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Inverse Trigonometric Functions
Inverse Trigonometric Functions and Identities
University of the Philippines Manila
April 30, 2014
Mathematics 14
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Inverse Trigonometric Functions
Trigonometric Functions
Recall: Trigonometric Functions are not one-to-one
We restrict each of their domains such that:
1 the circular function is one-to-one on the restricted domain
2 the range of the restricted trigonometric function is the same
Mathematics 14
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Inverse Trigonometric Functions
Mathematics 14
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Inverse Trigonometric Functions
Inverse sine function
Definition
For any x ∈ [−1, 1],
y = sin−1 x ⇔ x = sin y , y ∈−π
2,
π
2
dom sin−1 = [−1, 1] ran sin−1 =−pi
2 ,
pi
2
Mathematics 14
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Inverse Trigonometric Functions
Examples
1 sin−1
√ 3
2
2 sin−1 −1
2
3 sin−1
−√
2
2
4 sin−1 0
5 sin−
1(−1)6 sin−1 2
Mathematics 14
I
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Inverse Trigonometric Functions
Mathematics 14
I T i i F i
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Inverse Trigonometric Functions
Mathematics 14
I e se T i et i F ti s
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Inverse Trigonometric Functions
Inverse cosine function
DefinitionFor any x ∈ [−1, 1],
y = cos−1 x ⇔ x = cos y , y ∈ [0, π]
dom cos−1 = [−1, 1] ran sin−1 = [0, π]
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Inverse Trigonometric Functions
Examples
1 cos−1
√ 3
2
2 cos−1 −1
2
3 cos−1
−√
2
2
4 cos−1 0
5 cos−
1(−1)6 cos−1 2
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Inverse Trigonometric Functions
Other Inverse Trigonometric Functions
y = tan−1 x ⇔ x = tan y , x ∈ R, y ∈−π
2,
π
2
y = cot−1 x ⇔ x = cot y , x ∈ R, y ∈ (0, π)
y = sec−1 x ⇔ x = sec y , x ∈ (−∞,−1]
[1, +∞),y ∈
0,
π
2
π2
, π
y = csc−1 x ⇔ x = csc y , x ∈ (−∞,−1]
[1, +∞),y ∈
−π
2, 0
0,π
2
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Inverse Trigonometric Functions
Note
The sin−1, cos−1,tan−1, . . . functions are also denoted byArcsin,Arccos,Arctan,...
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Inverse Trigonometric Functions
Graph of the Inverse Tangent Function
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Inverse Trigonometric Functions
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Graph of the Inverse Cotangent Function
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Graph of the Inverse Secant Function
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Graph of the Inverse Cosecant Function
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Exercises
1 tan−1√ 3
3
2 tan−1(−1)3 cot−1(
−
√ 3)
4 Arccot 05 sec−1(−
√ 2)
6 sec−1(−2)7 Arccsc
√ 2
8 csc−1−2√ 33
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Inverse Trigonometric Identities
1 sin−1(sin x ) = x for every x ∈−π
2,
π
2
2 cos−1(cos x ) = x for every x ∈ [0, π]3 tan
−
1(tan x ) = x for every x ∈ −π2 , π24 cot−1(cot x ) = x for every x ∈ (0, π)5 sec−1(sec x ) = x for every x ∈
0,
π
2
π
2, π
6 csc
−
1(csc x ) = x for every x ∈ −π2 , 00, π2
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Inverse Trigonometric Functions
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Examples
1 tan−1
tan π
12
2 cos−1
cos 7π
8
3 sin−1
sin 5π
6
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Inverse Trigonometric Identities
1 sin(sin−1) = x for every x ∈ [−1, 1]2 cos(cos−1) = x for every x ∈ [−1, 1]3 tan(tan
−
1) = x for every x ∈ R4 cot(cot−1) = x for every x ∈ R5 sec(sec−1) = x for every x ∈ (−∞,−1]
[1, +∞)
6 csc(csc−1) = x for every x ∈ (−∞,−1][1, +∞)
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Examples
1 cos(cos−1 2
3)
2 tan(tan−
1(−4))3 cos(sin−1
√ 3
2 )
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Exercises1 Evaluate sin(cos−1
−3
5
)
2 Evaluate tan(csc−1 −√
5− tan−1 2
3)
3 Solve for x : cos−1x
4
+ tan−1
−√ 3
3
= cot−1 0
4 Solve for x : sec−1(−√
2)− tan−1(x − 2) = csc−1(−2)5
Solve for x : sin−1
(x )− cos−1
(x ) =
π
6
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