θsin

+ Counter clockwise

- clockwise

Initial Ray

Terminal Ray

Terminal Ray

Definition of an angle

Radian Measure

2

,0

2

3

r

r1 Radian

57.3 o

2

360o = 2π radians

180o = π radians

Definition of Radians

C= 2πr

C= 2π radii

C= 2π radians

6

3

2

3

2

6

5

2,0

6

7

3

4

2

3

3

5

6

11

Unit Circle – Radian Measure

61

6

2

6

3

64

65

66

4

2

4

3

2,0

4

5

2

3

4

7

Unit Circle – Radian Measure

44

43

42

41

6

4

3

2

3

2

4

3

6

5

2,0

6

7

4

5

3

4

2

3

3

5 4

76

11

Unit Circle – Radian Measure

Degrees

Converting Degrees ↔ Radians

Recall oo

180,180

Converts degrees to Radians

o180Converts Radians to degrees

36

5

180

25

18025

oo

oo

50180

18

5

more examples

4

54

3

Coterminal angles – angles with a common terminal ray

Initial Ray

Terminal Ray

4

54

3

Coterminal angles – angles with a common terminal ray

Initial Ray

Terminal Ray

4

13

Trigonometric Ratios

θReference Angle

Adjacent Leg

HypotenuseOpposite Leg

hypotenuse

legoppositesin

hypotenuse

legadjacentcos

legadjacent

legoppositetan

Basic ratio definitions

2,0

Circle Trigonometry Definitions

(x, y)

Radius = r

Adjacent Leg = x

Opposite Leg = y

r

ysin

r

xcos

x

ytan

reciprocal functions

2,0

Unit - Circle Trigonometry Definitions

(x, y)

Radius = 1

Adjacent Leg = x

Opposite Leg = y

yy

1

sin

xx

1

cos

x

ytan

1

2

1,

2

3

6

2

3,2

1

3

2

2

3,

2

1

3

2

2

1,

2

3

6

5

2,0

2

1,

2

3

6

7

2

3,

2

1

3

4

2

3

2

3,2

1

3

5

2

1,

2

3

6

11

Unit Circle – Trig Ratios sin cos tan

6

4

3

(+, +)

(-, -)

(-, +)

(+, -)

2

1

2

1

2

3

3

3

32

3

2

3

2

11

6

3

2

1

12

3

Reference AnglesSkip π/4’s

2

2,

2

2

4

2

2,0

2

3

2

2

2

2

1

Unit Circle – Trig Ratios sin cos tan

6

4

3

2

2

2

2 1

2

2,

2

2

4

3

2

2,

2

2

4

5

2

2,

2

2

4

7

(+, +)

(-, -)

(-, +)

(+, -)

2

2,0

2

3

Unit Circle – Trig Ratios sin cos tan

6

4

3

(+, +)

(-, -)

(-, +)

(+, -)

sin cos tan

2

2

3 -1

1

1

-1

0

0

0

0

0

0

Ø

Ø

(0, -1)

(0 , 1)

(1, 0)(-1, 0)

0 /2π

Quadrant Angles

View π/4’s

6

4

3

2

3

2

4

3

6

5

2,0

6

7

4

5

3

4

2

3

3

5 4

76

11

Unit Circle – Radian Measure

sin cos tan

6

4

3

2

1

2

1

2

3

3

3

32

3

2

2

2

2 1

(+, +)

(-, -)

(-, +)

(+, -)Degrees

1sin cos tan

2

2

3 -1

1

1

-1

0

0

0

0

0

0

Ø

Ø

0 /2π

Quadrant Angles

A unit circle is a circle with a radius of 1 unit. For every point P(x, y) on the unit circle, the value of r is 1. Therefore, for an angle θ in the standard position: