We have learned previously that in order for a function to have an inverse that is also a function, it must pass the horizontal line test. In other words, that function must be one-to-one, meaning that each point in the domain of the function must correspond to a unique point in that function's range.
When examining the graph of 
, or any other trigonometric function, we can see that the function is periodic, and therefore not one-to-one.
, or any other trigonometric function, we can see that the function is periodic, and therefore not one-to-one.
The inverse of would appear like so:
would appear like so:
This is not a function. However, by constraining the domain of our function to 
, The graph of becomes becomes one-to-one, and therefore its inverse, shown below, is also a function.
, The graph of becomes becomes one-to-one, and therefore its inverse, shown below, is also a function.
This is known as the inverse sine function, and is denoted by either 
or 
.
or .In the same vein, an inverse cosine function stems from the parent function 
, but unlike the sine function, the domain of the cosine function is restrained to 
in order to maintain the function as one-to-one. The result is displayed below.
, but unlike the sine function, the domain of the cosine function is restrained to 
in order to maintain the function as one-to-one. The result is displayed below.
By constraining the function 
by the interval 
, an inverse tangent function may also be defined.
by the interval 
, an inverse tangent function may also be defined.
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