Inverse Trigonometric Functions

• Domain, Range, and Graph of secinv(x)

Introduction

• Inverse Trigonometric Functions
• Inverse of secant function (sec⁻¹x)
• Properties and characteristics

Definition

• Inverse of secant function:
  • sec⁻¹x or arcsecx
• It is the function that extracts the angle whose secant is given.
• sec⁻¹[sinθ/cosθ] = θ

Domain and Range

• The domain of sec⁻¹x is (-∞, -1] ∪ [1, ∞)
• The range is [0, π] ∪ (π, 2π]

Graph of secinv(x)

• The graph of sec⁻¹x is the reflection of the graph of secx with respect to the line y = x.
• It is a non-periodic function.

Graphical Representation

• Graph of sec⁻¹x:
  • Increasing function from -∞ to -1
  • Decreasing function from 1 to ∞
• Points of interest:
  • (1, 0) - Minima
  • (-1, π) - Maxima

Important Points

• Minimum value of secinv(x) = 0
• Maximum value of secinv(x) = π

Key Features

• Domain: (-∞, -1] ∪ [1, ∞)
• Range: [0, π] ∪ (π, 2π]
• Reflection of sec(x) graph
• Non-periodic function

Example 1

• Find the value of sec⁻¹(-2).
• Solution:
  • The angle whose secant is -2 is π + π/3.
  • Therefore, sec⁻¹(-2) = π + π/3 = 4π/3.

Example 2

• Determine the domain and range of sec⁻¹x.
• Solution:
  • The domain is (-∞, -1] ∪ [1, ∞).
  • The range is [0, π] ∪ (π, 2π].

Properties of sec⁻¹x

• sec⁻¹(-x) = sec⁻¹x
• sec⁻¹(1/x) = cos⁻¹(1/x)
• sec⁻¹(x) + cos⁻¹(x) = π/2
• sec⁻¹x = sin⁻¹(1/√(x²-1))
• sec⁻¹(√x) = sec⁻¹x

Derivatives of sec⁻¹x

• d/dx(sec⁻¹x) = 1/(|x|√(x²-1)), |x| > 1
• d/dx(sec⁻¹x) = -1/(|x|√(x²-1)), |x| < 1

Integration of sec⁻¹x

• ∫sec⁻¹x dx = xsec⁻¹x + ln|x + √(x²-1)| + C
• ∫sec⁻¹x dx = xsec⁻¹x - ln|x - √(x²-1)| + C

Trigonometric Equations

• sec⁻¹x = y, then x = secy
• sec⁻¹(secx) = x
• sec(sec⁻¹x) = x

Trigonometric Identities

• sec⁻¹x = sin⁻¹(1/x)
• sec⁻¹x = cos⁻¹(1/x)
• sec⁻¹x = tan⁻¹(√(x²-1))

Derivation of sec⁻¹x

• Let y = sec⁻¹x, then x = secy
• Differentiating both sides with respect to x:
  • 1 = secy * tan y
  • tan y = 1/secy = cos y
  • y = π/4 or 5π/4

Example 3

• Find the value of sec⁻¹tan⁻¹(3).
• Solution:
  • Let y = tan⁻¹(3)
  • tan y = 3
  • Using the identity tan⁻¹x = π/4 + tan⁻¹(1/x):
    • y = π/4 + tan⁻¹(1/3)
  • x = secy = sec(π/4 + tan⁻¹(1/3))

Example 4

• Evaluate ∫sec⁻¹(2x) dx.
• Solution:
  • Let y = 2x, then x = y/2
  • ∫sec⁻¹(2x) dx = ∫sec⁻¹y/2 * (1/2) dy
  • Using the integration formula, the result is:
    • (1/2)(ysec⁻¹y - ln|y + √(y²-4)|) + C
  • Substituting back y = 2x, the final result is obtained.

