Projects - Cengage Learning

C-68 Appendix C
This secant function has an inverse, denoted sec 1 x or arcsec, with
domain of sec 1 range of the restricted secant (q, 1] [1, q)
and
range of sec 1 domain of the restricted secant 30, p 2 2 3p, 3p 2 2
The graph of y sec 1 x is shown in Figure A(ii) and can be obtained by
reflecting the graph in Figure A(i) about the line with equation y x. (Try it.)
What is sec 1 x? Start with the equation y sec x. Interchange x and y to
get x sec y. What does the y in this last equation represent? The last equation
tells us that y sec 1 x is the unique number in the domain of the restricted
secant, 30, p 2 2 3p, 3p 2 2 whose secant is x.
EXAMPLE 1 Values of Some Expressions Involving the Inverse Secant
(a) sec 1 p
2 the unique number in 30, p whose secant is 2
(b) arcsec 2 2 3p, 3p
the unique number in 30, p 2 2 2 2
3p, 3p 2 2
3
whose secant is
12 5p 4
(c) sec[sec 1 (5)] the secant of (the unique number in
whose secant is 5) 5
30, p 2 2 3p, 3p 2 2
(d) sec1 1sec 2 32 the unique number in 30, p 2 2 3p, 3p 2 2 whose secant is
sec 2 3 2 3
y
5
¨
2
Figure B
œ„„ 55
Figure C
3
~
¨
œ„„ 21
8
y
¨
x
x
EXAMPLE 2 Evaluating Expressions Involving the Inverse Secant
(a) sin3sec 1 1 5 224
5
Let u sec 1 1 5 22, then u is in 30, p 2 2 and sec u 2. Using the right triangle
labeled in Figure B, we have
(b)
Alternatively, using identities,
So
sin c sec 1 a 5 bd sin u
opposite
2 hypotenuse 121
5
cos u 1
sec u 1
52 2 5
sin u 21 cos 2 u 21 (25) 2 121
5
Why the positive
square root?
tan3sec 1 8
1 324
Let u sec 1 8
1 32 and let ũ be its reference angle. Then u is in
Quadrant III and sec u 8 3. Using the right triangle labeled in Figure C,
we have
8
tan c sec ũ opposite
adjacent 155
1 a bd tan u tan
3 3