Stractura Inc. Case Analysis

Inverse trigonometric functions (Sect. 7.6)

Today: Derivatives and integrals.
Review: Definitions and properties. Derivatives. Integrals.

Last class: Definitions and properties.
Domains restrictions and inverse trigs. Evaluating inverse trigs at simple values. Few identities for inverse trigs.

Review: Definitions and properties
Remark: On certain domains the trigonometric functions are invertible. y
1

y = sin(x)

y
1

y = cos(x)

y y = tan(x)

−π/2

π/2

x

0

π/2

π

−π/2

π/2

x

x

−1

−1

y

y = csc(x)

y

y = sec(x)

y

y = cot(x)

1 −π/2 0 −1 π/2

1

x

0 −1

π/2

π

x

0

π/2

π

x

Review: Definitions and properties
Remark: The graph of the inverse function is a reflection of the original function graph about the y = x axis. y π/2

y = arcsin(x)

y π y = arccos(x)

y π/2 y = arctan(x)

−1

1

x

π/2

x
−π/2

−π/2

−1

0

1

x y π

y π/2 y = arccsc(x)

y

y = arcsec(x) π y = arccot(x)

−1

0

1

x

π/2

π/2

−π/2 −1 0 1

x

0

x

Review: Definitions and properties
Theorem
For all x ∈ [−1, 1] the following identities hold, arccos(x) + arccos(−x) = π, arccos(x) + arcsin(x) = π . 2

Proof: y arccos(−x) 1 1 x = sin(π/2−θ) arccos(x) θ −x = cos (π−θ) π−θ θ x = cos (θ) π/2 − θ θ arccos(x)

y

arcsin(x)

x

x = cos (θ)

x

Review: Definitions and properties
Theorem
For all x ∈ [−1, 1] the following identities hold, arcsin(−x) = − arcsin(x), arctan(−x) = − arctan(x), arccsc(−x) = −arccsc(x).

Proof: y π/2

y = arcsin(x)

y π/2 y = arctan(x)

y π/2 y = arccsc(x)

−1

1

x
−π/2

x

−1

0

1

x

−π/2

−π/2

Inverse trigonometric functions (Sect. 7.6)

Today: Derivatives and integrals.
Review: Definitions and properties. Derivatives. Integrals.

Derivatives of inverse trigonometric functions
Remark: Derivatives inverse functions can be computed with