Basic Concepts of inverse Trigonometric Functions

  1. Introduction:

    • Inverse trigonometric functions are functions that "reverse" the effect of trigonometric functions.
    • Denoted by sin1(x), cos1(x), tan1(x), cot1(x), sec1(x), and csc1(x).
  2. Domain and Range:

    • The domain of inverse trigonometric functions corresponds to the range of the corresponding trigonometric functions.
    • The range of inverse trigonometric functions depends on the principal values and is often restricted to ensure single-valuedness.
  3. Principal Values:

    • Each inverse trigonometric function has principal values that restrict its range to ensure a unique value for a given input.
    • For sin1(x) and cos1(x): [-π/2, π/2].
    • For tan1(x): (π2,π2).
    • For cot1(x): (0,π).
    • For sec1(x) and csc1(x): [0,π][π,2π].
  4. Graphs of Inverse Trigonometric Functions:

    • The graphs of inverse trigonometric functions exhibit specific characteristics due to their restricted domains and principal values.
    • sin1(x) and cos1(x) have domain [1,1] and range [0,π].
    • tan1(x) has an asymptote at ±π2.
    • cot1(x) has an asymptote at x=0 and π.
    • sec1(x) and csc1(x) have asymptotes at [1,1] and [1,1] respectively.
  5. Trigonometric Identities:

    • Inverse trigonometric functions satisfy certain identities, such as:
      • sin(sin1(x))=x.
      • cos(cos1(x))=x.
      • tan(tan1(x))=x.
      • cot(cot1(x))=x.
      • sec(sec1(x))=x.
      • csc(csc1(x))=x.
  6. Inverse Trigonometric Formulas:

    • The inverse trigonometric functions have various identities and formulas, e.g.:
      • sin1(x)+cos1(x)=π2.
      • tan1(x)+cot1(x)=π2.
      • sec1(x)+csc1(x)=π2.
  7. Applications:

    • Inverse trigonometric functions are widely used in solving trigonometric equations, analyzing periodic phenomena, and in various scientific and engineering applications.
  8. Special Values:

    • Memorize the special values of inverse trigonometric functions, e.g., sin1(0)=0, cos1(1)=0, tan1(0)=0, etc.