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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

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Inverse Trigonometric Functions

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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

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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

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I

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Inverse Trigonometric Functions

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I T i i F i

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Inverse Trigonometric Functions

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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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g

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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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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