Trigonometric Inverse Functions

Notes:

Remember inverse functions?

  • Inverse functions are one-to-one functions: f(x) and g(x) such that… f(g(x)) = x and g(f(x)) = x

Remember which functions had inverse functions?

  • Only one-to-one functions have inverse functions.
  • Remember that these are functions that pass the horizontal line test.

To define the inverse function of f(x) = sin(x), we pretty clear have a problem… The function fails the horizontal line test spectacularly!

To define an inverse function for f(x) = sin(x), we’ll have to restrict the domain. So we restrict it down to [(-π / 2), (π / 2)].

Now, g(x) = sin-1(x) will be an angle between (-π / 2) and (π / 2).

But let us define the inverse function of f(x) = cos(x) now.

g(x) = cos-1(x) is an angle y between 0 and π such that cos(y) = x.

Basically: - Cos-1(x) = y <===> cos(y) = x

Domain: [-1, 1]

Range: [0, π]

Other Stuff to Note:

  • g(x) = sin-1(x) is also written as g(x) = arcsin(x)

Rest of the notes are math problems, check out my Math generator app :).