1. MATH 31 LESSONS Chapters 6 & 7: Trigonometry
1. Trigonometry Basics

2. Section 6.1: Functions of Related Values
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3. A. Standard Position
Angles are created by rotating an “arm” from the
positive x-axis, which is called the initial arm.
Where the angle ends is called the terminal arm.
Counterclockwise angles are positive.
Clockwise angles are negative.

4. e.g.
Sketch the angle +200 in standard position. y x

5. y x
start
Angles in standard position are measured from the positive x-axis, which is the initial arm.

6. 90 y
180
0, 360 x
start
270
It is useful to include the major angles at each axis. The clockwise direction is positive.

7. 90 y
200
180
0, 360 x
start
270
Now, we sweep the arm 200 counterclockwise, since it is a positive angle.

8. e.g.
Sketch the angle 310 in standard position. y x

9. y x
start
Angles in standard position are measured from the positive x-axis, which is the initial arm.

10. y x
start
0, 360
270
180
90
It is useful to include the major angles at each axis. This time, we want the negative angles (clockwise).

11. Now, we sweep the arm 310 clockwise, since it is a negative angle.y x
start
0, 360
270
180
90
310
Now, we sweep the arm 310 clockwise, since it is a negative angle.

12. Radians
Another measure of angle, apart from degrees, is radians.
To convert from radians to degrees (and vice versa),
use the following information:
2  radians = 360
or,  radians = 180

13. y 
0, 2 x
These are the major angles at each axis in radians.

14. Converting from Radians to Degrees
e.g. Convert radians to degrees.

15. Since  radians = 180, you can simply substitute 180 wherever you see .

16. Converting from Radians to Degrees
e.g. Convert 252 to radians.

17. degrees goes on the bottom, since it must cancel
Recall,  radians = 180

19. B. Trig Ratios
When we define the trigonometric ratios, we will use
a circle (rather than a triangle).
In this way, we can deal with angles that are
bigger than 180 (as well as negative angles).

20. Primary Trig Ratios
Consider a circle of radius r. y x r

21. We consider any point P
that is on the circumference
of the circle.
Its general coordinates
will be (x, y).
P (x, y)

22. We can create a triangle
with height y and base x.
The hypotenuse will be r,
since it represents the
radius of the circle. r y x
P (x, y) 

23. Using the Pythagorean
theorem, r y x
P (x, y) 
r is the radius of the circle, so it must always be positive (r > 0)

24. We can now use
“Soh Cah Toa” to define
each primary trig ratio. r y x
P (x, y) 

25. “Soh Cah Toa”
P (x, y) r y  x

26. “Soh Cah Toa”
P (x, y) r y  x

27. “Soh Cah Toa”
P (x, y) r y  x

28. Reciprocal Trig Ratios

29. Ex. 1 Evaluate cos  and csc  if
Answer in exact values. Do not find the angle.
Try this example on your own first.
Then, check out the solution.

30.  Determine the quadrant of the angley x  x
The angle is in quadrant 2, where x is negative and y is positive

31.  Sketch the triangle
Thus, x = -12 and y = 5
Remember, x is negative and y is positive

32. r
x = -12
y = 5 

33.  Find r r
x = -12
y = 5 

34.  Find cos 
r = 13
x = -12
y = 5 

35.  Find csc 
r = 13
x = -12
y = 5 

36. C. Reference and Coterminal Angles
Reference Angle
The reference angle is the acute angle (< 90)
between the terminal arm and the nearest x-axis.
Reference angles are always positive.

37. e.g.
What is the reference angle for 260?

38. First, we sketch the angle.y x
start
0, 360
90
180
270
260
First, we sketch the angle.

39. The reference angle is the angle between the terminal arm
and the nearest x-axis. y x
180
80
270
The reference angle is 80

40. Coterminal Angles
Two angles that have the same terminal arm
are called coterminal angles.

41. y x  y x 
 and  are coterminal.

42. For any given reference angle (e.g. 50),
Note:
For any given reference angle (e.g. 50),
there are an infinite number of coterminal angles. y
50 x

43. For any given reference angle (e.g. 50),
Note:
For any given reference angle (e.g. 50),
there are an infinite number of coterminal angles. y x
50

44. The smallest positive angle to the terminal arm (130)
is called the principal angle. y
130
50 x

45. The next positive angle with the same terminal arm is constructed by adding 360 to the principal angle. y
130 + 360 = 490
50 x

46. The first negative angle with the same terminal arm is constructed by subtracting 360 from the principal angle. y
50 x
130 - 360 = -230

47. In general, we can find all coterminal angles by adding
or subtracting multiples of 360 from the principal angle.
i.e. If 1 and 2 are coterminal, then
2 = 1 + (360) n , where n  I
n belongs to the integers.
Thus, n  ... , -3, -2, -1, 0, 1, 2, 3, ...

