Sullivan Algebra and Trigonometry: Section 6.4

Trig Functions of General AnglesObjectives of this Section

• Find the Exact Value of the Trigonometric Functions for General Angles

• Determine the Sign of the Trigonometric Functions of an Angle in a Given Quadrant

• Use Coterminal Angles to Find the Exact Value of a Trigonometric Function

• Find the Reference Angle of a General Angle

• Use the Theorem of Reference Angles

Let be any angle in standard position, and

let denote the coordinates of any point,

except the origin (0, 0), on the terminal side

of . If denotes the distance from

(0, 0) to ( , then the

are defined as the ratios

a b

r a b

a b

,

, )

2 2

six trigonometric

functions of

sin cos tan

csc sec cot

b r a r b a

r b r a a b

provided no denominator equals 0.

(a, b)

rx

y

Find the exact value of each of the six trigonometric functions of a positive angle if (-2, 3) is a point on the terminal side.

(-2, 3)

x

y

a b 2 3,

r a b 2 2 2 22 3 13( )

sin br

313

3 1313

cos ar

213

2 1313

tan

ba

32

32

csc rb

133

sec ra

132

cot ab

23

P= (1, 0)

sin sin0 001

0 br

cos cos0 011

1 ar

tan tan0 001

0 ba

csc csc0 010

rb

sec sec0 011

1 ra

cot cot0 010

ab

P= (a, b)

x

y

P= (0,1)

sin sin2

9011

1 br

cos cos2

9001

0 ar

tan tan2

9010

ba

csc csc2

9011

1 rb

sec sec2

9010

ra

cot cot2

9001

0 ab

x

y

sin

cos

tan

csc

sec

cot

0

1

0

1

Not defined

Not defined

180( radians)

1

0

1

Not defined

Not defined

0

270 3 2( radians)

x

y

(a, b)

a < 0, b < 0, r > 0

r

a > 0, b > 0, r > 0a < 0, b > 0, r > 0

a > 0, b < 0, r > 0

I (+, +)

All positive

II ,

sin , csc 0 0

All others negative

III ,

tan , cot 0 0

All others negative

IV ,

cos , sec 0 0

All others negative

x

y

Two angles in standard position are said to be coterminal if they have the same terminal side.

x

y

Let denote a nonacute angle that lies in a quadrant. The acute angle formed by the terminal side of and either the positive x-axis or the negative x-axis is called the reference angle for .

Reference Angle

Finding the reference angle

1.

.

Add / subtract multiples of 360 2

until you obtain an angle between

0 and 360 0 and 2 radians

2. Determine the quadrant in which the terminal side of the angle formed by the angle lies.

180

180

360

2

x

y

Reference Angles

If is an angle that lies in a quadrant and if

is its reference angle, then

sin sin csc csc tan tan cos cos

cot cot sec sec

where the + or sign depends on the

quadrant in which lies.

Find the exact value of each of the following trigonometric functions using reference angles:

(a) cos 570 (b) tan16

3

(a) 570 360 210 in Quadrant III, so cos < 0

210 180 30

cos cos210 303

2

b 16

3 2

163

63

103

103

63

43

is in Quadrant III, so tan > 0

43 3

tan tan16

3 33

2