pair (a,b) of the
function corresponds to an
ordered pair (b,a) of the
inverse.
(b,a)
y=x
a
a
b
b
(a,b)
Inverses of Trig Functions
Do the inverses of the trig
functions exist?
– NO!
Note:
1 radian  57°
2 radians  115°
4 radians  229°
6 radians  344°
Try the following on a
calculator.
sin-1(sin(1))
sin-1(sin(2))
sin-1(sin(4))
sin-1(sin(6))
sin-1(sin(-1))
sin-1(sin(-2))
sin-1(sin(-4))
sin-1(sin(-6))
Inverses of Trig Functions
Do the inverses of the
trig functions exist?
– NO!
Note:
1 radian  57°
2 radians  115°
4 radians  229°
6 radians  344°
Try the following on a
calculator.
sin-1(sin(1)) = 1
sin-1(sin(2))  1.14
sin-1(sin(4))  -0.86
sin-1(sin(6))  -0.28
sin-1(sin(-1)) = -1
sin-1(sin(-2))  -1.14
sin-1(sin(-4))  -0.86
sin-1(sin(-6))  -0.28
Inverses of Trig Functions
Do the inverses of the
trig functions exist?
– NO!
Note:
1 radian  57°
2 radians  115°
4 radians  229°
6 radians  344°
Try the following on a
calculator.
cos-1(cos(1))
cos-1(cos(2))
cos-1(cos(4))
cos-1(cos(6))
cos-1(cos(-1))
cos-1(cos(-2))
cos-1(cos(-4))
cos-1(cos(-6))
Inverses of Trig Functions
Do the inverses of the
trig functions exist?
– NO!
Note:
1 radian  57°
2 radians  115°
4 radians  229°
6 radians  344°
Try the following on a
calculator.
cos-1(cos(1)) = 1
cos-1(cos(2)) = 2
cos-1(cos(4))  2.28
cos-1(cos(6))  0.28
cos-1(cos(-1)) = 1
cos-1(cos(-2)) = 2
cos-1(cos(-4))  2.28
cos-1(cos(-6))  0.28
Inverses of Trig Functions
Try the following on a
calculator.
sin(sin-1(1))
sin(sin-1(2))
sin(sin-1(-1))
sin(sin-1(-2))
sin(sin-1(0.5))
sin(sin-1(-0.5))
Inverses of Trig Functions
Try the following on a
calculator.
sin(sin-1(1)) = 1
sin(sin-1(2)) = error
sin(sin-1(-1)) = -1
sin(sin-1(-2)) = error
sin(sin-1(0.5)) = 0.5
sin(sin-1(-0.5)) = -0.5
Try the following on a
calculator.
cos(cos-1(1))
cos(cos-1(2))
cos(cos-1(-1))
cos(cos-1(-2))
cos(cos-1(0.5))
cos(cos-1(-0.5))
Inverses of Trig Functions
Try the following on a
calculator.
sin(sin-1(1)) = 1
sin(sin-1(2)) = error
sin(sin-1(-1)) = -1
sin(sin-1(-2)) = error
sin(sin-1(0.5)) = 0.5
sin(sin-1(-0.5)) = -0.5
Try the following on a
calculator.
cos(cos-1(1)) = 1
cos(cos-1(2)) = error
cos(cos-1(-1)) = -1
cos(cos-1(-2)) = error
cos(cos-1(0.5)) = 0.5
cos(cos-1(-0.5)) = -0.5
y = sin-1x = arcsin x
y = sin x
(-3/2, 1)
1
(/2, 1)
0 .8
0 .6
0 .4
-2
-6
0 .2
-4
-
-2
0
-0 .2
2
-0 .4

