We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.

1. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
Sec on 3.6
Inverse Trigonometric Func ons
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
March 28, 2011
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Notes
Announcements
Midterm has been returned. Please see
FAQ on Blackboard (under ”Exams and
Quizzes”)
Quiz 3 this week in recita on on
Sec on 2.6, 2.8, 3.1, 3.2
Quiz 4 April 14–15 on Sec ons 3.3, 3.4,
3.5, and 3.7
Quiz 5 April 28–29 on Sec ons 4.1, 4.2,
4.3, and 4.4
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Notes
Objectives
Know the defini ons, domains, ranges,
and other proper es of the inverse
trignometric func ons: arcsin, arccos,
arctan, arcsec, arccsc, arccot.
Know the deriva ves of the inverse
trignometric func ons.
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2. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
Outline
Inverse Trigonometric Func ons
Deriva ves of Inverse Trigonometric Func ons
Arcsine
Arccosine
Arctangent
Arcsecant
Applica ons
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.
Notes
What is an inverse function?
Defini on
Let f be a func on with domain D and range E. The inverse of f is the
func on f−1 defined by:
f−1 (b) = a,
where a is chosen so that f(a) = b.
So
f−1 (f(x)) = x, f(f−1 (x)) = x
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Notes
What functions are invertible?
In order for f−1 to be a func on, there must be only one a in D
corresponding to each b in E.
Such a func on is called one-to-one
The graph of such a func on passes the horizontal line test:
any horizontal line intersects the graph in exactly one point if at
all.
If f is con nuous, then f−1 is con nuous.
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3. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
Graphing the inverse function
y y=x
If b = f(a), then f−1 (b) = a.
So if (a, b) is on the graph of f,
then (b, a) is on the graph of f−1 .
On the xy-plane, the point (b, a) (b, a)
is the reflec on of (a, b) in the
line y = x. (a, b)
Therefore:
.
x
Fact
The graph of f−1 is the reflec on of the graph of f in the line y = x.
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Notes
arcsin
Arcsin is the inverse of the sine func on a er restric on to
[−π/2, π/2].
y
y=x
arcsin
. x
π π sin
−
2 2
The domain of arcsin is [−1, 1]
[ π π]
The range of arcsin is − ,
2 2
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.
Notes
arccos
Arccos is the inverse of the cosine func on a er restric on to [0, π]
arccos
y
y=x
cos
. x
0 π
The domain of arccos is [−1, 1]
The range of arccos is [0, π]
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4. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
arctan
y=x
Arctan is the inverse of the tangent func on a er restric on to
y
(−π/2, π/2).
π
2 arctan
. x
3π π π 3π
− −
2 2− π 2 2
2
( π π ∞)
The domain of arctan is (−∞, )
The range of arctan is − ,
2 2 tan
π π
lim arctan x = , lim arctan x = −
. x→∞ 2 x→−∞ 2
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Notes
arcsec 3π
2
Arcsecant is the inverse of secant a er restric on to x
y=
[0, π/2) ∪ [π, 3π/2). y
π
2
. x
3π π π 3π
− −
2 2 2 2
The domain of arcsec is (−∞, −1] ∪ [1, ∞)
[ π ) (π ]
The range of arcsec is 0, ∪ ,π
2 2
π 3π
lim arcsec x = , lim arcsec x = sec
x→∞ 2 x→−∞ 2
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Notes
Values of Trigonometric Functions
x 0 π/6 π/4 π/3 π/2
√ √
sin x 0 1/2 2/2 3/2 1
√ √
cos x 1 3/2 2/2 1/2 0
√ √
tan x 0 1/ 3 1 3 undef
√ √
cot x undef 3 1 1/ 3 0
√ √
sec x 1 2/ 3 2/ 2 2 undef
√ √
csc x undef 2 2/ 2 2/ 3 1
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5. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
Check: Values of inverse trigonometric functions
Example Solu on
Find π
arcsin(1/2) 6
arctan(−1)
( √ )
2
arccos −
2
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Notes
Caution: Notational ambiguity
sin2 x =.(sin x)2 sin−1 x = (sin x)−1
sinn x means the nth power of sin x, except when n = −1!
The book uses sin−1 x for the inverse of sin x, and never for
(sin x)−1 .
