Transcript Document
Barnett/Ziegler/Byleen
Precalculus: Functions & Graphs, 4th Edition
Chapter Five
Trigonometric Functions
Copyright © 1999 by the McGraw-Hill Companies, Inc.
Wrapping Function
v
v
2
2
1
v
2
1
1
3
(1, 0)
0
(1, 0)
0
u
(1, 0)
0
u
–1
–1
–2
–2
v
u
–3
–2
–1
v
|x|
A(1, 0)
P
0
u
A(1, 0)
u
0
|x|
P
(a) x > 0
(b) x < 0
5-1-48
Circular Functions
If x is a real number and (a, b) are the
coordinates of the circular
point W(x), then:
v
sin x = b
1
csc x = b
cos x = a
1
sec x = a
b
tan x = a
a
a 0 cot x = b
b0
a0
(a, b)
W(x)
(1, 0)
u
b0
5-2-49
Angles
Terminal side
Terminal side
Initial side
Initial side
(a) positive
(b) negative
Terminal
side
IV
(a) is a quadrantal
angle
I
II
Initial side
Initial side
x
III
y
II
I
Terminal
side
III
Initial side
(c) and coterminal
y
y
II
Terminal side
x
Terminal
side
I
x
Initial side
IV
(b) is a third-quadrant
angle
III
IV
(c) is a second-quadrant
angle
5-3-50-1
Angles
180 °
(a) Straight angle
1
( 2 rotation)
90 °
(b) Right angle
1
( 4 rotation)
(c) Acute angle
(0° < < 90°)
(d) Obtuse angle
(90° < < 180°)
5-3-50-2
Radian Measure
s
s
= r radians
Also, s = r
r
r
O
s =r
r
r
= r = 1 radian
O
r
1 radian
5-3-51
Trigonometric Functions with
Angle Domains
If q is an angle with radian measure x, then the value of each trigonometric
function at q is given by its value at the real number x.
Trigonometric
Function
Circular
Function
b
(a, b)
sin
= sin x
cos
= cos x
tan
= tan x
csc
= csc x
sec
= sec x
cot
= cot x
W(x)
x rad
x units
arc length
a
(1, 0)
5-4-52
Trigonometric Functions with Angle
Domains Alternate Form
b
If q is an arbitrary angle in standard
position in a rectangular coordinate
system and P(a, b) is a point r units
from the origin on the terminal
side of q, then:
b
a
P ( a, b
a
)
b
r
r = a2 + b2 > 0; P(a, b) is an
arbitrary point on the terminal
side of ,
(a, b) (0, 0)
b
r
b
a
b
sin = r
a
cos = r
b
tan = a , a 0
a
P(a, b )
r
csc = b , b 0
r
sec = a , a 0
a
cot = b , b 0
a
r
a
b
P ( a, b )
5-4-53
Reference Triangle and Reference Angle
1. To form a reference triangle for , draw a
perpendicular from a point P(a, b) on the terminal side
of to the horizontal axis.
b
2. The reference angle is the acute angle (always taken
positive) between the terminal side of and the
horizontal axis.
a
a
b
(a, b) (0, 0)
is always positive
P(a, b)
5-4-54
30—60 and 45 Special Triangles
30 °
( /6)
45 °
( /4)
2
2
3
1
45 °
( /4)
1
60 °
( /3)
1
5-4-55
Trigonometric Functions with Angle
Domains Alternate Form
b
If q is an arbitrary angle in standard
position in a rectangular coordinate
system and P(a, b) is a point r units
from the origin on the terminal
side of q, then:
b
a
P ( a, b
a
)
b
r
b
b
sin = r
a
cos = r
b
tan = a , a 0
b
r
a
r = a2 + b2 > 0; P(a, b) is an
arbitrary point on the terminal
side of ,
(a, b) (0, 0)
a
P(a, b )
r
csc = b , b 0
r
sec = a , a 0
a
cot = b , b 0
a
r
a
b
P ( a, b )
5-5-53
/2
a
b
b
P(cos x , sin x )
(0, 1)
1
Graph of y = sin x
x
Period: 2
b
a
(–1, 0)
0
a
(1, 0)
2
Domain: All real numbers
Range: [–1, 1]
y = sin x = b
(0, –1)
3 /2
Symmetric with respect to the origin
y
1
–2
–
0
2
3
4
x
-1
5-6-56
/2
a
b
b
P(cos x , sin x )
(0, 1)
1
Graph of y = cos x
x
b
a
(–1, 0)
0
a
Period: 2
2
(1, 0)
Domain: All real numbers
Range: [–1, 1]
Symmetric with respect to the
y axis
y = cos x = a
(0, –1)
3 /2
y
1
–2
–
0
2
3
4
x
-1
5-6-57
Graph of y = tan x
y
Period:
Domain: All real numbers
except /2 + k ,
k an integer
1
–2
–
5
2
–
–
3
2
–
2
–1
2
Range: All real numbers
2
0
3
2
5
2
x
Symmetric with respect to
the origin
Increasing function
between asymptotes
Discontinuous at
x = /2 + k , k an integer
5-6-58
Graph of y = cot x
y
Period:
Domain: All real numbers
except k ,
k an integer
3
–
2
–2
–
2
1
–
2
0
–1
Range: All real numbers
3
2
2
x
Symmetric with respect to
the origin
Decreasing function
between asymptotes
Discontinuous at
x = k , k an integer
5-6-59
Graph of y = csc x
y
y = csc x
=
y = sin x
1
sin x
1
–2
–
0
2
x
–1
Period: 2
Domain: All real numbers except k , k an integer
Range: All real numbers y such that y –1 or y 1
Symmetric with respect to the origin
Discontinuous at x = k , k an integer
5-6-60
Graph of y = sec x
y
y = sec x
=
1
cos x
1
–2
–
0
–1
y = cos x
2
x
Period: 2
,
Domain: All real numbers except /2 + k
Discontinuous at x =
k an integer
/2 + k, k an integer
Symmetric with respect to the y axis
Range: All real numbers y such that y –1 or y 1
5-6-61
Graphing y = A sin(Bx + C) and y = A cos(BX + C)
Step 1.
