In order for a function to have an inverse it must be One-to-one and pass the Horizontal Line Test. It is obvious that the graph
does not pass this test. So it may appear illogical to use inverse trigonometric functions, but they help make equations much easier. Below is each Trigonometric Function Graph and its Inverse.
The inverse sine function is defined by
if and only if 
where
and 
*Domain of
is 
*Range of
is 
* Notice the domain and range are reversed on the interval of the graph y = sin x above*
Remember when evaluating for inverse sine functions it the arcsin of x represents an angle measure
Therefore,
Other Inverse Trigonometric Functions
a.
if and only if
Domain Range
b.
if and only if 
Domain Range
All Real Numbers
Evaluating Inverse Trigonometric Functions
1.
Therefore,
2.
Therefore,
Using a Calculator to Approximate Inverse Trigonometric Functions
* Make sure your calculator is in radians otherwise you will not find the correct answer
1.
Calculator Keystrokes
From the display, it follows that
Compositions of Functions
a. If
and 
are both true,
are true.b. If
and 
are both true,
and 
are true.c. If x is a real number and
are both true,
are true.These compositions follow the same rules learned earlier when finding the composites of functions and allow us to use these trigonometry functions as identities. It is important to note that any value outside of either interval makes these identities false, thus it is important to stay within the boundaries of the intervals.
Using Inverse Properties
1.
because 
is contained within
, the inverse property holds, and 
For Extra Help
https://www.youtube.com/watch?v=JGU74wbZMLg - Arcsin
https://www.youtube.com/watch?v=eTDaJ4ebK28 - Arccos
https://www.youtube.com/watch?v=Idxeo49szW0 - Arctan
These links will direct you to video explanations from Khan Academy that should help you further understand Inverse Trigonometric Functions.
No comments:
Post a Comment