1. Vocabulary: Initial side & terminal side: Terminal side Terminal side
(6 – 1) Angle and their Measure
Learning target: To convert between decimals and degrees, minutes, seconds forms
To find the arc length of a circle
To convert from degrees to radians and from radians to degrees
To find the area of a sector of a circle
To find the linear speed of an object traveling in circular motion
Initial side
Vocabulary: Initial side & terminal side:
Terminal side
Terminal side
Terminal side
Initial side
Initial side

2. Drawing an angle 360 90 150 200 -90 -135
Positive angles: Counterclockwise
Negative angles: Clockwise
Drawing an angle
360
90
150
200
-90
-135

3. Another unit for an angle: Radian
Definition: An angle that has its vertex at the center of a circle and that intercepts an arc on the circle equal in length to the radius of the circle has a measure of one radian. r
One radian r

4. From Geometry:
Therefore: using the unit circle r = 1
 = 180
So, one revolution 360 = 2

5. Converting from degrees to radians & from radians to degrees
Degrees Radians
Radians Degrees
Degree multiply by
Radian multiply by
I do: Convert from degrees to radians or from radians to degrees.
(a) -45 (b)

6. You do: Convert from degrees to radians or from radians to degrees.
(c) radians
(d) 3 radians
(a) 90
(b) 270

7. Special angles in degrees & in radians0
30
45
60
90
180
360
Radians

8. Finding the arc length & the sector area of a circle
Arc length (s):
is the central angle. S r
Area of a sector (A):
Important: is in radians.

9. (ex) A circle has radius 18. 2 cm
(ex) A circle has radius 18.2 cm. Find the arc length and the area if the central angle is 3/8.
Arc length:
Area of the sector:

10. You do: (ex) A circle has radius 18. 2 cm
You do: (ex) A circle has radius 18.2 cm. Find the arc length and the area if the central angle is 144.
Convert the degrees to radians
Arc length:
Area of the sector: 2.

11. To find the exact values of the trig functions in different quadrants
(6 – 2) Trigonometric functions & Unit circle
Learning target: To find the values of the trigonometric functions using a point on the unit circle
To find the exact values of the trig functions in different quadrants
To find the exact values of special angles
To use a circle to find the trig functions
Vocabulary:
Unit circle is a circle with center at the origin and the radius of one unit.

12. Unit circle

13. Recall: trig ratio from Geometry
SOH
CAH
TOA

14. Also, Two special triangles2 1 2 60 1 90 30 45 1 1 1 45 90 1

15. Using the unit circle r y x

16. Finding the values of trig functions
Now we have six trig ratios.

17. Find the exact value of the trig ratios.
Sin is positive when  is in QI.  = = =

18. Sin is positive when  is in QII

19. Sin is negative when  is in QIII

20. Sin is negative when  is in QIV

21. cos is positive when  is in QI
cos is negative when  is in QII
cos is negative when  is in QIII
cos is positive when  is in QIV

22. tan is positive when  is in QI (+, +)
cos is negative when  is in QII(-, +)
cos is negative when  is in QIII(-, -)
cos is positive when  is in QIV(+, -)

23. Find the exact values of the trig ratios.

24. Learning target: To learn domain & range of the trig functions
(6 – 3) Properties of trigonometric functions
Learning target: To learn domain & range of the trig functions
To learn period of the trig functions
To learn even-odd-properties
Signs of trig functions in each quadrant
 in Q.
sin
cos
tan
csc
sec
cot I + II -
III IV

25. (sin)(csc) = 1
(cos)(sec) = 1
(tan)(cot) = 1

26. The formula of a circle with the center at the origin and the radius 1 is:
Therefore,

27. Fundamental Identities:
(1) Reciprocal identities:
(2) Tangent & cotangent identities:
(3) Pythagorean identities:

28. Even-Odd Properties

29. Co-functions:

31. Find the period, domain, and range
y = sinx
Period: 2
Domain: All real numbers
Range:  y  1

32. y = cosx
Period: 2
Domain: All real numbers
Range:  y  1

33. y = tanx
Period: 
Domain: All real number but
Range:  < y <

34. y = cotx
Period: 
Domain: All real number but
Range:  < y <

35. y = cscx
y = cscx
y = sinx
Period: 
Domain: All real number but
Range: - < y  -1
or 1 y < 

36. y = secx
Period: 
Domain: All real number but
Range: - < y  -1
or 1 y < 

37.  Summary for: period, domain, and range of trigonometric functions
y = sinx 2
All real #’s
-1  y  1
y = cosx
y = tanx 
All real #’s but
- < y <
y = cotx
y = cscx
- < y  -1
or 1 y < 
y = secx

38. To find amplitude and period of sinusoidal function
(6 – 4) Graph of sine and cosine functions
Learning target: To graph y = a sin (bx) & y = a cos (bx) functions using transformations
To find amplitude and period of sinusoidal function
To graph sinusoidal functions using key points
To find an equation of sinusoidal graph
Sine function:
Notes: a function is defined as: y = a sin(bx – c) + d
Period :
Amplitude: a

39. Period and amplitude of y = sinx graph

40. Graphing a sin(bx – c) +d
a: amplitude = |a| is the maximum depth of the graph above half and below half.
bx – c : shifting along x-axis
Set 0  bx – c  2 and solve for x to find the starting and ending point of the graph for 1 perid.
d: shifting along y-axis
Period: one cycle of the graph

41. I do (ex) : Find the period, amplitude, and sketch the graph y = 3 sin2x for 2 periods.
Step 1:a = |3|, b = 2, no vertical or horizontal shift
Step 2: Amplitude: |3| Period:
Step 3: divide the period into 4 parts equally.
Step 4: mark one 4 points, and sketch the graph

