Download a, b

Barnett/Ziegler/Byleen
Precalculus: A Graphing Approach
Chapter Five
Trigonometric Functions
Copyright © 2000 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
Copyright © 2000 by the McGraw-Hill Companies, Inc.
5-1-44
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0
(a, b)
b0
W(x)
(1, 0)
a0
a
cot x = b
u
b0
Copyright © 2000 by the McGraw-Hill Companies, Inc.
5-2-45
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
Copyright © 2000 by the McGraw-Hill Companies, Inc.
5-3-46(a)
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°)
Copyright © 2000 by the McGraw-Hill Companies, Inc.
5-3-46(b)
Radian Measure
s

s
 = r radians
Also, s = r 
r
r
O
s =r
r
r
 = r = 1 radian
O
r
Copyright © 2000 by the McGraw-Hill Companies, Inc.
1 radian
5-3-47
Trigonometric Functions with
Angle Domains
If  is an angle with radian measure x, then the value of each trigonometric
function at  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
Copyright © 2000 by the McGraw-Hill Companies, Inc.
x units
arc length
a
(1, 0)
5-4-48
Trigonometric Functions with Angle
Domains Alternate Form
If  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 , then:
b
b
b
a
P(a, b)
r

b
b

a
r

r
a
a
r
sec  = a , a  0
a
cos  = r
a
,a0
b
a
b
P(a, b)
P(a, b)
r
csc  = b , b  0
b
sin  = r
tan  =
a
r = a2 + b2 > 0; P(a, b) is an
arbitrary point on the terminal
side of , (a, b)  (0, 0)
a
cot  = b , b  0
Copyright © 2000 by the McGraw-Hill Companies, Inc.
5-4-49
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)
Copyright © 2000 by the McGraw-Hill Companies, Inc.
5-4-50
30— 60  and 45  Special Triangles
30 °
(  /6)
45 °
(  /4)
2
2
3
1
45 °
(  /4)
1
60 °
(  /3)
1
Copyright © 2000 by the McGraw-Hill Companies, Inc.
5-4-51
Right Triangle Ratios
sin  
Opp
Hyp
csc  
Hyp
Opp
cos  
Adj
Hyp
sec  
Hyp
Adj
tan  
Opp
Adj
cot  
Adj
Opp
Hyp
Opp

Adj
0° <  < 90°
Copyright © 2000 by the McGraw-Hill Companies, Inc.
5-5-52
 /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
Copyright © 2000 by the McGraw-Hill Companies, Inc.
5-6-53
 /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
Copyright © 2000 by the McGraw-Hill Companies, Inc.
5-6-54
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
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5-6-55
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
Copyright © 2000 by the McGraw-Hill Companies, Inc.
5-6-56
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
Copyright © 2000 by the McGraw-Hill Companies, Inc.
5-6-57
Graph of y = sec x
y
y = sec x
=
1
cos x
y = cos x
1
–2
–

0
–1
2
x
Period: 2
,
Domain: All real numbers except /2 + k
k an integer
Range: All real numbers y such that y  –1 or y  1
Symmetric with respect to the y axis
Discontinuous at x =
Copyright © 2000 by the McGraw-Hill Companies, Inc.
/2 + k, k an integer
5-6-58
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.


Copyright © 2000 by the McGraw-Hill Companies, Inc.
5-7-59
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
DOMAIN f –1
5. f [f –1(y)] = y
f –1[f(x)] = x
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
Copyright © 2000 by the McGraw-Hill Companies, Inc.
x = f –1( y )
x
5-9-60
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
Copyright © 2000 by the McGraw-Hill Companies, Inc.
5-9-61
Inverse Cosine Function
y
1
x

–1
Cosine function
y
= arccos x
y = cos x
1
0
–1
y
y = cos–1 x

(–1,  )
(0,1)
  ,0 
2 

2


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
Copyright © 2000 by the McGraw-Hill Companies, Inc.
5-9-62
Inverse Tangent Function
y
y = tan x

2
1

–
2
3
–
2
3
2
Tangent function
x
–1
y
y = tan–1x
= arctanx
y = tan x
–

2
  , 1
4 
1
 –  , –1
 4


2
–1
y

2
1 ,  
4

–1
x
1
x
 –1 , –   – 
4

2
  
DOMAIN = – 2 , 2 


RANGE = (– ,)
Restricted tangent function
DOMAIN = (– ,  )
  
RANGE = – 2 , 2 


Inverse tangent function
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5-9-63