Look at the graph of y = sin(x)
An Inverse involves switching the coordinates; in it the
the domain becomes the range,
and the range becomes the domain.
also remember that inverse may not be functions
Trigonometric Inverse and Their Graphs
Unless we limit the domain before
We find the inverse!
Why that?
It is the right half of unit circle and
uses the 0 radian angle still
Principal values
The unique solutions of a trig function, if the values
of the function are limited to 2 adjacent quadrants.
y = sin(x) if and only if y = sin-1 (x) and


x
2
2
y = cos(x) if and only if y = cos-1 (x) and
0 x 
y = tan(x) if and only if y = tan-1 (x) and


x
2
2
Inverse of sine, Inverse cosine and Inverse
tangent are also called: Arcsine, arccosine, and
arctangent
y  sin ( x)
y  arcsin(x)
y  cos ( x)
y  arccos(x)
y  tan ( x)
y  arctan(x)
1
1
example
1
Write the inverse of
y = Arccos(2x)
y = ½ cos(x)
Function
y = Sin (x)
Domain
(-/2)  x  (/2)
Range
-1 y  1
Find the principal values of each
Function
Domain
And the table under
it if you get time. Range
y = Sin
y = Arcsin
(x)(x) -1 x  1
y = Arcsin (x)
y =cos(x)
y = Arccos (x)
0 x  
-1 x  1
(-/2)  y  (/2)
-1 y  1
0 y  
y = tan(x)
(-/2)  x  (/2)
All real numbers
y = Arctan (x)
All real numbers
(-/2)  y  (/2)
Examples on the next slide use others until they have copied all of this
1
Work with the principal values
More examples?
1.)
2.)
Using counter example to “prove”

 3 
cos Arc tan 3  Arc sin   
 2 

tan tan

1
 3   
3
=1
 3
Application example
it does not work:
Another Ferris wheel problem
Determine whether Sin -1 (sin x) = x is true or
false for all values of x. If false give a counter
example.
False when x = (2)/3, sin(x )=
3/2. but Sin-1 (3/2) =(3). Which
does not equal x.
The Ferris wheel was introduced at the
World Columbian Exposition in Chicago in
1893. The first Ferris Wheel had a
diameter of 250 feet and a maximum height
of 264 feet. It cost 50 cents per ride and
made a complete revolution every 5.15
minutes. If a car is heading up and passes
the midline point at 9:41 am, when will that
car reach the top for the first time?
9:42:17.25
2