Inverse Trigonometric Functions - Other properties of inverse trig functions

  • Recap of Inverse Trigonometric Functions

    • Definition: Inverse function of a trigonometric function
    • Range of inverse trig functions
  • Domain and Range of Inverse Trigonometric Functions

    • Domain: Range of corresponding trig functions
    • Range: Restricted domains of inverse trig functions
  • Basic Properties of Inverse Trig Functions

    • Identity Property: sin^(-1)(sin(x)) = x
    • Identity Property: cos^(-1)(cos(x)) = x
    • Identity Property: tan^(-1)(tan(x)) = x
  • Odd and Even Properties of Inverse Trig Functions

    • Odd Property: sin^(-1)(-x) = -sin^(-1)(x)
    • Even Property: cos^(-1)(-x) = cos^(-1)(x)
    • Odd Property: tan^(-1)(-x) = -tan^(-1)(x)
  • Pythagorean Identities of Inverse Trig Functions

    • Pythagorean Identity: sin^(-1)(x) = cos^(-1)(√(1 - x^2))
    • Pythagorean Identity: cos^(-1)(x) = sin^(-1)(√(1 - x^2))
  • Examples: Identifying Domain and Range

    • Example 1: Find the domain and range of sin^(-1)(x)
    • Example 2: Find the domain and range of cos^(-1)(x)
  • Complementary Angles Property

    • Complementary Angles Property: sin^(-1)(x) + cos^(-1)(x) = π/2
    • Complementary Angles Property: tan^(-1)(x) + cot^(-1)(x) = π/2
  • Examples: Complementary Angles Property

    • Example 1: Find the value of sin^(-1)(3/5) + cos^(-1)(4/5)
    • Example 2: Find the value of tan^(-1)(2) + cot^(-1)(3)
  • Composition of Inverse Trig Functions

    • Composition Property: sin(sin^(-1)(x)) = x
    • Composition Property: cos(cos^(-1)(x)) = x
    • Composition Property: tan(tan^(-1)(x)) = x
  • Examples: Composition Property

    • Example 1: Prove that sin(sin^(-1)(x)) = x
    • Example 2: Prove that tan(tan^(-1)(x)) = x
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Inverse Trigonometric Functions - Other properties of inverse trig functions Recap of Inverse Trigonometric Functions Definition: Inverse function of a trigonometric function Range of inverse trig functions Domain and Range of Inverse Trigonometric Functions Domain: Range of corresponding trig functions Range: Restricted domains of inverse trig functions Basic Properties of Inverse Trig Functions Identity Property: sin^(-1)(sin(x)) = x Identity Property: cos^(-1)(cos(x)) = x Identity Property: tan^(-1)(tan(x)) = x Odd and Even Properties of Inverse Trig Functions Odd Property: sin^(-1)(-x) = -sin^(-1)(x) Even Property: cos^(-1)(-x) = cos^(-1)(x) Odd Property: tan^(-1)(-x) = -tan^(-1)(x) Pythagorean Identities of Inverse Trig Functions Pythagorean Identity: sin^(-1)(x) = cos^(-1)(√(1 - x^2)) Pythagorean Identity: cos^(-1)(x) = sin^(-1)(√(1 - x^2)) Examples: Identifying Domain and Range Example 1: Find the domain and range of sin^(-1)(x) Example 2: Find the domain and range of cos^(-1)(x) Complementary Angles Property Complementary Angles Property: sin^(-1)(x) + cos^(-1)(x) = π/2 Complementary Angles Property: tan^(-1)(x) + cot^(-1)(x) = π/2 Examples: Complementary Angles Property Example 1: Find the value of sin^(-1)(3/5) + cos^(-1)(4/5) Example 2: Find the value of tan^(-1)(2) + cot^(-1)(3) Composition of Inverse Trig Functions Composition Property: sin(sin^(-1)(x)) = x Composition Property: cos(cos^(-1)(x)) = x Composition Property: tan(tan^(-1)(x)) = x Examples: Composition Property Example 1: Prove that sin(sin^(-1)(x)) = x Example 2: Prove that tan(tan^(-1)(x)) = x