4.2, 4.4 – The Unit Circle, Trig Functions

The unit circle is defined by the equation x2 + y2 = 1.

It has its center at the origin and radius 1.

(0 , 1)

(1 , 0) 1

(0 , 1)

(1 , 0)

4.2, 4.4 – The Unit Circle, Trig Functions

If the point (x , y) lies on the terminal side of θ, the six trig functions of θ can be defined as follows:

(x , y)

y θ x

x

yθtan

y

xθcot

x

rθsec

y

rθ csc

r

yθsin

r

xθ cos

A reference triangle is made by “dropping” a perpendicular

line segment to the x-axis.

r2 = x2 + y2

r(− , +)

(− , −) (+ , −)

4.2, 4.4 – The Unit Circle, Trig Functions

Evaluate the six trig functions of an angle θ whose terminal side contains the point (−5 , 2).

(−5 , 2)

2

−5 5

2θtan

2

5θcot

5

29θsec

2

29θ csc

29

292

29

2θsin

29

295

29

5θ cos

29

4.2, 4.4 – The Unit Circle, Trig Functions

For a unit circle (radius 1)

1 (1 , 0)

1

(x , y)

sin = y

cos = x

tan = x

y

4.2, 4.4 – The Unit Circle, Trig Functions

1

(1 , 0) 1

3π

2

3 ,

2

1

4.2, 4.4 – The Unit Circle, Trig Functions

4.2, 4.4 – The Unit Circle, Trig Functions

Find the six trig functions of 0º

(1 , 0)x

yθtan

y

xθcot

x

rθsec

y

rθ csc

r

yθsin

r

xθ cos

r = 1

undef.0

1

01

0

11

1

undef.0

1

01

0

11

1

4.2, 4.4 – The Unit Circle, Trig Functions

Deg. Rad. Sin Cos Tan

0º 0 0 1 0

30º

45º 1

60º

90º 1 0 undef.

180º 0 −1 0

270º −1 0 undef.

360º 2 0 1 0

Summary

21

22

23

21

22

23

33

33π

6π

4π

2π

23π

4.2, 4.4 – The Unit Circle, Trig Functions

Basic Trig Identities

θtan

1θcot

θ cos

1θsec

θsin

1θ csc

Reciprocal Quotient Pythagoreansin2θ + cos2θ = 1

tan2θ + 1 = sec2θ

cot2θ + 1 = csc2θθsin

θ cosθcot

θ cos

θsin θtan

Cofunctionsinθ = cos(90

θ)

tanθ = cot(90 θ)

secθ = csc(90 θ)

Evencos(θ) = cos θ

sec(θ) = sec θ

Oddsin(θ) = sin θ

tan(θ) = tan θ

cot(θ) = cot θ

csc(θ) = csc θ

4.2, 4.4 – The Unit Circle, Trig Functions

Use trig identities to evaluate the six trig functions of an

angle θ if cos θ = and θ is a 4th quadrant angle.

sin2θ = 1 − cos2θ5

4

θcos1θsin 2

2541

25161

25

1625

25

9

53

4

3

54

53

θtan

3

4θcot

4

5θsec

3

5θ csc

5

3θsin

5

4θ cos

4

5 −3

4.2, 4.4 – The Unit Circle, Trig Functions

For any angle θ, the reference angle for θ, generally written θ', is always positive, always acute, and always

made with the x-axis.

θ θ'

4.2, 4.4 – The Unit Circle, Trig Functions

For any angle θ, the reference angle for θ, generally written θ', is always positive, always acute, and always

made with the x-axis.

θ'

θ

4.2, 4.4 – The Unit Circle, Trig Functions

For any angle θ, the reference angle for θ, generally written θ', is always positive, always acute, and always

made with the x-axis.

θ θ'

4.2, 4.4 – The Unit Circle, Trig Functions

Find the reference angles for α and β below.

α = 217º β = 301º

α' = 217º − 180º = 37º

β' = 360º − 301º = 59º

37º 59º

4.2, 4.4 – The Unit Circle, Trig Functions

The trig functions for any angle θ may differ from the trig functions of the reference angle θ' only in sign.

θ = 135º

θ' = 180º − 135º = 45º

sin 135º = sin 45º

=

=

cos 135º = −

tan 135º = −1

22

22

22

θ θ'

4.2, 4.4 – The Unit Circle, Trig Functions

A function is periodic if

f(x + np) = f(x)

for every x in the domain of f,every integer n,

and some positive number p (called the period).

0 , 2π

sine & cosine period = 2π

secant & cosecant period = 2π

tangent & cotangent period = π

4.2, 4.4 – The Unit Circle, Trig Functions

sin =

sin =

sin =

3π

23

3π 3tan =

tan =

tan =

π23π 2

3

π43π 2

3

π3π

π23π

3

3

Find the exact value of each.

7

sin300 cot4

cos( 240 ) csc4