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Appendix C C-69<br />
Alternatively, using identities,<br />
So<br />
8<br />
sec u <br />
3<br />
tan u 2sec 2 u 1 2(83) 2 1 155<br />
3<br />
Why the positive<br />
square root?<br />
Now we construct a second version of the inverse secant by restricting the<br />
secant’s domain in a different way. We’ll use a capital S to distinguish this version<br />
from our previous version of a restricted secant. Let<br />
y Sec x<br />
with<br />
domain 30, p 2 2 1 p 2 , p4 and range (q, 1] [1, q)<br />
Then Sec x is one-to-one as can be seen from its graph in Figure D(i). So this<br />
Secant function has an inverse denoted Sec 1 or Arcsec, with<br />
domain of Sec 1 range of the restricted Secant (q, 1] [1, q)<br />
and<br />
range of Sec 1 domain of the restricted Secant 30, p 2 2 1 p 2 , p4<br />
The graph of y Sec 1 x is shown in Figure D(ii) and can be obtained by<br />
reflecting the graph in Figure D(i) about the line with equation y x. (Try it.)<br />
What is Sec 1 x? This time, y Sec 1 x the unique number in the restricted<br />
domain of Secant, 30, p 2 2 1 p 2 , p4, whose Secant is x.<br />
y<br />
y<br />
(1, p)<br />
p<br />
y Sec 1 x<br />
(p, 1)<br />
1<br />
p<br />
2<br />
p<br />
(p, 1)<br />
x<br />
1<br />
p<br />
2<br />
(1, 0)<br />
x<br />
Figure D<br />
(i) The graph of y Sec x for x in<br />
p p<br />
[ 0, ) ∪ ( , p<br />
2 2 ]<br />
(ii) The graph of y Sec 1 x
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