Vanier College Continuing Education
Math 536 Upgrading (201-009-50)
Section: 2120, Semester: Winter 2005
§7.6 Inverses of the Trigonometric Functions (p478)
Review §5.1 Inverse Function
A function f is called a one-to-one function if it never taken on the same value
twice; that is, f ( x1 )  f ( x2 ) whenever x1  x2
The inverse of a function f, denoted by f -1, is defined by f 1 (a)  b if f (b)  a
if f is one-to-one.
The domain of f -1 is range of f and the range of f -1 is domain of f.
Example
Determine whether the function is one-to-one
f ( x)  4  x
1)
2)
3)
4)
5)
f ( x)  x 2
f ( x)  sin x
f ( x)  e x
f ( x)  log 10 x
Horizontal Line Test
A solution is one-to-one if and only if no horizontal line intersects its graph more
than once
A.
Restricting Ranges to Define Inverse Functions
A function like f(x) = x2 does not have an inverse function, but by restricting the
domain of f to nonnegative numbers, we have a new squaring function, f (x) = x2,
x0
x  0, that has an inverse function, f 1 ( x)  x
1
File: f2/f23.doc
Date: 4/29/2005
B.
Definition of the inverse of the trigonometric function
a. Define a new trigonometric function by restricting the domain of f(x)
y  f ( x)  sin x and D  [ / 2,  / 2]
Range   1,1
y  f ( x)  cos x and D  [0,  ]
Range  [1,1]
y  f ( x)  tan x and D  ( / 2,  / 2)
Range  R
y  f ( x)  cot x and D  (0,  )
Range  R
y  f ( x)  sec x and D  [0,  / 2)  [ ,3 / 2)
Range  (,1]  [1, )
y  f ( x)  csc x and D  (0,  / 2]  ( , / 2] Range  (,1]  [1, )
b. Define inverse of the new trigonometric function.
y  f 1 ( x)  sin 1 x  arcsin x
D  [1,1]
Range  [ / 2,  / 2]
y  f 1 ( x)  cos 1 x  arccos x
Range  [0,  ]
y f
1
y f
1
y f
1
D  [1,1]
x  arctan x
DR
Range  ( / 2,  / 2)
1
( x)  cot x  arc cot x
DR
Range  (0,  )
1
D  (,1]  [1, ) Range  [0,  / 2)  [ ,3 / 2)
( x)  tan
1
( x)  sec x  arc sec x
y  f ( x)  csc x  arc csc x
D  (,1]  [1, ) Range  (0,  / 2]  ( , / 2]
Note:
1. arcsin(x) is an angle whose sin is x;
arccos(x) is an angle whose cos is x;
arctan(x) is an angle whose tan is x;
arccot(x) is an angle whose cot is x;
arcsec(x) is an angle whose sec is x;
arccsc(x) is an angle whose csc is x.
1
1
Example 1
Find sin 1
1
2
Example 2
Find arcsin
2
2
Example 3
 1
Find cos 1   
 2
Example 4
Find arctan 1
Example 5
Find cos 1  0.925678 in degrees, using a calculator.
2
File: f2/f23.doc
Date: 4/29/2005
C.
Graph of the inverse of the trigonometric function
1. Graph the functions y = sin x and y = arcsin x
2. Graph the functions y = cos x and y = arccos x
3. Graph the functions y = tan x and y = arctan x
3
File: f2/f23.doc
Date: 4/29/2005
4. Graph the functions y = cot x and y = arccot x
5. Graph the functions y = sec x and y = arcsec x (option)
y= sec(x)
y=arcsec(x)
4. Graph the functions y = csc x and y = arccsc x (option)
y=csc(x)
y=arccsc(x)
4
File: f2/f23.doc
Date: 4/29/2005