1. Trigonometry θ

2. + Counter clockwise - clockwise Definition of an angle Terminal Ray
Initial Ray
- clockwise
Emphasis direction of angle and sign
Terminal Ray

3. Coterminal angles – angles with a common terminal ray
Initial Ray
Find second measure by difference from 2π.

4. Coterminal angles – angles with a common terminal ray
Initial Ray
Find negative measure by adding 2π to previous negative angle.

5. Radian Measure

6. Definition of Radians r r C= 2πr C= 2π radii C= 2π radians
360o = 2π radians r
180o = π radians
1 Radian
 57.3 o r
Use Circumference formula C = 2π r to obtain radian measure of entire circle.

7. Unit Circle – Radian Measure
Bottom half is done in similar manner.

8. Unit Circle – Radian Measure

9. Unit Circle – Radian Measure
Click on degrees to see circle as degrees
Degrees

10. Converting Degrees ↔ Radians
Converts degrees to Radians
Recall
Converts Radians to degrees
Examples of converting between angle measures
more examples

11. Trigonometric Ratios

12. Basic ratio definitions
Hypotenuse
Opposite Leg
Reference Angle θ
Adjacent Leg
Review basic triangle definitions

13. Circle Trigonometry Definitions
(x, y)
Radius = r
Opposite Leg = y
Adjacent Leg = x
Wait on clicks until after new definition comes in
reciprocal functions

14. 1 Unit - Circle Trigonometry Definitions (x, y) Radius = 1
Opposite Leg = y
Adjacent Leg = x 1
Unit circle rather than any radius, definitions are just coordinates of endpoint of terminal ray

15. (-, +) (+, +) (+, -) (-, -) Unit Circle – Trig Ratios sin cos tan
Develops basic chart – show how triangle is just adjusted by quadrant signs.
(-, -)
(+, -)
Skip π/4’s
Reference Angles

16. Unit Circle – Trig Ratios
sin
cos
tan
(-, +)
(+, +)
(-, -)
(+, -)

17. (-, +) (+, +) (-, -) (+, -) (0 , 1) (-1, 0) (1, 0) (0, -1)
Unit Circle – Trig Ratios
sin
cos
tan
(-, +)
(+, +)
(0 , 1)
Quadrant Angles
(-1, 0)
(1, 0)
sin
cos
tan
/2π 1 1 Ø
(0, -1)
(-, -)
(+, -) -1 -1 Ø
View π/4’s

18. (-, +) (+, +) (-, -) (+, -) 1 Unit Circle – Radian Measure sin cos tan
Quadrant Angles
sin
cos
tan 1
/2π 1 1 Ø
(-, -)
(+, -) -1
Degrees -1 Ø

19. A unit circle is a circle with a radius of 1 unit
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:

20. Graphing Trig Functions
f ( x ) = A sin bx

23. Amplitude is the height of graph measured from middle of the wave.
Center of wave
f ( x ) = A sin bx

24. f ( x ) = cos x
A = ½ , half as tall

25. f ( x ) = sin x
A = 2, twice as tall

26. Period of graph is distance along horizontal axis for graph to repeat (length of one cycle)
f ( x ) = A sin bx

27. f ( x ) = sin x
B = ½ , period is 4π

28. f ( x ) = cos x
B = 2, period is π

30. The End Trigonometry Hipparchus, Menelaus, Ptolemy
Special Right Triangles The Pythagoreans
Graphs Rene’ DesCartes

31. Reference Angle Calculation
4th Quadrant Angles
3rd Quadrant Angles
2nd Quadrant Angles
Return

32. Unit Circle – Degree Measure 90
120 60
135 45
150 30
180
0/360
210
330
225
315
240
300
270
Return

33. (-, +) (+, +) (-, -) (+, -) 1 Unit Circle – Degree Measure sin cos tan90 30
(+, +)
120 60 45
135 45 60
150 30
Quadrant Angles
180
0/360
sin
cos
tan 1
210
330
0/360 1
225
315
240
300 90 1 Ø
(-, -)
(+, -)
180 -1
270
Return
270 -1 Ø

34. Ex. # 3
Ex. # 4
Ex. # 5
Ex. # 6
return

35. Circle Trigonometry Definitions – Reciprocal Functions
(x, y)
Radius = r
Opposite Leg = y
Adjacent Leg = x
Wait on clicks until after new definition comes in
return

36. Unit Circle – Radian Measure1