The document provides an overview of trigonometric functions, including how to evaluate them for any angle, reference angles, and their applications with real-world examples. It covers key concepts such as quadrantal angles, the unit circle, periodic functions, and includes exercises for practicing conversions between degrees and radians. Additionally, it explains methods for finding the values of trigonometric functions based on the coordinates of points related to those angles.

1. Five-Minute Check (over Lesson 4-2)
Then/Now
New Vocabulary
Key Concept: Trigonometric Functions of Any Angle
Example 1: Evaluate Trigonometric Functions Given a Point
Key Concept: Common Quadrantal Angles
Example 2: Evaluate Trigonometric Functions of Quadrantal Angles
Key Concept: Reference Angle Rules
Example 3: Find Reference Angles
Key Concept: Evaluating Trigonometric Functions of Any Angle
Example 4: Use Reference Angles to Find Trigonometric Values
Example 5: Use One Trigonometric Value to Find Others
Example 6: Real-World Example: Find Coordinates Given a Radius and an Angle
Key Concept: Trigonometric Functions on the Unit Circle
Example 7: Find Trigonometric Values Using the Unit Circle
Key Concept: Periodic Functions
Example 8: Use the Periodic Nature of Circular Functions

2. Over Lesson 4-2
Write 62.937˚ in DMS form.
A. 62°54'13"
B. 63°22'2"
C. 62°54'2"
D. 62°56'13.2"

3. Over Lesson 4-2
Write 96°42'16'' in decimal degree form to the
nearest thousandth.
A. 96.704o
B. 96.422o
C. 96.348o
D. 96.259o

4. Over Lesson 4-2
Write 135º in radians as a multiple of π.
A.
B.
C.
D.

5. Over Lesson 4-2
A. 240o
B. –60o
C. –120o
D. –240o
Write in degrees.

6. Over Lesson 4-2
Find the length of the intercepted arc with
a central angle of 60° in a circle with a radius of
15 centimeters. Round to the nearest tenth.
A. 7.9 cm
B. 14.3 cm
C. 15.7 cm
D. 19.5 cm

7. You found values of trigonometric functions for
acute angles using ratios in right triangles.
(Lesson 4-1)
• Find values of trigonometric functions for any angle.
• Find values of trigonometric functions using the unit
circle.

8. • quadrantal angle
• reference angle
• unit circle
• circular function
• periodic function
• period

10. Evaluate Trigonometric Functions Given a
Point
Let (–4, 3) be a point on the terminal side of an
angle θ in standard position. Find the exact values
of the six trigonometric functions of θ.
Pythagorean Theorem
x = –4 and y = 3
Use x = –4, y = 3, and r = 5 to write the six
trigonometric ratios.
Take the positive square root.

11. Evaluate Trigonometric Functions Given a
Point
Answer:

12. Let (–3, 6) be a point on the terminal side of an
angle Ө in standard position. Find the exact values
of the six trigonometric functions of Ө.
A.
B.
C.
D.

14. Evaluate Trigonometric Functions of
Quadrantal Angles
A. Find the exact value of cos π. If not defined,
write undefined.
The terminal side of π in standard position lies on the
negative x-axis. Choose a point P on the terminal side
of the angle. A convenient point is (–1, 0) because
r = 1.

15. Evaluate Trigonometric Functions of
Quadrantal Angles
Answer: –1
x = –1 and r = 1
Cosine function

16. Evaluate Trigonometric Functions of
Quadrantal Angles
B. Find the exact value of tan 450°. If not defined,
write undefined.
The terminal side of 450° in standard position lies on
the positive y-axis. Choose a point P(0, 1) on the
terminal side of the angle because r = 1.

17. Evaluate Trigonometric Functions of
Quadrantal Angles
Answer: undefined
y = 1 and x = 0
Tangent function

18. Evaluate Trigonometric Functions of
Quadrantal Angles
C. Find the exact value of . If not defined,
write undefined.
The terminal side of in standard position lies
on the negative y-axis. The point (0, –1) is convenient
because r = 1.

19. Evaluate Trigonometric Functions of
Quadrantal Angles
Answer: 0
x = 0 and y = –1
Cotangent function

20. A. –1
B. 0
C. 1
D. undefined
Find the exact value of sec If not defined, write
undefined.

22. Find Reference Angles
A. Sketch –150°. Then find its reference angle.
A coterminal angle is –150° + 360° or 210°. The
terminal side of 210° lies in Quadrant III. Therefore, its
reference angle is 210° – 180° or 30°.
Answer: 30°

23. Find Reference Angles
Answer:
The terminal side of lies in Quadrant II. Therefore,
its reference angle is .
B. Sketch . Then find its reference angle.

