1.6

Trigonometric Functions

What you’ll learn about…

Radian Measure Graphs of Trigonometric Functions Periodicity Even and Odd Trigonometric Functions Transformations of Trigonometric Graphs Inverse Trigonometric Functions

…and why

Trigonometric functions can be used to model periodic behavior and applications such as musical notes.

Radian Measure

The radian measure of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle.

Radian Measure

An angle of measure θ is placed in standard position at the center of circle of radius r,

Trigonometric Functions of θ

The six basic trigonometric functions of are

defined as follows:

sine: sin cosecant: csc

cosine: cos secant: sec

tangent: tan cotangent: cot

y r

r y

x r

r xy x

x y

q

q q

q q

q q

= =

= =

= =

Graphs of Trigonometric Functions When we graph trigonometric functions in the coordinate plane, we

usually denote the independent variable (radians) by x instead of θ .

Angle Convention

Angle Convention: Use Radians

From now on in this book, it is assumed that all angles are measured in radians

unless degrees or some other unit is stated explicitly. When we talk about the angle

we 3

pmean radians ( which is 60°), not degrees.

3 3

When you do calculus, keep your calculator in radian mode.

p p

Periodic Function, Period

( )( ) ( )

A function is if there is a positive number such

that for every value of . The smallest value

of p is the of .

The functions cos , sin , sec and csc are periodic wi

f x p

f x p f x x

f

x x x x

periodic

period

+ =

th

period 2 . The functions tan and cot are periodic with

period .

x xp

p

Even and Odd Trigonometric Functions The graphs of cos x and sec x are even functions

because their graphs are symmetric about the y-axis.

The graphs of sin x, csc x, tan x and cot x are odd functions.

cosy x= siny x=

Example Even and Odd Trigonometric Functions

Show that csc is an odd function.x

( )( )1 1

csc cscsin sin

x xx x

- = = -- -

Transformations of Trigonometric Graphs The rules for shifting, stretching, shrinking and reflecting

the graph of a function apply to the trigonometric functions.

( )( )y a f b x c d= + +

Vertical stretch or shrink

Reflection about x-axis

Horizontal stretch or shrink

Reflection about the y-axisHorizontal shift

Vertical shift

Example Transformations of Trigonometric Graphs

( )Determine the period, domain, range and draw the graph of

2sin 4y x p=- +

[-5, 5] by [-4,4]

( )

We can rewrite the function as 2sin 44

2The period of sin is . In our example 4,

2so the period is = . The domain is .

4 2The graph is a basic sin curve w

y x

y a bx bb

x

p

p

p p

æ öæ ö÷ç ÷ç=- + ÷÷ç ç ÷÷ç ÷ç è øè ø

= =

- ¥ ¥,

ith an amplitude of 2. Thus, the range is [ 2, 2].

The graph of the function is shown together with the graphof the sin function.x

-

Inverse Trigonometric Functions

None of the six basic trigonometric functions graphed in Figure 1.42 is one-to-one. These functions do not have inverses. However, in each case, the domain can be restricted to produce a new function that does have an inverse.

The domains and ranges of the inverse trigonometric functions become part of their definitions.

Inverse Trigonometric Functions

1

1

1

1

1

1

cos 1 1 0

sin 1 1 2 2

tan 2 2

sec 1 0 ,2

csc 1 , 02 2

cot 0

y x x y

y x x y

y x x y

y x x y y

y x x y y

y x x y

Function Domain Range

pp p

p p

pp

p p

p

-

-

-

-

-

-

= - £ £ £ £

= - £ £ - £ £

= - ¥ < <¥ - < <

= ³ £ £ ¹

= ³ - £ £ ¹

= - ¥ < <¥ < <

Inverse Trigonometric Functions The graphs of the six inverse trigonometric functions are shown

here.

Example Inverse Trigonometric Functions

1 1Find the measure of sin in degrees and in radians.

2- æ ö÷ç- ÷ç ÷çè ø

1

1

1Put the calculator in degree mode and enter sin .

2

The calculator returns 30

1Put the calculator in radian mode and enter sin .

2

The calculator returns .52359877556 radians.

-

-

æ ö÷ç- ÷ç ÷çè ø

-

æ ö÷ç- ÷ç ÷çè ø

-

°.

This is the same as radians.6

p-

Quick Quiz Sections 1.4 – 1.6

2

You should solve the following problems without using a graphing caluclator.

1. Which of the following is the domain of ( ) log 3 ?

(A) ,

(B) ,3

(C) 3,

(D) [ 3, )

(E) ( ,3]

f x x

Quick Quiz Sections 1.4 – 1.6

2

You should solve the following problems without using a graphing caluclator.

1. Which of the following is the domain of ( ) log 3 ?

(

(C) 3,

A) ,

(B) ,3

(D) [ 3, )

(E) ( ,3]

f x x

Quick Quiz Sections 1.4 – 1.6

2. Which of the following is the range of ( ) 5cos 3?

(A) ,

(B) 2,4

(C) 8,2

(D) 2,8

2 8(E) ,

5 5

f x x

Quick Quiz Sections 1.4 – 1.6

2. Which of the following is the range of ( ) 5cos 3?

(A) ,

(B) 2,4

(C) 8,2

2 8(E) ,

5

(D) 2,8

5

f x x

Quick Quiz Sections 1.4 – 1.6

3

3. Which of the following gives the solution of tan -1 in ?2

(A) 4

(B) 4

(C) 33

(D) 4

5(E)

4

x x

Quick Quiz Sections 1.4 – 1.6

3

3. Which of the following gives the solution of tan -1 in ?2

(A) 4

(B) 4

(C) 33

5(E)

D)

4

( 4

x x