1. Section 4 Inverses of the Trigonometric Functions
MTH 112 Elementary Functions Chapter 6 Trigonometric Identities, Inverse Functions, and Equations
Section 4
Inverses of the Trigonometric Functions

2. 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.

3. 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?
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.

4. 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.
y = x
(a,b)
(b,a) b a

5. Inverses of Trig Functions
Do the inverses of the trig functions exist?
NO!
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))
Note:
1 radian  57°
2 radians  115°
4 radians  229°
6 radians  344°

6. Inverses of Trig Functions
Do the inverses of the trig functions exist?
NO!
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
Note:
1 radian  57°
2 radians  115°
4 radians  229°
6 radians  344°

7. Inverses of Trig Functions
Do the inverses of the trig functions exist?
NO!
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))
Note:
1 radian  57°
2 radians  115°
4 radians  229°
6 radians  344°

8. Inverses of Trig Functions
Do the inverses of the trig functions exist?
NO!
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
Note:
1 radian  57°
2 radians  115°
4 radians  229°
6 radians  344°

9. 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))

10. 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))

11. 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

12. y = sin-1x = arcsin x  - 2 -2 y = sin x Begin with y = sin x.
(/2, 1)
(3/2, -1)
(-/2, -1)
(-3/2, 1)
y = sin x
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.

13. y = sin-1x = arcsin x y = sin-1x  x = sin y, where y  [-/2, /2]
Therefore …
y = sin-1x  x = sin y, where y  [-/2, /2]
Domain: [-1, 1]
Range: [-/2, /2]

14. y = cos-1x = arccos x -/2 /2 3/2 -3/2 y = cos x
(2, 1)
(, -1)
(-, -1)
(-2, 1)
(0, 1)
y = cos x
Begin with y = cos x.
Restrict the domain to [0, ].
This gives a 1-1 function with the same range as the original function.

15. y = cos-1x = arccos x y = cos-1x  x = cos y, where y  [0, ]
Therefore …
y = cos-1x  x = cos y, where y  [0, ]
Domain: [-1, 1]
Range: [0, ]

16. y = tan-1x = arctan x /2 -/2 3/2 -3/2 y = tan x
(-, 0)
(, 0)
(0, 0)
(-2, 0)
(2, 0)
y = tan x
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.

17. y = tan-1x = arctan x y = tan-1x  x = tan y, where y  (-/2, /2)
Therefore …
y = tan-1x  x = tan y, where y  (-/2, /2)
Domain: (-, )
Range: (-/2, /2)

18. Likewise, the other three …
y = sec-1x = arcsec x  x = sec y
Domain: (-, -1]  [1, )
Range: [0, /2)  (/2, ]
y = csc-1x = arccsc x  x = csc y
Range: [-/2, 0)  (0, /2]
y = cot-1x = arccot x  x = cot y
Domain: (-, )
Why not (0, )? Because of calculators!

19. 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)

20. 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)

21. 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].

22. 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.

23. 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].

24. 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.

25. 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

26. Composition of Trig Functions with Inverse Trig Functions
What about something like sin(cos-1x)
This can always be evaluated without a calculator.
Begin by expressing x as a fraction … x = a/b
 = cos-1x = cos-1(a/b)  a b
Therefore, cos  = a/b
Therefore, sin(cos-1x) = sin  =

27. 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
cos-1x
Therefore, x 1

28. 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.
The hypotenuse will then be
tan-1x
Therefore, x 1

29. Composition of Trig Functions with Inverse Trig Functions
What about something like sin(cos-1x)
This can always be evaluated without a calculator.
y = sin(cos-1x)
y2 = sin2(cos-1x)
y2 = 1 - cos2(cos-1x) = 1 – [cos (cos-1x)] 2
y2 = 1 - x2
Therefore …

30. Composition of Trig Functions with Inverse Trig Functions
Other combinations?