§7.6–The inverse trigonometric functions
Mark R. Woodard
Furman U
Fall 2010
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
1 / 11
Outline
1
The inverse sine function
2
The inverse cosine function
3
The inverse tangent function
4
The other inverse trig functions
5
Miscellaneous problems
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
2 / 11
The inverse sine function
Definition
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
3 / 11
The inverse sine function
Definition
The sine function is one-to-one on [−π/2, π/2] and has range [−1, 1]
on this domain.
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
3 / 11
The inverse sine function
Definition
The sine function is one-to-one on [−π/2, π/2] and has range [−1, 1]
on this domain.
We define sin−1 to be the inverse of sine on this domain. It follows
that sin−1 has domain [−1, 1] and range [−π/2, π/2].
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
3 / 11
The inverse sine function
Definition
The sine function is one-to-one on [−π/2, π/2] and has range [−1, 1]
on this domain.
We define sin−1 to be the inverse of sine on this domain. It follows
that sin−1 has domain [−1, 1] and range [−π/2, π/2].
Cancellation equations
Because of these restrictions, we must be a little careful with the inverse
relationships:
sin sin−1 (x) = x − 1 ≤ x ≤ 1
sin−1 sin(x) = x − π/2 ≤ x ≤ π/2
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
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The inverse sine function
Problem
Evaluate the following:
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
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The inverse sine function
Problem
Evaluate the following:
sin sin−1 (.3)
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
4 / 11
The inverse sine function
Problem
Evaluate the following:
sin sin−1 (.3)
sin−1 sin(14π/3) .
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
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The inverse sine function
Problem
Evaluate the following:
sin sin−1 (.3)
sin−1 sin(14π/3) .
Problem
Find cos 2 sin−1 (1/4) .
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
4 / 11
The inverse sine function
Problem
Evaluate the following:
sin sin−1 (.3)
sin−1 sin(14π/3) .
Problem
Find cos 2 sin−1 (1/4) .
Problem
Evaluate cos sin−1 (x) .
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
4 / 11
The inverse sine function
Problem
Evaluate the following:
sin sin−1 (.3)
sin−1 sin(14π/3) .
Problem
Find cos 2 sin−1 (1/4) .
Problem
Evaluate cos sin−1 (x) .
Solution
√
Let θ = sin−1 (x). Then cos(θ) = ± 1 − x 2 . Since cosine
is nonnegative
√
−1
for θ ∈ [−π/2, π/2], it follows that cos(sin (x)) = 1 − x 2 .
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
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The inverse sine function
Theorem
Dx sin−1 (x) = √
Mark R. Woodard (Furman U)
1
.
1 − x2
§7.6–The inverse trigonometric functions
Fall 2010
5 / 11
The inverse sine function
Theorem
Dx sin−1 (x) = √
1
.
1 − x2
Proof.
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
5 / 11
The inverse sine function
Theorem
Dx sin−1 (x) = √
1
.
1 − x2
Proof.
Let y = sin−1 (x).
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
5 / 11
The inverse sine function
Theorem
Dx sin−1 (x) = √
1
.
1 − x2
Proof.
Let y = sin−1 (x).
Then sin(y ) = x. By the chain rule, cos(y )y 0 = 1 hence y 0 =
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
1
cos(y ) .
Fall 2010
5 / 11
The inverse sine function
Theorem
Dx sin−1 (x) = √
1
.
1 − x2
Proof.
Let y = sin−1 (x).
Then sin(y ) = x. By the chain rule, cos(y )y 0 = 1 hence y 0 =
√
However, cos(y ) = 1 − x 2 , which gives the desired result.
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
1
cos(y ) .
Fall 2010
5 / 11
The inverse sine function
Theorem
Dx sin−1 (x) = √
1
.
1 − x2
Proof.
Let y = sin−1 (x).
Then sin(y ) = x. By the chain rule, cos(y )y 0 = 1 hence y 0 =
√
However, cos(y ) = 1 − x 2 , which gives the desired result.
1
cos(y ) .
Problem
Find the derivative of y = x sin−1 (x 2 ).
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
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The inverse cosine function
Definition
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
6 / 11
The inverse cosine function
Definition
The cosine function is one-to-one on the interval [0, π] and has range
[−1, 1] on that domain.
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
6 / 11
The inverse cosine function
Definition
The cosine function is one-to-one on the interval [0, π] and has range
[−1, 1] on that domain.
Let cos−1 denote the inverse of the cosine function restricted to the
domain [0, π]. Thus the domain of cos−1 is [−1, 1] and its range is
[0, π].
