5
Chapter
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Trigonometric Functions
q1/ \n,
E,i
Vocabulary Check
FiIl in the blanks.
Function
l. y :
Alternative Notation
,,
y:
3.
In
rl
l,
il'
I'
i;
ri
ir
ll
ri
il
ii
i:
\_.
l-7, find the exact value of each
expression without using a calculator.
1. (a) arcsin j
2. (a) arccos)
(b) arcsin 0
(b) arccos 0
ii'
arctanf
+. rul
"or-'/-4\
\ 2/
(b) arctan(-
1)
(b) arctan
6
(D) arcsrn
l-
v
o. ral arccos/- 1)
v5
(a) Use
i'
ii
x
;i
;riji
-
0.8
-
0.6
-0.4
-0.2
iil
'j ,,r
-0.2
0
0.2
0.4
0.6
0.8
hand-drawn graph in part O).
graph.
Numerical and Graphical Analysis Consider
functiony = arctanx.
(a) Use a graphing urility to complete
the
the table.
I
'l!
'
rlllr
x
l,ilrl
v
r;l
ilrll
0
0.2
0.4
0.6
0.8
x
1
-10
-8 -6 -4 -2
v
li
r,
lli
illl
!rlt
rjli,
llti
.r:l
ii,,
:ii
l1
,,'ll
r.1l
,i
-0.4
utility to complete.the table.
v
ti
0.6
(d) Determine any intercepts and symmetry of the
arcsinx.
-t
-
",,",-'(-f)
rll,
i;lii
0.8
(b) Plot the points from the table in part (a) and gaph
the function. @o not ur" a gruphiog utility.i
(c) Use a graphing utility to graph the inverse cosine
function and compare thi result with your
2
a graphing
-
!a
f
funcfionY:
-l
v
x
-
the
table.
8. Numerical and Graphical Analysis Consider the
I
9. Numerical and Graphical Analysis Consider
function;l = arccosr.
(a) Use a graphing utility to complete the
5. (a) arctanCJI)
I
'I
.l
1
c),t"-'(-/)
7. (a) sin-l
li,'
<x<
x
S. 1a;
\2/
lli
-1
arctanx
Exercises
t
--(v(
2-J-
coS ,x
):
:
1i
Range
arcsinx
i
I
t'.
I
rlii;i
x
(b) Plot
the points from the table in part (a) and graph
the function. (Do not use a graphing utitity.)
(c) Use a graphing utility to graph the inverse sine
function and compare the result with your
hand-drawn graph in part (b).
(d) Determine any intercepts and symmeha of the
graph.
0
2
4
6
8
10
v
(b) Plot
the points from the table in parr (a) and graph
the function. @o not or" u gruphirrg utility.)
(c) Use a graphing utility to graph rhe inverse
tangent function and compare the result witb
your hand-drawn graph in part (b).
(d) Detennine
the horizontal asymptotes of the gaph.
5.6
Section
In Exercises 11 and L2, determine the missing coordi_
nates
of the points on the graph of the function.
fl.
t2.
lnverse Tiigonometric Functions
457
33. coslarccos(- 0.3)]
34. sinfarcsin(-0.1)]
35. arcsin(sin 3n)
36.
arccos(c*t)
ss.
'i"-'(ri"f)
37.
o,,-'(,* 112\
/
'i-\
In
Exercises 39-46, find the exact value of the
expressiorr. U-se a graphing utility to verify your
result. Qlint: Make a sketch of a riglrt triangle.)
F,tl'
In Exercises t3-24, use a calculator to approximate
the value of the expression. Round your answer to two
decimal places.
13.
cos-r 0.75
15.
arcsin(-0.75)
14. sin-I 0.56
16. arccos(-0.7)
17. arctan(-6)
19.
18. arctan(-
sin-l 0.19
23.
tan-II.32
1g)
24. tan-r 5.9
i . i:.':
Ii^'-oercises 25 and 26, use a graphing utility to graph
,f'9, and J = x in the same viewing window to verify
geometrically
(Be
that
= talx,
26. f (x): sinx,
25.
g is the inverse function of /.
sure to properly restrict the domain of
f (x)
e(x) : arctanx
g(x) : arcsinx
47. cot(arctanx)
49. sinfarccos(x + 2)]
1,1
31.
ffi(***;)
52. cot(arctan -,/
ss.
"r.(*"t*f)
54. cos(arcsin*
,4
l'*,
32. tan(arctan
35)
1)]
- ft)
rl
In Exercises 55 and 56, use a,graphing utility to graph
and g in the same viewing window io o*#y that the
"f
two functions are equal. Explain why they are equal.
Identify any asymptotes of the graphs.
55.
/(x) :
sin(arctan
2x),
g(x) =
2x
^9
xl
bo
3l-38, use the properties of inverse
to
find
the exact value of the expression.
I , ,rs
0.7)
\
30'
Exercises
sin(arcsin
-
Jl * 4x2
JT=V
s6. f (x): tu"(*""",;),
e@) :
10
fu'
,/Z\
28.
)/ l.*, ,/
Arrl^rd
Iu
48. sin(arctan.r)
50. secfarcsin(x
sr.
l,)
!
In Exercises 27-30, use an
inverse trigonometric
*rite 0 as a function of x.
,4
+0. cot(arctan f)
triangle, as demonstrated in Example 7.)
tunction to
2e'
40. sec(arcsin J)
42. cscfarctan(- f)]
++. tanfarcsin(-f)
jfi' In Exercis es 47-54rwrite an algebraic expression that
is equivalent to the expressiori. (Ilizr; Sketch a right
20. cos-r 0.21
22. arcsin(-0.125)
21. arccos(-0.51)
39. sin(arctan f)
41. cos(arcsin ff)
as. s"clarctan(-!)]
45. sinfarcco'(-3)]
\co
R\
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orl
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I
Il'
:r""-;
f
rc.
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