Find the amplitude | A |.
Step 2.
Solve Bx + C = 0 and Bx + C = 2 :
Bx + C = 0
C
x = –B
and
Bx + C = 2
C 2
x = –B + B
Phase shift
C
Phase shift = – B
Period
2
Period = B
The graph completes one full cycle as Bx + C varies from 0 to
2 — that is, as x varies over the interval
C
C 2
– , –
B+ B
B
Step 3.
C
C 2
Graph one cycle over the interval – B , – B + B .
Step 4.
Extend the graph in step 3 to the left or right as desired.
5-7-62
Graph of y = tan x
y
Period:
Domain: All real numbers
except /2 + k ,
k an integer
1
–2
–
5
2
–
–
3
2
–
2
–1
2
Range: All real numbers
2
0
3
2
5
2
x
Symmetric with respect to
the origin
Increasing function
between asymptotes
Discontinuous at
x = /2 + k , k an integer
5-8-58
Graph of y = cot x
y
Period:
Domain: All real numbers
except k ,
k an integer
3
–
2
–2
–
2
1
–
2
0
–1
Range: All real numbers
3
2
2
x
Symmetric with respect to
the origin
Decreasing function
between asymptotes
Discontinuous at
x = k , k an integer
5-8-59
Graph of y = csc x
y
y = csc x
=
y = sin x
1
sin x
1
–2
–
0
2
x
–1
Period: 2
Domain: All real numbers except k , k an integer
Range: All real numbers y such that y –1 or y 1
Symmetric with respect to the origin
Discontinuous at x = k , k an integer
5-8-60
Graph of y = sec x
y
y = sec x
=
1
cos x
1
–2
–
0
–1
y = cos x
2
x
Period: 2
,
Domain: All real numbers except /2 + k
Discontinuous at x =
k an integer
/2 + k, k an integer
Symmetric with respect to the y axis
Range: All real numbers y such that y –1 or y 1
5-8-61
Facts about Inverse Functions
For f a one-to-one function and f–1 its inverse:
1. If (a, b) is an element of f, then (b, a) is an element of f–1, and conversely.
2. Range of f = Domain of f–1
Domain of f = Range of f–1
3.
DOMAIN f
RANGE f
f
x
f ( x)
f –1( y )
y
f –1
RANGE f –1
5. f[f–1(y)] = y
f–1[f(x)] = x
DOMAIN f –1
4. If x = f–1(y), then y = f(x) for y in the
domain of f–1 and x in the domain
of f, and conversely.
y
y = f (x)
for y in the domain of f–1
for x in the domain of f
x = f –1( y)
x
5-9-63
Inverse Sine Function
y
–
1
2
–1
2
Sine function
y
y = sin x
–
– , –1
2
x
, 1
2
1
2 (0,0)
2
–1
DOMAIN = – 2 , 2
RANGE = [–1, 1]
Restricted sine function
x
y
y = sin –1 x
= arcsin x
1 ,
2
2
(0,0)
–1
–1 , –
2
x
1
–
2
DOMAIN = [–1, 1]
RANGE = – 2 , 2
Inverse sine function
5-9-64
Inverse Cosine Function
y
1
x
–1
Cosine function
y
= arccos x
y = cos x
1
0
–1
y
y = cos –1 x
(0,1)
,0
2
2
(–1, )
x
2
0 ,
2
(1,0)
( , –1)
–1 0
DOMAIN = [0, ]
RANGE = [–1, 1]
Restricted cosine function
1
x
DOMAIN = [–1, 1]
RANGE = [0, ]
Inverse cosine function
5-9-65
Inverse Tangent Function
y
y = tan x
2
1
3
–
2
–
2
3
2
Tangent function
x
–1
y
y = tan –1x
= arctan x
y = tan x
–
2
, 1
4
x
2
1
– , –1
4
–1
y
2
1 ,
4
–1
1
x
–1 , – –
4
2
DOMAIN = – 2 , 2
RANGE = (– ,)
Restricted tangent function
DOMAIN = (– , )
RANGE = – 2 , 2
Inverse tangent function
5-9-66