x
4
2
6
-0 .6
-0 .8
-1
(-/2, -1)
(3/2, -1)
Begin with y = sin x.
Restrict the domain to [-/2, /2].
This gives a 1-1 function with the same range as the
original function.
y = sin-1x = arcsin x
Therefore …
• y = sin-1x  x = sin y, where y  [-/2, /2]
• Domain: [-1, 1]
• Range: [-/2, /2]
1.5
1
0.5
-1
-0.8 -0.6 -0.4 -0.2 0
-0.5
-1
-1.5
0.2
0.4 x 0.6
0.8
1
y = cos-1x = arccos x
(-2, 1)
(2, 1)
(0, 1)
1
0 .8
0 .6
-6
-3/2
-4
-/2
0 .4
-2
0
-0 .2
0 .2
-0 .4
/2
2
3/2
x 4
6
-0 .6
-0 .8
y = cos x
(-, -1)
-1
(, -1)
Begin with y = cos x.
Restrict the domain to [0, ].
This gives a 1-1 function with the same range as the
original function.
y = cos-1x = arccos x
Therefore …
• y = cos-1x  x = cos y, where y  [0, ]
• Domain: [-1, 1]
• Range: [0, ]
3
2.5
2
1.5
1
0.5
-1
-0.8 -0.6 -0.4 -0.2 0
0.2
0.4 x 0.6
0.8
1
y = tan-1x = arctan x
2
y = tan x
1
(-2, 0)
-6
(-, 0)
-
-4
3/2
(0, 0)
-/2
-2
0
/2
2
(, 0)
(2, 0)
4
3/2
6
-1
-2
Begin with y = tan x.
Restrict the domain to (-/2, /2).
This gives a 1-1 function with the same range as the
original function.
y = tan-1x = arctan x
Therefore …
• y = tan-1x  x = tan y, where y  (-/2, /2)
• Domain: (-, )
• Range: (-/2, /2)
1.5
1
0.5
-4
-2
0
-0.5
-1
-1.5
2 x
4
Likewise, the other three …
3
y = sec-1x = arcsec x  x = sec y
5.2
2
– Domain: (-, -1]  [1, )
– Range: [0, /2)  (/2, ]
5.1
1
5.0
1
2
1
2
5.1
1
5.0
0
1-
2-
5.01-
– Domain: (-, -1]  [1, )
– Range: [-/2, 0)  (0, /2]
0
1-
2-
y = csc-1x = arccsc x  x = csc y
5.15.1
y = cot-1x = arccot x  x = cot y
1
5.0
4
2
0
2-
4-
5.015.1-
– Domain: (-, )
– Range: [-/2, 0)  (0, /2]
• Why not (0, )? Because of calculators!
sec-1x, csc-1x & cot-1x
On a Calculator
y = sec-1x




x = sec y
x = 1/cos y
cos y = 1/x
y = cos-1(1/x)
sec-1x, csc-1x & cot-1x
On a Calculator
Therefore …
• y = sec-1x  y = cos-1(1/x)
• y = csc-1x  y = sin-1(1/x)
• y = cot-1x  y = tan-1(1/x)
Composition of Trig Functions
with Inverse Trig Functions
Under what conditions does sin-1(sin x) = x ?
– The range of Inverse Sine function is [-/2, /2]
– Therefore, x [-/2, /2].
Composition of Trig Functions
with Inverse Trig Functions
sin-1(sin x) = x  x [-/2, /2]
cos-1(cos x) = x  x [0, ]
tan-1(tan x) = x  x (-/2, /2)
What about something like ..
sin-1(sin 5/6) = ? (5/6 is not in the above interval)
= sin-1(1/2)
= /6
• It can still be evaluated, it’s just not equal to 5/6.
Composition of Trig Functions
with Inverse Trig Functions
Under what conditions does sin(sin-1x) = x ?
– The domain of Inverse Sine function is [-1, 1]
– Therefore, x [-1, 1].
Composition of Trig Functions
with Inverse Trig Functions
sin(sin-1x) = x  x [-1, 1]
cos(cos-1x) = x  x [-1, 1]
tan(tan-1x) = x  x (-, )
What about something like ..
sin(sin-1(5/3)) = ?
(5/3 is not in the above interval)
• It can not be evaluated, because 5/3 is not in the domain
of the inverse sine function.
Composition of Trig Functions
with Inverse Trig Functions
What about something like ...
sin-1(cos x)
– If cos x is a known value, evaluate it and then find
the angle whose sine is this value.
– If cos x is not a known value, use a calculator.
Example:
sin-1(cos(4/3))
= sin-1(-1/2)
= -/6
Composition of Trig Functions
with Inverse Trig Functions
What about something like ...
sin(cos-1x)
– This can always be evaluated without a calculator.
– Remember: cos-1x represents an acute angle (if x > 0)!
• Draw a triangle where cos-1x is one of the acute angles.
• Using the definition, if the adjacent side is x, then the
hypotenuse will be 1.
• The opposite side will then be 1  x 2
1
1 x2
cos-1x
x
Therefore,


sin cos 1 x  1  x 2
Composition of Trig Functions
with Inverse Trig Functions
Another variation ... same approach: csc(tan-1x)
– tan-1x represents an acute angle (if x > 0)!
• Draw a triangle where tan-1x is one of the acute angles.
• Using the definition, if the opposite side is x, then the
adjacent side will be 1.
2
• The hypotenuse will then be 1  x
1 x2
Therefore,
x
tan-1x
1
2
1

x
csc tan 1 x 
x


Composition of Trig Functions
with Inverse Trig Functions
sin(cos 1 x)  1  x 2
cos(sin 1 x)  1  x 2
tan(sec 1 x)  x 2  1
sec(tan 1 x)  1  x 2
cot(csc 1 x)  x 2  1
csc(cot 1 x)  1  x 2
Other combinations?