I don't really understand why we sketch the graphs of cos^-1(cosx), cos(cos^-1(x)) , sin^-1(sinx) and sin(sin^-1(x)) the way they are in the textbook. I would appreciate it if someone could explain the reason behind sketching them that way!
Thanks in advance!
cos(cos^-1(x)), sin(sin^-1(x)) should be easy. Use the theorem
f(f^-1(x)) = x.
sin^-1(sinx), cos^-1(cosx) you need to examine by inspection to sketch and get the pattern. Use the theorem
f^-1(f(x)) = x but this only holds for the domain of the sin^-1 / cos^-1
As an excerise try to prove the theorems:
1. f(f^-1(x)) = x
2. f^-1(f(x)) = x
Observe for both theorems the argument of the function, in simple terms that is the input the output so long the domain constraint is satisfied.
Hint: For proving the theorems, the functions are defined differently in 1/2.
