CO4-Lesson 1 - Inverse Trigonometric Functions

MAPÚA UNIVERSITY MATH04 PreCalculus Course Outcome 4

• Lesson 1: Inverse Trigonometric Functions

Inverse Trigonometric Functions

• Definition: If 𝑓 is a one-to-one function with domain 𝑨 and range 𝑩, then its inverse 𝑓⁻¹ has domain 𝑩 and range 𝑨, defined by 𝑓⁻¹(𝑥) iff 𝑓(𝑦) = 𝑥.

Characteristics of Inverse Trigonometric Functions

• One-to-One Requirement: For a function to have an inverse, it must be one-to-one.

• Trigonometric Functions: Standard trigonometric functions are not one-to-one, thus do not initially have inverses.

• Domain Restriction: Trigonometric functions must have restricted domains to become one-to-one and achieve their inverses.

Inverse Sine Function and Its Graph

• Expression: 𝑦 = sin⁻¹(𝑥) iff 𝑥 = sin(𝑦).

• Domain: [−1, 1]

• Range: [−𝜋/2, 𝜋/2]

• Graph characteristics:

  • -𝜏/2 to 𝜏/2 is the essential portion.

Inverse Cosine Function and Its Graph

• Expression: 𝑦 = cos⁻¹(𝑥) iff 𝑥 = cos(𝑦).

• Domain: [−1, 1]

• Range: [0, 𝜋]

• Graph features:

  • Function is defined on a range between 0 and 𝜋.

Inverse Tangent Function and Its Graph

• Expression: 𝑦 = tan⁻¹(𝑥) iff 𝑥 = tan(𝑦).

• Domain: (−∞, ∞)

• Range: (−𝜋/2, 𝜋/2)

• Graph spans all real values for 𝑥,

  • Essential range between −𝜋/2 and 𝜋/2.

Inverse Cotangent Function and Its Graph

• Expression: 𝑦 = cot⁻¹(𝑥) iff 𝑥 = cot(𝑦).

• Domain: (−∞, ∞)

• Range: (0, 𝜋)

Inverse Secant Function and Its Graph

• Expression: 𝑦 = sec⁻¹(𝑥) iff 𝑥 = sec(𝑦).

• Domain: (−∞, -1] U [1, ∞)

• Range: [0, 𝜋], y ≠ 𝜋/2

Inverse Cosecant Function and Its Graph

• Expression: 𝑦 = csc⁻¹(𝑥) iff 𝑥 = csc(𝑦).

• Domain: (−∞, -1] U [1, ∞)

• Range: [−𝜋/2, 𝜋/2], y ≠ 0

Domain and Range of Inverse Functions

• Various inverses with specific domains/ranges:

  • y = sin⁻¹(𝑥): Domain [−1, 1], Range [−𝜋/2, 𝜋/2]

  • y = cos⁻¹(𝑥): Domain [−1, 1], Range [0, 𝜋]

  • y = tan⁻¹(𝑥): Domain (−∞, ∞), Range (−𝜋/2, 𝜋/2)

Evaluating Inverse Trigonometric Functions

• Process: Find an angle such that the trigonometric function of that angle equals 𝑥 within the range of the inverse function.

  • Example: sin⁻¹(1/2) = 𝜋/6 because sin(𝜋/6) = 1/2.

• Evaluation Guide: Use images for common special angles to help derive values.

Special Angles and Values

• Remember trigonometric values for frequently referenced angles in evaluation:

  • 30°: sin = 1/2, cos = √3/2;

  • 45°: sin = √2/2, cos = √2/2;

  • 60°: sin = √3/2, cos = 1/2;

• Reference expanded tables for angles.

Compositions of Functions

• Inverse function properties:

  • 𝑓(𝑓⁻¹(𝑥)) = 𝑥 and 𝑓⁻¹(𝑓(𝑥)) = 𝑥.

• Caution with specific ranges: arcsin(sin(3𝜋/2)) ≠ 3𝜋/2; valid only if y is in the interval [−𝜋/2, 𝜋/2].

Practice Examples

• Various evaluations showcasing the use of inverse formulas, considering the domains and special angles for accurate results.

• Application of properties involving inverse identities in specific situations:

  • sin⁻¹(sin(5π/3)) and solving compositions examples.