1
I use csc x for and arcsin x for the inverse of sin x.
sin x
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Notes
Outline
Inverse Trigonometric Func ons
Deriva ves of Inverse Trigonometric Func ons
Arcsine
Arccosine
Arctangent
Arcsecant
Applica ons
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6. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
The Inverse Function Theorem
Theorem (The Inverse Func on Theorem)
Let f be differen able at a, and f′ (a) ̸= 0. Then f−1 is defined in an
open interval containing b = f(a), and
1
(f−1 )′ (b) =
f′ (f−1 (b))
In Leibniz nota on we have
dx 1
=
dy dy/dx
.
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Notes
Illustrating the IFT
Example
Use the inverse func on theorem to find the deriva ve of the
square root func on.
Solu on (Newtonian nota on)
√
Let f(x) = x2 so that f−1 (y) = y. Then f′ (u) = 2u so for any b > 0
we have
1
(f−1 )′ (b) = √
2 b
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Notes
Illustrating the IFT
Example
Use the inverse func on theorem to find the deriva ve of the
square root func on.
Solu on (Leibniz nota on)
If the original func on is y = x2 , then the inverse func on is defined
by x = y2 . Differen ate implicitly:
dy dy 1
1 = 2y =⇒ = √
dx dx 2 x
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7. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
The derivative of arcsine
Let y = arcsin x, so x = sin y. Then
dy dy 1 1
cos y = 1 =⇒ = =
dx dx cos y cos(arcsin x)
To simplify, look at a right triangle:
√
cos(arcsin x) = 1 − x2
1
So x
Fact
d 1 y = arcsin x
arcsin(x) = √ .√
dx 1 − x2 1 − x2
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Notes
Graphing arcsin and its derivative
1
√
The domain of f is [−1, 1], 1 − x2
but the domain of f′ is
(−1, 1) arcsin
lim− f′ (x) = +∞
x→1
| . |
lim f′ (x) = +∞ −1 1
x→−1+
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Notes
Composing with arcsin
Example
Let f(x) = arcsin(x3 + 1). Find f′ (x).
Solu on
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8. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
The derivative of arccos
Let y = arccos x, so x = cos y. Then
dy dy 1 1
− sin y = 1 =⇒ = =
dx dx − sin y − sin(arccos x)
To simplify, look at a right triangle:
√
sin(arccos x) = 1 − x2
1 √
So 1 − x2
Fact
d 1 y = arccos x
arccos(x) = − √ .
dx 1 − x2 x
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Notes
Graphing arcsin and arccos
arccos Note
(π )
cos θ = sin −θ
arcsin 2
π
=⇒ arccos x = − arcsin x
2
| . |
−1 1 So it’s not a surprise that their
deriva ves are opposites.
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Notes
The derivative of arctan
Let y = arctan x, so x = tan y. Then
dy dy 1
sec2 y = 1 =⇒ = = cos2 (arctan x)
dx dx sec2 y
To simplify, look at a right triangle:
1
cos(arctan x) = √
1 + x2
√
So 1 + x2 x
Fact
d 1 y = arctan x
.
arctan(x) = 1
dx 1 + x2
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9. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
Graphing arctan and its derivative
y
π/2
arctan
1
1 + x2
. x
−π/2
The domain of f and f′ are both (−∞, ∞)
Because of the horizontal asymptotes, lim f′ (x) = 0
x→±∞
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Notes
Composing with arctan
Example
√
Let f(x) = arctan x. Find f′ (x).
Solu on
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Notes
The derivative of arcsec
Try this first.
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10. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Notes
Another Example
Example
Let f(x) = earcsec 3x . Find f′ (x).
Solu on
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Notes
Outline
Inverse Trigonometric Func ons
Deriva ves of Inverse Trigonometric Func ons
Arcsine
Arccosine
Arctangent
Arcsecant
Applica ons
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Notes
Application
Example
One of the guiding principles of most
sports is to “keep your eye on the
ball.” In baseball, a ba er stands 2 ft
away from home plate as a pitch is
thrown with a velocity of 130 ft/sec
(about 90 mph). At what rate does
the ba er’s angle of gaze need to
change to follow the ball as it crosses
home plate?
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11. . V63.0121.001: Calculus ISec on 3.6: Inverse Trigonometric Func ons
. . March 28, 2011
Solu on Notes
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Notes
Summary
y y′ y y′
1 1
arcsin x √ arccos x − √
1−x 2 1 − x2
1 1
arctan x arccot x −
1 + x2 1 + x2
1 1
arcsec x √ arccsc x − √
x x2 − 1 x x2 − 1
Remarkable that the deriva ves of these transcendental
func ons are algebraic (or even ra onal!)
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Notes
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