Inverse Trigonometric Functions

Radians and Degrees

The document discusses inverse trigonometric functions, their ranges in both radians and degrees, and the necessity of restricting the domains of original trigonometric functions to define their inverses.

  • Radians and Degrees Conversion:

    • It implicitly references the conversion between radians and degrees, crucial in trigonometry.

  • Arcsin(x) or sin⁻¹(x):

    • Defines arcsin(x) as the inverse of sin(x).
    • Range in radians: −π2≤arcsin(x)≤π2-{\pi \over 2} \le arcsin(x) \le {\pi \over 2}
    • Range in degrees: −90∘≤arcsin(x)≤90∘-90^\circ \le arcsin(x) \le 90^\circ

  • Arccos(x) or cos⁻¹(x):

    • Defines arccos(x) as the inverse of cos(x).
    • Range in radians: 0≤arccos(x)≤π0 \le arccos(x) \le \pi
    • Range in degrees: 0∘≤arccos(x)≤180∘0^\circ \le arccos(x) \le 180^\circ

  • Arctan(x) or tan⁻¹(x):

    • Defines arctan(x) as the inverse of tan(x).
    • Range in radians: -{\pi \over 2} < arctan(x) < {\pi \over 2}
    • Range in degrees: -90^\circ < arctan(x) < 90^\circ

Inverse Trigonometric Functions

  • Non-invertibility of Trigonometric Functions:

    • Trigonometric functions are not inherently invertible due to multiple inputs yielding the same output (e.g., sin(0) = sin(π) = 0).
    • Example: sin(0)=sin(π)=0sin(0) = sin(\pi) = 0, questioning what sin−1(0)sin^{-1}(0) should be.

  • Domain Restriction for Inverse Functions:

    • To define inverse trigonometric functions, the domain of the original functions must be restricted to intervals where they are invertible (one-to-one).
    • These restricted domains determine the ranges of the corresponding inverse functions.

  • Ranges of Inverse Trigonometric Functions:

    • The value from the appropriate range is returned by the inverse trigonometric function.