42. y = 3 sin2x a = |3|
P:  3 -3

43. y = cos x

44. Graphing a cos (bx – c) +d
a: amplitude = |a| is the maximum depth of the graph above half and below half.
bx – c : shifting along x-axis
Set 0  bx – c  2 and solve for x to find the starting and ending point of the graph for 1 perid.
d: shifting along y-axis
Period: one cycle of the graph

45. We do: Find the period, amplitude, and sketch the graph
y = 2 cos(1/2)x for 1 periods.
Step 1:a = |2|, b = 1/2, no vertical or horizontal shift
Step 2: Amplitude: |2|
Period:
Step 3: divide the period into 4 parts equally.
Step 4: mark the 4 points, and sketch the graph

46. 2 -2

47. You do: Find the period, amplitude, and sketch the graph
y = 3 sin(1/2)x for 1 periods.

48. I do: Find the period, amplitude, translations, symmetric, and
I do: Find the period, amplitude, translations, symmetric, and sketch the graph
y = 2 cos(2x - ) - 3 for 1 period.
Step 1:a = |2|, b = 2
Step 2: Amplitude: |2| Period:
Step 3: shift the x-axis 3 units down.
Step 4: put 0  2x –   2 , and solve for x to find the beginning point and the ending point.
Step 5: divide one period into 4 parts equally.
Step 6: mark the 4 points, and sketch the graph.

49. y = 2 cos(2x - ) – 3
a: |2| Horizontal shift: /2  x  3/2,
P:  Vertical shift: 3 units downward

50. We do: Find the period, amplitude, translations, symmetric, and
We do: Find the period, amplitude, translations, symmetric, and sketch the graph
y = -3 sin(2x - /2) for 1 period.
Step 1: graph y = 3 sin(2x - /2) first
Step 2:a = |3|, b = 2, no vertical shift
Step 3: Amplitude: |3| Period:
Step 4: put 0  2x – /2  2 , and solve for x to find the beginning point and ending point.
Step 5: divide one period into 4 parts equally.
Step 6: mark the 4 points, and sketch the graph with a dotted line.
Step 7: Start at -3 on the starting x-coordinates.

51. y = -3 sin(2x - /2)
a = 3
P = 
/4  x  5/4
No vertical shift 3 -3

52. You do: Find the period, amplitude, translations, symmetric, and
You do: Find the period, amplitude, translations, symmetric, and sketch the graph
y = 3 cos(/4)x + 2 for 1 period.
Step 1: graph y = 3 cos(/4)x first
Step 2:a = |3|, b = /4
Step 3: Shift 2 units upward
Step 4: Amplitude: |3| Period:
Step 5: Step 5: divide one period into 4 parts equally.
Step 6: mark the 4 points, and sketch the graph with a dotted line.

54. To graph functions of the form y = a csc(bx) + c and y = a sec(bx) + c
(6 – 5) Graphing tangent, cotangent, cosecant, and secant functions
Learning target: To graph functions of the form y = a tan(bx) + c and y = a cot(bx) + c
To graph functions of the form y = a csc(bx) + c and y = a sec(bx) + c

55. The graph of a tangent function
Period: 
Domain: All real number but
Range:  < y <
interval:

56. Tendency of y = a tan(x) graph
y = ½ tan(x)
y = 2 tan(x)
y = tan(x)

57. To graph y = a tan(bx + c):
The period is and
(2) The phase shift is
(3) To find vertical asymptotes for the graph:
solve for x that shows the one period

58. I do: Find the period and translation, and sketch the graph
y = ½ tan (x + /4)
a = ½ , b = 1,
c = /4
P =
-3/4
/4
Interval:
One half of the interval is the zero point.

59. We do: Find the period and translation, and sketch the graph
Graph first
a = b = ½
c = /3
P =
Interval:
- /2< (1/2)x + /3 < /2

60. a = 1
P = 2
Interval:
-5/3 < x < /3

61. You do: Find the period and translation, and sketch the graph
Interval:

62. interval: 0 < x <  The graph of a cotangent function y = cot(x)
Period: 
interval:
0 < x < 
Domain: All real number but
Range:  < y <

63. The tendency of y = a cot(x)
As a gets smaller, the graph gets closer to the asymptote.

64. Graphing cosecant functions
Period: 
Interval: 0 < x < 
Domain: all real numbers, but x  n
Range: |y|  1 or
y  -1 or y  1
(-, -1]  [1, )

65. Step 1: y = cos(x), graph y = sin(x)
Step 2: draw asymptotes x-intercepts
Step 3: draw a parabola between each asymptote with the vertex at y = 1

66. Graphing secant functions
Period: 
Interval: /2 < x < 3/2
Domain: all real numbers, but
Range: |y|  1 or
y  -1 or y  1
(-, -1]  [1, )

67. Graphing secant functions
Step 1: graph y = cos(x)
Step 2: draw asymptotes x-intercepts
Step 3: draw a parabola between each asymptote with the vertex at y = 1

68. I do (ex) Find the period, interval, and asymptotes and sketch the graph.
Graph y = sin(2x - )
Period: P = 2/|b|
Interval: 0 <2x -  < 2
draw the asymptotes
Draw a parabola between the asymptotes 1 -1

69. You do: Find the period, interval, and asymptotes and sketch the graph.
Graph y = cos(x - /2)
Period: P = 2/|b|
Interval: 0 <x - /2 < 2
draw the asymptotes
Draw a parabola between the asymptotes