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
6 / 11
The inverse cosine function
Definition
The cosine function is one-to-one on the interval [0, π] and has range
[−1, 1] on that domain.
Let cos−1 denote the inverse of the cosine function restricted to the
domain [0, π]. Thus the domain of cos−1 is [−1, 1] and its range is
[0, π].
The cancellation equations
cos cos−1 (x) = x
cos−1 cos(x) = x
Mark R. Woodard (Furman U)
−1≤x ≤1
0≤x ≤π
§7.6–The inverse trigonometric functions
Fall 2010
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The inverse cosine function
Problem
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
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The inverse cosine function
Problem
Evaluate cos−1 cos(14π/3)
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
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The inverse cosine function
Problem
Evaluate cos−1 cos(14π/3)
√
Show that sin cos−1 (x) = 1 − x 2
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
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The inverse cosine function
Problem
Evaluate cos−1 cos(14π/3)
√
Show that sin cos−1 (x) = 1 − x 2
Theorem
Dx cos−1 (x) = − √
Mark R. Woodard (Furman U)
1
1 − x2
§7.6–The inverse trigonometric functions
Fall 2010
7 / 11
The inverse cosine function
Problem
Evaluate cos−1 cos(14π/3)
√
Show that sin cos−1 (x) = 1 − x 2
Theorem
Dx cos−1 (x) = − √
1
1 − x2
Proof.
Let y = cos−1 (x). Then cos(y ) = x and, by taking derivatives,
√
0 = 1; hence y 0 = −1/ sin(y ). Since sin(y ) =
− sin(y )y √
1 − x 2,
0
2
y = −1/ 1 − x , as was to be shown.
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
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The inverse tangent function
Definition
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
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The inverse tangent function
Definition
The tangent is one-to-one on the interval (−π/2, π/2) and has range
(−∞, ∞) on this domain.
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
8 / 11
The inverse tangent function
Definition
The tangent is one-to-one on the interval (−π/2, π/2) and has range
(−∞, ∞) on this domain.
Let tan−1 be the inverse of the tangent function on this restricted
domain. Thus the domain of tan−1 is (−∞, ∞) and its range is
(−π/2, π/2).
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
8 / 11
The inverse tangent function
Definition
The tangent is one-to-one on the interval (−π/2, π/2) and has range
(−∞, ∞) on this domain.
Let tan−1 be the inverse of the tangent function on this restricted
domain. Thus the domain of tan−1 is (−∞, ∞) and its range is
(−π/2, π/2).
The cancellation equations
tan tan−1 (x) = x
tan−1 tan(x) = x
Mark R. Woodard (Furman U)
−∞<x <∞
− π/2 < x < π/2.
§7.6–The inverse trigonometric functions
Fall 2010
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The inverse tangent function
Problem
Show that sec2 tan−1 (x) = 1 + x 2
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
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The inverse tangent function
Problem
Show that sec2 tan−1 (x) = 1 + x 2
Theorem
Dx tan−1 (x) =
Mark R. Woodard (Furman U)
1
,
1 + x2
x ∈ R.
§7.6–The inverse trigonometric functions
Fall 2010
9 / 11
The inverse tangent function
Problem
Show that sec2 tan−1 (x) = 1 + x 2
Theorem
Dx tan−1 (x) =
1
,
1 + x2
x ∈ R.
Proof.
Let y = tan−1 (x). Then tan(y ) = x and, by the chain rule, sec2 (y )y 0 = 1.
Thus y 0 = 1/ sec2 (y ) = 1/(1 + x 2 ), as was to be shown.
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
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The other inverse trig functions
The other inverse trig functions
The remaining inverse trig functions (cot−1 , sec−1 , and csc−1 ) are defined
in similar ways. See the text for a summary of these functions, their
definitions, and derivatives.
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
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Miscellaneous problems
Problem
Find y 0 in each case:
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
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Miscellaneous problems
Problem
Find y 0 in each case:
y = tan−1 (e x )
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
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Miscellaneous problems
Problem
Find y 0 in each case:
y = tan−1 (e x )
√
y = 1 − x 2 sin−1 (x)
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
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Miscellaneous problems
Problem
Find y 0 in each case:
y = tan−1 (e x )
√
y = 1 − x 2 sin−1 (x)
y = sin−1 (x) + cos−1 (x)
Mark R. Woodard (Furman U)
§7.6–The inverse trigonometric functions
Fall 2010
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