Arccosine
The arccosine (notation: arccos or cos^{-1}) is the inverse function of the cosine. That is:
As the arcsine and sine are inverse functions, their composition is the identity. That is:
Properties of Arccosine
- Domain (x): The domain for arcsin x is from −1 to 1,
- Range (θ): The range, or output for arcsin x is all angles from 0 to π radians.
To define the inverse function of a function, it must necessarily be bijective (or one-to-one). The cosine function is not injective in the set of reals. By convention, the codomain is restricted to the interval to make the cosine function bijective.
That is, since none of the six trigonometric functions are one-to-one, they must be restricted to have inverse functions. Thus the domains of the trigonometric functions are restricted appropriately so that they become one-to-one functions and their inverse can be determined.
- Symmetry: none.
- Increasing-decreasing behaviour: decresing.
- Continuity: continuous for all x in domain.
- The derivative of arccosine function:
- The integral of arccosine function:
- End behaviour: The limits as x approaches ±∞ do not exist.
Arccosine Table
The arccosine of the most common values is:
Graphical Representation of the Arccosine Function
To better understand the graph of the arccosine, let’s first see the graphical representation of the sine function:
As we see in the graph above the cosine is periodic, it is not one-to-one and the graph of the cosine function fails the horizontal line test. Hence the cosine does not have an inverse unless we restrict its domain. So, by convention, the domain of the cosine is usually restricted to the interval (0, π).
The graph of the arccosine function is symmetric to that of the cosine function to the bisector line of the first and third quadrants (y = x).
Your graph shows the sine function, not the cosine. Please revise it for confusion.
Thank you Mohammed. We have corrected it.