Sullivan Algebra and Trigonometry: Section 6.5

Unit Circle Approach; Properties of the Trig Functions

Objectives of this Section

• Find the Exact Value of the Trigonometric Functions Using the Unit Circle

• Determine the Domain and Range of the Trigonometric Functions

• Determine the Period of the Trigonometric Functions

• Use Even-Odd Properties to Find the Exact Value of the Trigonometric Functions

The unit circle is a circle whose radius is 1 and whose center is at the origin.

Since r = 1:

s r

becomes

s

(0, 1)

(-1, 0)

(0, -1)

(1, 0)

s

y

x

(0, 1)

(-1, 0)

(0, -1)

(1, 0)

t

y

x

P = (a, b)

Let t be a real number and let P = (a, b) be the point on the unit circle that corresponds to t.

The sine function associates with t the y-coordinate of P and is denoted by

sin t b

The cosine function associates with t the x-coordinate of P and is denoted by

cost a

as defined is the,0 If functiontangent a

tan tba

as defined is the,0 If functioncosecant b

csc tb

1

as defined is the,0 If functionsecant a

sec ta

1

as defined is the,0 If functioncotangent b

cot tab

functions.

ometricsix trigon theof eexact valu theFind

. toscorrespond that circleunit on thepoint the

be 4

15,

4

1let andnumber real a be Let

t

Pt

4

15,41, ba

sin t b 154

cost a 14

tan tba

15

41

415

4

15,41, ba

csc tb

1 115

4

415

4 1515

sec ta

1 11

44

cot tab

1

415

4

115

1515

(0, 1)

(-1, 0)

(0, -1)

(1, 0)

t

y

x

P = (a, b)

If radians, the six

are

defined as

t trigonometric

functions of the angle

sin sin

cos cos

tan tan

t

t

t

csc csc

sec sec

cot cot

t

t

t

y

x

r

a

b

For an angle in standard position, let

be any point on the terminal side of . Let

equal the distance from the origin to . Then

P a b

r

P

( , )

0 ,tan cos sin aa

b

r

a

r

b

0 ,cot 0 ,sec 0 ,csc bb

aa

a

rb

b

r

Given that sec = and sin > 0, find the

exact value of the remaining five trigonometric

functions.

52

P=(a,b)

(5, 0)

x y2 2 25

sec , ,

52

5 2ra

r aso

a b r bbr

2 2 2 with > 0 since sin > 0

2 52 2 2b

4 252 b

b2 21

b 21

a b r 2 21 5, ,

sin br

215

cos ar

25

tan

ba

212

212

csc rb

521

5 2121

cot ab

221

2 2121

(0, 1)

(-1, 0)

(0, -1)

(1, 0)

t

y

x

P = (a, b)

The domain of the sine function is the set of all real numbers.

The domain of the cosine function is the set of all real numbers.

The domain of the tangent function is the set of all real numbers except odd multiples of

2 90 .

The domain of the secant function is the set of all real numbers except odd multiples of

2 90 .

The domain of the cotangent function is the set of all real numbers except integral multiples of 180 .

The domain of the cosecant function is the set of all real numbers except integral multiples of 180 .

Let P = (a, b) be the point on the unit circle that corresponds to the angle . Then, -1 < a < 1 and -1 < b < 1.

sin b

1 1sincos a

1 1cos

sin 1 cos 1

Range of the Trigonometric Functions

11

sin

1csc

b

1cscor 1csc

1secor 1sec

11

cos

1sec

a

cot

tan

A function is called if there is a

positive number such that whenever is

in the domain of , so is , and

f

p

f p

periodic

f p f p

If there is a smallest such number p, this smallest value is called the (fundamental) period of f.

Periodic Properties

sin sin

cos cos

tan tan

2

2

csc csc

sec sec

cot cot

2

2

4

7cot (b) 390sec (a)

of eexact valu theFind

3

3230sec36030sec390sec (a)

14

3cot

4

3cot

4

7cot )b(

Even-Odd Properties

sin sin

cos cos

tan tan

csc csc

sec sec

cot cot

4cot (b) 30sin (a)

of eexact valu theFind

2

130sin30sin )a(

14

cot4

cot (b)