Barnett/Ziegler/ByleenCollege Algebra with Trigonometry, 7th Edition

Chapter Six

Trigonometric Functions

Copyright © 2001 by the McGraw-Hill Companies, Inc.

θ

Initial side

Terminal side

β

Initial side

Terminal side

α

Initial side

Terminal side

x

y

III

III IV

θInitial side

Terminal side

x

y

III

III IV

θx

y

III

III IV

θ

Initial side

Terminal side

θInitial side

Terminal side

(a) θ positive (b) θ negative (c) and β coterminal

(a) θ is a quadrantal (b) θ is a third-quadrant (c) θ is a second-quadrant angle angle angle

Angles

6-1-58-1

180° 90°

θθ

(a) Straight angle

(12 rotation)

(b) Right angle

(14 rotation)

(c) Acute angle

(0° < θ < 90°)(d) Obtuse angle

(90° < θ < 180°)

Angles

6-1-58-2

θ = sr radians

Also, s = rθ

s

rO

θ

r

O r

r s = r

1 radian

θ= rr = 1 radian

Radian Measure

6-1-59

sin θ = OppHyp csc θ =

HypOpp

cos θ = AdjHyp sec θ =

HypAdj

tan θ = OppAdj cot θ =

AdjOpp

0˚ < θ < 90°

Trigonometric Functions of Acute Angles

6-2-60

θa

b

a

br

P ( a, b )

a

b

θ

P(a, b)

r

a

b

r

θa

b

a

b

P ( a, b )

For an arbitrary angle θ :

sin θ = br

cos θ = ar

tan θ = ba , a

? 0

r = a2 + b2 > 0; P(a, b) is anarbitrary point on the terminalside of θ,(a, b) ? (0, 0)

csc θ = rb , b ? 0

secθ = ra , a

? 0

cot θ = ab , b ? 0

Trigonometric Functions with Angle Domains

6-3-61

QUADRANT QUADRANT QUADRANT QUADRANT

I II III IV

ar

b r a b r a b r a

++

+ + – + + – – + +

sin x =

br

csc x = rb

+ + – –

cos x =

ar

sec x = ra

+ – – +

tan x =

ba

cot x = ab

+ – + –

b

–

Signs of the Trigonometric Functions

(a, b)

(a, b)(a, b)

(a, b)

+++

++

++

––

–

6-3-62

1. To form a reference triangle for θ , draw a perpendicular from a point P(a, b) on the terminal side of θ to the horizontal axis.

2. The reference angle is the acute angle (always taken positive) between the terminal side of θ and the horizontal axis.

a

b

θ

a

b

P(a, b)

(a, b) ? (0, 0) is always positive

Reference Triangle and Reference Angle

6-4-63

3

1

2

30°

60°

(/6)

(/3)

2

1

1

45°

45°

(/4)

(/4)

30—60 and 45 Special Triangles

6-4-64

a

b

0

U

xP(a, b)

(1, 0)a

b

0

U(1, 0)

P(a, b)x = 0 a

b

0

U

(1, 0)

P(a, b)

|x|

1. For x > 0: 2. For x = 0: 3. For x < 0:

y = sin x = b y = cos x = a y = tan x = ba , a ? 0

y = csc x = 1b , b ? 0 y = sec x =

1a , a

? 0 y = cot x = ab , b

? 0

In all cases, we define:

Where y is the dependent variable and x is the independent variable.

Circular Functions

6-5-65

Circular Function Trigonometric Function

sin x = b = b1 = sin (x radians)

cos x = a = a1 = cos ( x radians)

tan x = ba (a 0) = tan (x radians)

csc x = 1b (b 0) = csc ( x radians)

sec x = 1a (a 0) = sec (x radians)

cot x = ab (b 0) = cot (x radians)

x rad2

/2

3 /2

(1, 0)(–1, 0)

(0, –1)

(0, 1)

0 cos x

sin x

r= 1

P(cos x, sin x)ab b

a

x units

Circular Functions and Trigonometric Functions

6-5-66

2

/2

3 /2

(1, 0)(–1, 0)

(0, –1)

(0, 1)

