For a function to have an inverse, it must
pass the Horizontal Line Test. Y=sinx does not pass this
test because different values of x yield the same y-value.
However, if you take only a segment by restricting the
domain to the interval of [-/2
x /2] the
properties of the test hold true for this segment.
1. On the interval [-/2,
/2], y=sinx
is increasing.
2. On the interval [-/2,
/2], y=sinx
takes its full range of values from –1 to 1
3. On the interval [-/2,
/2], y=sinx
passes the Horizontal Line Test
With this restricted domain, y=sinx has an inverse
function known as an inverse sine function. This is
written as y= arcsin x or y= sin –1 x. This functions
range is [-1,1] and its range is [-/2,
/2].
The inverse sine function is defined by y= arcsin x
only if siny = x, where -1
x
1 and -/2
y /2. The
domain is [-1,1] and the range is [-/2,
/2].
Other Inverse functions
Arcsine:
By definition, y=arcsinx and siny=x are
equal. Therefore both graphs are the same. From siny=x
you can assign values to make a table of values for y=arcsinx.
From this, you can graph y=arcsinx. Its range is from
[-/2,
/2]. The
graph below is y=arcsinx’s entire graph. Because
it has to be a function, its domain and range have been
shifted to make it so that for every value of x there
is only one value of y.
Arccosine
The cosine function is decreasing on the
interval of 0
x . The graph
of y=cosx shows this interval. This interval has an inverse
function, denoted as y=arcos x or y= cos –1 x. The
values for these two functions are the same and there
values are:
Arctangent
Like arcsine and arccosine, you can find the inverse
tangent function my restricting the domain of y=tanx to
the interval (-/2,
/2).
Definitions of Inverse Functions
Y=arcsin x only if siny=x. Domain= -1x1.
Range= -/2
y /2.
Y=arccos x only if cosy=x. Domain= -1x1.
Range= 0
y .
Y=arctan only if tany=x. Domain= -
x .
Range= -/2
y /2.
/2
x 
