Section 4.7
INVERSE trigonometric functions:
What does an inverse function do?
For a function to have an inverse, the function must be one­to­one.
Find an inverse by swapping x and y, then solve for y.
Graphically, a function and its inverse are symmetric over the line y = x.
NOTE: The domain and range of a function switch roles for the inverse function.
So...for the function f(x) = x3 we have an inverse function f­1(x) = ∛x
Consider another well known function and inverse pair: f(x) = x2 and f ­1(x) = √x
f(x) = x2 Domain: Range:
f ­1(x) = √x
Domain: Range:
Moral of the story...SOME functions have to be "fixed" in order to create an inverse function.
As a result, you must be careful when the function and its inverse are acting on each other.
f ­1(x) = sin­1x = arcsin x
f(x) = sin x
Domain
Range
Domain
Range
How about the inverse cosine function? y = cos­1x
y = cos x
Domain Domain Range Range Here is the cosine function graph (with restricted domain) together with the inverse cosine graph.
How is the inverse tangent graph created?
y = tan­1x
y = tan x
Domain Range
Domain Range
What quadrants do the inverse functions give as outputs?
sin­1x = arcsin x If x is positive, sin­1x will output an angle in ___________
If x is negative, sin­1x will output an angle in __________
cos­1x = arccos x
If x is positive, cos­1x will output an angle in ___________
If x is negative, cos­1x will output an angle in __________
tan­1x = arctan x If x is positive, tan­1x will output an angle in ___________
If x is negative, tan­1x will output an angle in __________
What you do need to know is what quadrants the outputs of the inverse functions will be.
In what quadrant will the answer be?
sin­10.5
sin­1(­0.4664)
arccos ­0.8
cos­1 0.8425
tan­1­10.3
arctan 4.2
Be careful when trig functions and their inverses are put together.
sin­1(sin 330o) tan­1(tan (­3π/4))
cos (sin­1­0.5)