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- Inverse trigonometric functions
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Key concepts
- An inverse function is one that “undoes” another function. The domain of an inverse function is the range of the original function and the range of an inverse function is the domain of the original function.
- Because the trigonometric functions are not one-to-one on their natural domains, inverse trigonometric functions are defined for restricted domains.
- For any trigonometric function
if
then
However,
only implies
if
is in the restricted domain of
See
[link] .
- Special angles are the outputs of inverse trigonometric functions for special input values; for example,
See
[link] .
- A calculator will return an angle within the restricted domain of the original trigonometric function. See
[link] .
- Inverse functions allow us to find an angle when given two sides of a right triangle. See
[link] .
- In function composition, if the inside function is an inverse trigonometric function, then there are exact expressions; for example,
See
[link] .
- If the inside function is a trigonometric function, then the only possible combinations are
if
and
if
See
[link] and
[link] .
- When evaluating the composition of a trigonometric function with an inverse trigonometric function, draw a reference triangle to assist in determining the ratio of sides that represents the output of the trigonometric function. See
[link] .
- When evaluating the composition of a trigonometric function with an inverse trigonometric function, you may use trig identities to assist in determining the ratio of sides. See
[link] .
Section exercises
Verbal
Why do the functions
and
have different ranges?
The function
is one-to-one on
thus, this interval is the range of the inverse function of
The function
is one-to-one on
thus, this interval is the range of the inverse function of
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Why must the domain of the sine function,
be restricted to
for the inverse sine function to exist?
In order for any function to have an inverse, the function must be one-to-one and must pass the horizontal line test. The regular sine function is not one-to-one unless its domain is restricted in some way. Mathematicians have agreed to restrict the sine function to the interval
so that it is one-to-one and possesses an inverse.
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Source:
OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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