Summary

• Inverse of secant function is sec⁻¹x or arcsecx.
• It has a domain of (-∞, -1] ∪ [1, ∞) and a range of [0, π] ∪ (π, 2π].
• The graph of sec⁻¹x is the reflection of the graph of secx.
• There are properties, derivatives, and integration formulas for sec⁻¹x.
• Trigonometric equations and identities involving sec⁻¹x can be derived and used.

Recap

• Inverse Trigonometric Functions
• Definitions, properties, and characteristics of sec⁻¹x
• Domain, range, and graph representation
• Derivatives and integration formulas of sec⁻¹x
• Trigonometric equations and identities involving sec⁻¹x

Domain and Range of sec⁻¹x

• The domain of sec⁻¹x is (-∞, -1] ∪ [1, ∞).
• This means that the input x should be either less than -1 or greater than 1.
• The range of sec⁻¹x is [0, π] ∪ (π, 2π].
• This means that the output y lies between 0 and π, excluding π, and also between π and 2π.

Graphical Representation of sec⁻¹x

• The graph of sec⁻¹x is the reflection of the graph of sec x about the line y = x.
• As sec x is a periodic function, the graph of sec⁻¹x is not periodic.
• It is a non-periodic function that exhibits increasing and decreasing behavior.

Graph of sec⁻¹x

• The graph of sec⁻¹x starts from negative infinity and approaches -π/2 as x approaches -∞.
• At x = 1, the graph has a vertical asymptote.
• It reaches a minimum value of 0 at x = -1.
• It has another vertical asymptote at x = -1.
• At x = ∞, the graph approaches π/2.

Key Features of sec⁻¹x

• Domain: (-∞, -1] ∪ [1, ∞)
• Range: [0, π] ∪ (π, 2π]
• Non-periodic function
• Reflection of sec x graph
• Vertical asymptotes at x = -1 and x = 1
• Minimum value at x = -1, which is 0
• Approaches π/2 as x approaches ∞ and -π/2 as x approaches -∞

Derivatives of sec⁻¹x

• The derivative of sec⁻¹x can be found using the chain rule:
  • d/dx(sec⁻¹x) = 1/(|x|√(x²-1)), for |x| > 1
  • d/dx(sec⁻¹x) = -1/(|x|√(x²-1)), for |x| < 1
• Here, the absolute value of x is taken to ensure the non-negative values in the denominator.

Integration of sec⁻¹x

• The integral of sec⁻¹x can be found by integrating the expression:
  • ∫sec⁻¹x dx = xsec⁻¹x - ln|x - √(x²-1)| + C
  • ∫sec⁻¹x dx = xsec⁻¹x + ln|x + √(x²-1)| + C
• The constant of integration (C) is added at the end.

Trigonometric Equations

• Inverse trigonometric equations involving sec⁻¹x can be solved by applying the properties of sec⁻¹x.
• For example:
  • sec⁻¹x = y, then x = secy
  • sec⁻¹(secx) = x
  • sec(sec⁻¹x) = x
• These equations can help us find the value of x when the value of the inverse secant function is given, or vice versa.

Trigonometric Identities

• There are several trigonometric identities involving sec⁻¹x:
  • sec⁻¹x = sin⁻¹(1/x)
  • sec⁻¹x = cos⁻¹(1/x)
  • sec⁻¹x = tan⁻¹(√(x²-1))
• These identities relate sec⁻¹x to other inverse trigonometric functions such as sin⁻¹x, cos⁻¹x, and tan⁻¹x.

Summary

• Inverse trigonometric functions provide a way to find angles when the trigonometric ratios are given.
• The domain of sec⁻¹x is (-∞, -1] ∪ [1, ∞), and the range is [0, π] ∪ (π, 2π].
• The graph of sec⁻¹x is the reflection of the graph of sec x about the line y = x.
• The derivatives and integrals of sec⁻¹x can be found using specific formulas.
• Trigonometric equations and identities involving sec⁻¹x can be derived and used for problem-solving.

Recap

• Domain and range of sec⁻¹x
• Graphical representation of sec⁻¹x
• Key features of sec⁻¹x
• Derivatives and integrals of sec⁻¹x
• Trigonometric equations and identities involving sec⁻¹x