082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

122 Chapter 3 The trigonometric functions
sin x
1
–1 1 x
sin –1 x
– p –2
p –2 x
–1
Figure3.10
Thefunctionsinx isone-to-one [ ] ifthe
domainis restrictedto − π 2 , π .
2
–1
p –2
– p –2
Figure3.11
Theinverse sine
function, sin −1 x.
1
x
Figure3.12
A single input produces many
outputvalues.Thisisnotafunction.
Notethaty = sinxisamany-to-onefunction.Ifthedomainisrestrictedto[−π/2, π/2]
then the resulting function isone-to-one. This isshown inFigure 3.10.
Recall from Section 2.3 that a one-to-one function has a corresponding inverse. So
if the domain of y = sinx is restricted to [−π/2,π/2], then an inverse function exists.
A graph ofy = sin −1 x is shown in Figure 3.11. Without the domain restriction, a
one-to-many graph would result as shown in Figure 3.12. To denote the inverse sine
function clearly, we write
y=sin −1 x − π 2 y π 2
Example3.2 Useascientific calculatortoevaluate
(a) sin −1 (0.3169)
(b) sin −1 (−0.8061)
Solution (a) sin −1 (0.3169) = 0.3225
(b) sin −1 (−0.8061) = −0.9375
A word of warning about inverse trigonometric functions is needed. The calculator returns
a value of 0.3225 for sin −1 (0.3169). Note, however, that sin(0.3225 ± 2nπ) =
0.3169, n = 0,1,2,3,..., so there are an infinite number of values of x such that
sinx = 0.3169. Only one of these values is returned by the calculator. This is because
the domain ofy = sinx is restricted to ensure it has an inverse function. To ensure the
inverse functionsy = cos −1 x andy = tan −1 x can be obtained, restrictions are placed
on the domains of y = cosx and y = tanx. By convention, y = cosx has its domain
restrictedto[0,π] whereas withy = tanx the restrictionis (−π/2,π/2).