2.4.3: Composition of Trig Functions and Their Inverses

Overview

• Exploration of how to compose trigonometric functions with their inverses.
• Consideration of applying inverse trig functions to different trig functions.
• Example problem discussed: Finding ( \sin^{-1}(\cos(32)) ).

Trigonometric Functions and Their Inverses

• Fundamental property: ( f(f^{-1}(x)) = x ) for defined values.
• Applied to trigonometric functions:
  • ( \sin(\sin^{-1}(x)) = x )
  • ( \cos(\cos^{-1}(x)) = x )
  • ( \tan(\tan^{-1}(x)) = x )
• Conversely, ( f^{-1}(f(x)) = x ) only holds within restricted domains:
  • ( \sin^{-1}(\sin(x)) = x )
  • ( \cos^{-1}(\cos(x)) = x )
  • ( \tan^{-1}(\tan(x)) = x )

Composite Functions

• Composite functions involve one trig function with another different one.
• These functions become algebraic, not trigonometric.

Examples

1. ( \sin(\sin^{-1}22) ):

   • ( \sin^{-1}22 = 22 ), hence ( \sin(\sin^{-1}22) = 22 ).

2. ( \cos(\tan^{-1}13) ):

   • ( \tan^{-1}13 = 3 ), thus ( \cos(\tan^{-1}13) = 12 ).

3. ( \tan(\sin^{-1}12) ):

   • ( \tan(6) = 33 ).

4. ( \cos(\tan^{-1}1) ):

   • ( \cos(\tan^{-1}1) = 22 ).

5. ( \sin(\cos^{-1}22) ):

   • ( \sin(\cos^{-1}22) = 22 ).

Example Problem Solutions

• ( \sin^{-1}(\cos(32)) ):

  • First find ( \cos(32) = 0 ).
  • Then ( \sin^{-1}0 = 0 ).

• ( \cos^{-1}32 ):

  • Solution given as 6.

• ( \sin(\cos^{-1}15/13) ):

  • Solves to ( 12/13 ).

• ( \tan(\sin^{-1}6/11) ):

  • Solution involves geometric consideration of right triangle:
  • Third side calculated, final tan value given as ( 68/85 ).

Review Problems

• Without technology, find exact values:
  1. ( \sin(\sin^{-1}12) )
  2. ( \cos(\cos^{-1}32) )
  3. ( \tan(\tan^{-1}3) )
  4. ( \cos(\sin^{-1}12) )
  5. ( \tan(\cos^{-1}1) )
  6. ( \sin(\cos^{-1}22) )
  7. ( \sin^{-1}(\sin2) )
  8. ( \cos^{-1}(\tan4) )
  9. ( \tan^{-1}(\sin) )
  10. ( \sin^{-1}(\cos3) )
  11. ( \cos^{-1}(\sin4) )
  12. ( \tan(\sin10) )
  13. ( \sin(\cos^{-1}32) )
  14. ( \tan^{-1}(\cos2) )
  15. ( \cos(\sin^{-1}22) )

Vocabulary

• Composite function: A function formed by using the output of one function as the input of another.

Additional Resources

• Video and practice available for deeper understanding and application of compositions of trig functions and their inverses.