3.3. DERIVATIVES OF THE LIBRARY OF FUNCTIONS 97
Knowledge Box 3.8
Trigonometric derivatives
f .x/ f
0
.x/
sin.x/ cos.x/
cos.x/ sin.x/
tan.x/ sec
2
.x/
cot.x/ csc
2
.x/
sec.x/ sec.x/ tan.x/
csc.x/ csc.x/ cot.x/
3.3.3 INVERSE TRIGONOMETRIC FUNCTIONS
We are already somewhat familiar with inverse functions, like the log-exponential pair and the
square-square root pair, but the time has come for a formal definition.
Knowledge Box 3.9
Definition of an inverse function
A function g.x/ is the inverse of a function f .x/ on an interval
Πa; b if, for all x in Πa; b , we have
f .g.x// D g.f .x// D x:
e inverse of f .x/ is denoted f
1
.x/:
Example 3.41 Since g.x/ D
p
x only exists on the interval Π0; 1/, we have that g.x/ D
p
x
is an inverse of f .x/ D x
2
on the interval Π0; 1/.
˙
When g.x/ is an inverse of f .x/ on some interval, we have a special name for it. e inverse of
f .x/ is denoted:
f
1
.x/
98 3. LIMITS, DERIVATIVES, RULES, AND THE MEANING OF THE DERIVATIVE
which is read “the inverse of f .x/.” is notation is traditional but problematic because it can
be confused with the negative-first power of f .x/, i.e., its reciprocal. Usually the meaning of a
negative first power is clear from context. If in doubt, ask.
Knowledge Box 3.10
Computing inverse functions
Suppose that y D f .x/. If we can solve x D f .y/ for y D g.x/, then
g.x/ D f
1
.x/ on some interval.
Definition 3.2 A function has a universal inverse if there is a single function that is its inverse on
its entire domain.
Example 3.42 Suppose that
f .x/
D
x C 3
1 x
Find f
1
.x/.
Solution:
Since we have y D
x C 3
1 x
, we solve x D
y C 3
1 y
for y.
x D
y C 3
1 y
x.1 y/ D y C 3
x xy D y C 3
x 3 D xy C y
x 3 D y.x C 1/
3.3. DERIVATIVES OF THE LIBRARY OF FUNCTIONS 99
x 3
x C 1
D y
y D
x 3
x C 1
So, we have:
f
1
.x/ D
x 3
x C 1
Now let’s check that f .f
1
.x// D x:
x 3
x C 1
C 3
1
x 3
x C 1
D
x 3 C 3.x C 1/
x C 1 .x 3/
D
x 3 C 3x C 3
x x C 1 C 3
D
4x
4
D x
and we have verified that the inverse is correct.
˙
We now have a firm enough grasp of inverse functions to go to work on the inverse trigonomet-
ric functions. ere is an interesting feature of functions that permits them to have universal
inverses—they must pass the horizontal line test—similar to the vertical line test for being a
function. Any y-value where a horizontal line intersects the graph of a function in two places
is a place where inverse values are ambiguous.
Figure 3.5 shows the horizontal line test applied to the sine function. Clearly, it fails the test.
If you look back at the graphs of the other trigonometric functions in Section 2.3, you will see
that all of them egregiously fail the horizontal line test. For that reason inverses are defined for
only part of the domain of the trig functions. e following table gives the domain and range of
each of the inverse trigonometric functions.
100 3. LIMITS, DERIVATIVES, RULES, AND THE MEANING OF THE DERIVATIVE
1
-1
0
2
3
2
2
3
2
Figure 3.5: e sine function fails the horizontal line test.
Properties and formulas for trigonometric functions
Name Abbrev. Domain Range
inverse sine sin
1
Œ1; 1
h
2
;
2
i
inverse cosine cos
1
Œ1; 1 Œ0;
inverse tangent tan
1
.1; 1/
2
;
2
inverse cotangent cot
1
.1; 1/ .0; /
inverse secant sec
1
.1; 1 [ Œ1; 1/
h
0;
2
[
2
;
i
inverse cosecant csc
1
.1; 1 [ Œ1; 1/
h
2
; 0
[
0;
2
i
Sometimes an alternate notion is used for inverse trig functions. e prefix “arc” is added
to the function name instead of the exponent 1. So, for example, sin
1
is written arcsin.
Figures 3.6–3.8 show the graphs of the inverse trig functions, and Knowledge Box 3.11 gives
the formulas for their derivatives.
3.3. DERIVATIVES OF THE LIBRARY OF FUNCTIONS 101
-=2
=2
1-1
Figure 3.6: e inverse sine (light) and inverse cosine (dark) functions.
-=2
=2
=2
6-6
Figure 3.7: e inverse tangent (light) and inverse cotangent (dark) functions.
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