0

1

P(cos x, sin x)ab b

a

x

a

b

x

y

1

-1

0 2 3 4––2

Period: 2

Domain: All real numbers

Range: [–1, 1]

Symmetric with respect to the origin

Graph of y = sin x

y = sin x = b

6-6-67

2

/2

3 /2

(1, 0)(–1, 0)

(0, –1)

(0, 1)

0

1

P(cos x, sin x)ab b

a

x

a

b

x

y

1

-1

0 2 3 4––2

Period: 2

Domain: All real numbers

Range: [–1, 1]

Symmetric with respect to they axisy = cos x = a

Graph of y = cos x

6-6-68

x

y

2 –2 – 0

1

–1

52

32

2

–52

–32

–2

Period:

Domain: All real numbers except /2 + k , k an integer

Range: All real numbers

Symmetric with respect to the origin

Increasing function between asymptotes

Discontinuous at x = /2 + k , k an integer

Graph of y = tan x

6-6-69

2–2 – 0

–2

x

y

1

–1

32

2

–32

Period:

Domain: All real numbers except k , k an integer

Range: All real numbers

Symmetric with respect to the origin

Decreasing function between asymptotes

Discontinuous at x = k , k an integer

Graph of y = cot x

6-6-70

x

y

1

–1 2––2 0

y = sin x

y = csc x

sin x1=

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

Graph of y = csc x

6-6-71

x

y

1

–1 2––2 0

y = cos x

y = sec x

cos x1=

Period: 2

Domain: All real numbers except /2 + k, k an integer Symmetric with respect to the y axis

Discontinuous at x = /2 + k, k an integer Range: All real numbers y such that y –1 or y 1

Graph of y = sec x

6-6-72

Step 1. Find the amplitude | A |.

Step 2. Solve Bx + C = 0 and Bx + C = 2:

Bx + C = 0 and Bx + C = 2

x = –CB x = –

CB +

2B

Phase shift Period

Phase shift = –CB Period =

2 B

The graph completes one full cycle as Bx + C varies from 0 to2 — that is, as x varies over the interval

–CB , –

CB +

2B

Step 3. Graph one cycle over the interval

–CB , –

CB +

2B .

Step 4. Extend the graph in step 3 to the left or right as desired.

Graphing y = A sin(Bx + C) and y = A cos(BX + C)

6-7-73

DOMAIN f RANGE f

f

DOMAIN f –1RANGE f –1

–1f

x

y

f (x)

f ( y)–1

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. 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.

5. f[f–1(y)] = y for y in the domain of f–1

f–1[f(x)] = x for x in the domain of fx = f (y)–1

y = f (x)

y

x

Facts about Inverse Functions

6-9-74

x

y

1

–1

– 2

2

Sine function

–

2 , –1

2 , 1

x

y

1

–1 2

–2

(0,0)

y = sin x

–1, –

2

1, 

2

x

y

1–1

2

–2

(0,0)

y = sin x = arcsin x

–1

DOMAIN =

–2 ,

RANGE = [–1, 1]Restricted sine function

DOMAIN = [–1, 1]

RANGE =

–2 ,

2

Inverse sine function

2

Inverse Sine Function

6-9-75

x

y

1

–1

y = cos x

x

y

1

–1

(0,1)

( , –1)

02

2 ,0

–1

y = cos x = arccos x

–1

2

x

y

1

(1,0)

(–1, )

0

0,

2

Cosine function

DOMAIN = [0, ] DOMAIN = [–1, 1]RANGE = [–1, 1] RANGE = [0, ]Restricted cosine function Inverse cosine function

Inverse Cosine Function

6-9-76

x

y

1

–1

2

32

32

– –2

y = tan x

–

4, –1

4, 1

x

y

1

–1 2

–2

y = tan x

–1 , –

4

1, 

4

y = tan x = arctan x

–1

x

y

1

–1

2

–2

Tangent function

DOMAIN =

–2 ,

2

RANGE = (– , )Restricted tangent function

DOMAIN = (– , )

RANGE =

–2 ,

2

Inverse tangent function

Inverse Tangent Function

6-9-77