48. If 1 and 2 are coterminal, then
2 = 1 + (360) n , where n  I
or in radian form,
2 = 1 + 2 n , where n  I

49. D. Exact Trig Values (Using Special Triangles)
It is crucial that you remember the
exact values for the trig ratios of the following angles:
 30
 45
 60
To do so, we need to use special triangles.

50. 60 "Soh Cah Toa"
60 2 1
sin 60 =
cos 60 =
tan 60 =

51. 30 "Soh Cah Toa"
30 2 1
sin 30 =
cos 30 =
tan 30 = 1

52. 45 "Soh Cah Toa"
45 1
sin 45 = 1
cos 45 = 1
tan 45 = 1

53. E. CAST
This is a simple but effective way to remember the signs
of all trig ratios in each quadrant. C A S T 1 2 3 4

54. sin + cos + tan + S
All + 2 1 3 4 T C

55. C A
Sine + T 1 2 3 4
sin + cos  tan 

56. C A S
Tan + 1 2 3 4
sin  cos  tan +

57. Cos + A S T 1 2 3 4
sin  cos + tan 

58. If we define the trig circle with a radius of 1 unit,
F. Unit Circle
If we define the trig circle with a radius of 1 unit,
called the unit circle, then finding exact values for
the trig ratios is much more straightforward. 1 1

59. “Soh Cah Toa” 1
y = sin   x
The sine ratio is simply the y-coordinate.

60. “Soh Cah Toa” 1 y 
x = cos 
The cosine ratio is simply the x-coordinate.

61. “Soh Cah Toa”
P (x, y) r y  x
The tangent ratio remains y over x.

62. In general,
P (sin , cos ) r
y = sin  
x = cos 

63. Building the unit circle ...
(1, 0)
(0, 1)
(-1, 0)
(0, -1)
Start with the axes.

64. Next, add the special triangle ratios in quadrant 1.
(1, 0)
(0, 1)
(-1, 0)
(0, -1)
Next, add the special triangle ratios in quadrant 1.
Remember, x = cos  and y = sin 

65. It is crucial that you memorize this special triangle.
(1, 0)
(0, 1)
(-1, 0)
(0, -1)
It is crucial that you memorize this special triangle.

66. Ex. 2 Evaluate the following exactly (without your calculator):
Try this example on your own first.
Then, check out the solution.

67.  Convert to degrees (if you need to)
Recall,  radians = 180

69.  Evaluate the special angles using the unit circle
(1, 0)
(0, 1)
(-1, 0)

70. Recall, sine is the y-coordinate
(1, 0)
(0, 1)
(-1, 0)
90
Recall, sine is the y-coordinate

71. Recall, cosine is the x-coordinate
(1, 0)
(0, 1)
(-1, 0)
180
Recall, cosine is the x-coordinate

72. (1, 0)
(0, 1)
(-1, 0)
360
Recall, tangent is y / x

73.  Answer the question:

74. Ex. 3 Evaluate the following exactly (without your calculator):
Try this example on your own first.
Then, check out the solution.

75.  Convert to degrees (if you need to)
Recall,  radians = 180

76. Note that sin 2  = (sin ) 2

77.  Evaluate the special angles using the unit circle
(1, 0)
(0, 1)
(-1, 0)
These special angles can be read off the unit circle directly.

78.  Answer the question:

81. Ex. 4 Express the following as a function of its related acute
angle and then evaluate:
Try this example on your own first.
Then, check out the solution.

82.  First, sketch the angle
90 y
120
180 0 x
start

83.  Next, find the reference the angley
120
60 x
The reference angle is 60

84.  Using CAST, determine whether the trig ratio is
positive or negative y S A
60 x T C
Since the angle is in quadrant 2, sine is positive.

85.  Express the trig ratio in terms of the reference angle:

86. to evaluate the special angle exactly:
 Use the unit circle
to evaluate the special
angle exactly:
(1, 0)
(0, 1)
(-1, 0)
Recall, sine is the y-coordinate.

87. Ex. 5 Express the following as a function of its related acute
angle and then evaluate:
Try this example on your own first.
Then, check out the solution.

88.  First, convert to degrees and a primary trig ratio:
Recall,  radians = 180

90.  Next, sketch the angle
90 y
330
180
0, 360 x
start
270

91.  Next, find the reference the angley
330 x
30
The reference angle is 30

92.  Using CAST, determine whether the trig ratio is
positive or negative y A S x
30 T C
Since the angle is in quadrant 4, tangent is negative (only cosine is positive)

93.  Express the trig ratio in terms of the reference angle:

94. to evaluate the special angle exactly:
 Use the unit circle
to evaluate the special
angle exactly:
(1, 0)
(0, 1)
(-1, 0)
Recall, tangent is y / x.

96.  Answer the question: