The inverse trig functions are transcendental, but their derivatives are algebraic!

1. Section 3.5
Inverse Trigonometric
Functions
V63.0121, Calculus I
March 11–12, 2009
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. . . . . .

2. What functions are invertible?
In order for f−1 to be a function, there must be only one a in D
corresponding to each b in E.
Such a function is called one-to-one
The graph of such a function passes the horizontal line test:
any horizontal line intersects the graph in exactly one point if at
all.
If f is continuous, then f−1 is continuous.
. . . . . .

3. Outline
Inverse Trigonometric Functions
Derivatives of Inverse Trigonometric Functions
Arcsine
Arccosine
Arctangent
Arcsecant
. . . . . .

4. arcsin
Arcsin is the inverse of the sine function after restriction to
[−π/2, π/2].
y
.
. . . x
.
s
. in
π π
− −
. .
2 2
. . . . . .

5. arcsin
Arcsin is the inverse of the sine function after restriction to
[−π/2, π/2].
y
.
. . . x
.
s
. in
π π
− −
. .
2 2
. . . . . .

6. arcsin
Arcsin is the inverse of the sine function after restriction to
[−π/2, π/2].
y
.
. =x
y
. . . x
.
s
. in
π π
− −
. .
2 2
. . . . . .

7. arcsin
Arcsin is the inverse of the sine function after restriction to
[−π/2, π/2].
y
.
a
. rcsin
. . . x
.
s
. in
π π
− −
. .
2 2
The domain of arcsin is [−1, 1]
[ π π]
The range of arcsin is − ,
22
. . . . . .

8. arccos
Arccos is the inverse of the cosine function after restriction to [0, π]
y
.
c
. os
. . x
.
π
.
0
.
. . . . . .

9. arccos
Arccos is the inverse of the cosine function after restriction to [0, π]
y
.
c
. os
. . x
.
π
.
0
.
. . . . . .

10. arccos
Arccos is the inverse of the cosine function after restriction to [0, π]
y
.
. =x
y
c
. os
. . x
.
π
.
0
.
. . . . . .

11. arccos
Arccos is the inverse of the cosine function after restriction to [0, π]
a
. rccos
y
.
c
. os
. . x
.
π
.
0
.
The domain of arccos is [−1, 1]
The range of arccos is [0, π]
. . . . . .

12. arctan
Arctan is the inverse of the tangent function after restriction to
[−π/2, π/2].
y
.
. x
.
π π
3π 3π
−
− . .
. .
2 2
2 2
t
. an
. . . . . .

13. arctan
Arctan is the inverse of the tangent function after restriction to
[−π/2, π/2].
y
.
. x
.
π π
3π 3π
−
− . .
. .
2 2
2 2
t
. an
. . . . . .

14. arctan
Arctan is the inverse of the tangent function after restriction to
. =x
y
[−π/2, π/2].
y
.
. x
.
π π
3π 3π
−
− . .
. .
2 2
2 2
t
. an
. . . . . .

15. arctan
Arctan is the inverse of the tangent function after restriction to
[−π/2, π/2].
y
.
π
. a
. rctan
2
. x
.
π
−
.
2
The domain of arctan is (−∞, ∞)
( π π)
The range of arctan is − ,
22
π π
lim arctan x = , lim arctan x = −
2 x→−∞ 2
x→∞
. . . . . .

16. Outline
Inverse Trigonometric Functions
Derivatives of Inverse Trigonometric Functions
Arcsine
Arccosine
Arctangent
Arcsecant
. . . . . .

17. Theorem (The Inverse Function Theorem)
Let f be differentiable at a, and f′ (a) ̸= 0. Then f−1 is defined in an open
interval containing b = f(a), and
1
(f−1 )′ (b) = ′ −1
f (f (b))
. . . . . .

18. Theorem (The Inverse Function Theorem)
Let f be differentiable at a, and f′ (a) ̸= 0. Then f−1 is defined in an open
interval containing b = f(a), and
1
(f−1 )′ (b) = ′ −1
f (f (b))
“Proof”.
If y = f−1 (x), then
f(y) = x,
So by implicit differentiation
dy dy 1 1
f′ (y) = 1 =⇒ =′ = ′ −1
dx dx f (y) f (f (x))
. . . . . .

19. The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
= 1 =⇒ = =
cos y
dx dx cos y cos(arcsin x)
. . . . . .

20. The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
= 1 =⇒ = =
cos y
dx dx cos y cos(arcsin x)
To simplify, look at a right
triangle:
.
. . . . . .

21. The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
= 1 =⇒ = =
cos y
dx dx cos y cos(arcsin x)
To simplify, look at a right
triangle:
1
.
x
.
.
. . . . . .

22. The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
= 1 =⇒ = =
cos y
dx dx cos y cos(arcsin x)
To simplify, look at a right
triangle:
1
.
x
.
. = arcsin x
y
.
. . . . . .

23. The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
= 1 =⇒ = =
cos y
dx dx cos y cos(arcsin x)
To simplify, look at a right
triangle:
1
.
x
.
. = arcsin x
y
.√
. 1 − x2
. . . . . .

24. The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
= 1 =⇒ = =
cos y
dx dx cos y cos(arcsin x)
To simplify, look at a right
triangle:
√
cos(arcsin x) = 1 − x2 1
.
x
.
. = arcsin x
y
.√
. 1 − x2
. . . . . .

25. The derivative of arcsin
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
= 1 =⇒ = =
cos y
dx dx cos y cos(arcsin x)
To simplify, look at a right
triangle:
√
cos(arcsin x) = 1 − x2 1
.
x
.
So
d 1
arcsin(x) = √ . = arcsin x
y
1 − x2 .√
dx
. 1 − x2
. . . . . .

26. Graphing arcsin and its derivative
1
.√
1 − x2
a
. rcsin
.
| |
. .
−
.1 1
.
. . . . . .

27. The derivative of arccos
Let y = arccos x, so x = cos y. Then
dy dy 1 1
− sin y = 1 =⇒ = =
− sin y − sin(arccos x)
dx dx
. . . . . .

28. The derivative of arccos
Let y = arccos x, so x = cos y. Then
dy dy 1 1
− sin y = 1 =⇒ = =
− sin y − sin(arccos x)
dx dx
To simplify, look at a right
triangle:
√
sin(arccos x) = 1 − x2 √
1
.
. 1 − x2
So
d 1 . = arccos x
y
arccos(x) = − √ .
1 − x2
dx x
.
. . . . . .

29. Graphing arcsin and arccos
a
. rccos
a
. rcsin
.
| |
. .
−
.1 1
.
. . . . . .

30. Graphing arcsin and arccos
a
. rccos
Note
(π )
−θ
cos θ = sin
2
a
. rcsin
π
=⇒ arccos x = − arcsin x
2
. So it’s not a surprise that their
| |
. .
−
.1 1
. derivatives are opposites.
. . . . . .

31. The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = cos2 (arctan x)
= 1 =⇒ =
sec2 y
dx dx
. . . . . .

32. The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = cos2 (arctan x)
= 1 =⇒ =
sec2 y
dx dx
To simplify, look at a right
triangle:
.
. . . . . .

33. The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = cos2 (arctan x)
= 1 =⇒ =
sec2 y
dx dx
To simplify, look at a right
triangle:
x
.
.
1
.
. . . . . .

34. The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = cos2 (arctan x)
= 1 =⇒ =
sec2 y
dx dx
To simplify, look at a right
triangle:
x
.
. = arctan x
y
.
1
.
. . . . . .

35. The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = cos2 (arctan x)
= 1 =⇒ =
sec2 y
dx dx
To simplify, look at a right
triangle:
√
x
.
. 1 + x2
. = arctan x
y
.
1
.
. . . . . .

36. The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = cos2 (arctan x)
= 1 =⇒ =
sec2 y
dx dx
To simplify, look at a right
triangle:
1
cos(arctan x) = √
√
1 + x2
x
.
. 1 + x2
. = arctan x
y
.
1
.
. . . . . .

37. The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = cos2 (arctan x)
= 1 =⇒ =
sec2 y
dx dx
To simplify, look at a right
triangle:
1
cos(arctan x) = √
√
1 + x2
x
.
. 1 + x2
So
d 1
. = arctan x
y
arctan(x) =
.
1 + x2
dx
1
.
. . . . . .

38. Graphing arctan and its derivative
y
.
a
. rctan
1
. x
.
1 + x2
. . . . . .

39. Example
√
x. Find f′ (x).
Let f(x) = arctan
. . . . . .

40. Example
√
x. Find f′ (x).
Let f(x) = arctan
Solution
√ d√
d 1 1 1
·√
(√ )2
arctan x = x=
1+x 2 x
dx x dx
1+
1
=√ √
2 x + 2x x
. . . . . .

41. Recap
y′
y
1
√
arcsin x
1 − x2
1 Remarkable that the
arccos x − √
1 − x2 derivatives of these
1 transcendental functions
arctan x
1 + x2 are algebraic (or even
1
−
arccot x rational!)
1 + x2
1
√
arcsec x
x x2 − 1
1
arccsc x − √
x x2 − 1
. . . . . .