24. Find the reference angle for a 520o
angle.
A. 20°
B. 70°
C. 160°
D. 200°

26. Use Reference Angles to Find Trigonometric
Values
A. Find the exact value of .
Because the terminal side of  lies in Quadrant III, the
reference angle

27. Use Reference Angles to Find Trigonometric
Values
Answer:
In Quadrant III, sin θ is negative.

28. Use Reference Angles to Find Trigonometric
Values
B. Find the exact value of tan 150º.
Because the terminal side of θ lies in Quadrant II, the
reference angle θ' is 180o
– 150o
or 30o
.

29. Use Reference Angles to Find Trigonometric
Values
Answer:
tan 150° = –tan 30° In Quadrant II, tan θ is negative.
tan 30°

30. Use Reference Angles to Find Trigonometric
Values
C. Find the exact value of .
A coterminal angle of which lies in
Quadrant IV. So, the reference angle
Because cosine and secant are reciprocal functions
and cos θ is positive in Quadrant IV, it follows that
sec θ is also positive in Quadrant IV.

31. Use Reference Angles to Find Trigonometric
Values
In Quadrant IV, sec θ is positive.

32. Use Reference Angles to Find Trigonometric
Values
Answer:
CHECK You can check your answer by using a
graphing calculator.

33. A.
B.
C.
D.
Find the exact value of cos .

34. Use One Trigonometric Value to Find Others
To find the other function values, you must find the
coordinates of a point on the terminal side of θ. You
know that sec θ is positive and sin θ is positive, so θ
must lie in Quadrant I. This means that both x and y
are positive.
Let , where sin θ > 0. Find the exact
values of the remaining five trigonometric
functions of θ.

35. Use One Trigonometric Value to Find Others
Because sec  =
and x = 5 to find y.
Take the positive square root.
Pythagorean Theorem
r = and x = 5

36. Use One Trigonometric Value to Find Others
Use x = 5, y = 2, and r = to write the other five
trigonometric ratios.

37. Use One Trigonometric Value to Find Others
Answer:

38. Let csc θ = –3, tan θ < 0. Find the exact values o
the five remaining trigonometric functions of θ.
A.
B.
C.
D.

39. ROBOTICS A student programmed a 10-inch long
robotic arm to pick up an object at point C and
rotate through an angle of 150° in order to release
it into a container at point D. Find the position of
the object at point D, relative to the pivot point O.
Find Coordinates Given a
Radius and an Angle

40. Find Coordinates Given a
Radius and an Angle
Cosine ratio
 = 150° and r = 10
cos 150° = –cos 30°
Solve for x.

41. Find Coordinates Given a
Radius and an Angle
Sin ratio
θ = 150° and r = 10
sin 150° = sin 30°
Solve for y.
5 = y

42. Find Coordinates Given a
Radius and an Angle
Answer: The exact coordinates of D are .
The object is about 8.66 inches to the left of
the pivot point and 5 inches above the pivot
point.

43. CLOCK TOWER A 4-foot long minute hand on a
clock on a bell tower shows a time of 15 minutes
past the hour. What is the new position of the end
of the minute hand relative to the pivot point at 5
minutes before the next hour?
A. 6 feet left and 3.5 feet above the pivot point
B. 3.4 feet left and 2 feet above the pivot point
C. 3.4 feet left and 6 feet above the pivot point
D. 2 feet left and 3.5 feet above the pivot point

45. Find Trigonometric Values Using the Unit
Circle
Definition of sin t
sin t = y
Answer:
A. Find the exact value of . If undefined,
write undefined.
corresponds to the point (x, y) = on
the unit circle.
y = .
sin

46. Find Trigonometric Values Using the Unit
Circle
Answer:
cos t = x Definition of cos t
cos
corresponds to the point (x, y) = on the
unit circle.
B. Find the exact value of . If undefined,
write undefined.

47. Find Trigonometric Values Using the Unit
Circle
Definition of tan t.
C. Find the exact value of . If undefined,
write undefined.

48. Find Trigonometric Values Using the Unit
Circle
Simplify.
Answer:

49. Find Trigonometric Values Using the Unit
Circle
D. Find the exact value of sec 270°. If undefined,
write undefined.
270° corresponds to the point (x, y) = (0, –1) on the
unit circle.
Therefore, sec 270° is undefined.
Answer: undefined
Definition of sec t
x = 0 when t = 270°

50. A.
B.
C.
D.
Find the exact value of tan . If undefined, write
undefined.

52. Use the Periodic Nature of Circular Functions
cos t = x and x =
A. Find the exact value of .
Rewrite as the sum of a
number and 2π.
+ 2π map to the same
point (x, y) = on the
unit circle.

53. Use the Periodic Nature of Circular Functions
Answer:

54. B. Find the exact value of sin(–300).
sin (–300o
) = sin (60o
+ 360o
(–1)) Rewrite –300o
as the
sum of a number
and an integer
multiple of 360o
.
Use the Periodic Nature of Circular Functions
= sin 60o
60o
and 60o
+ 360o
(–1) map to the
same point
(x, y) =
on the unit

55. Use the Periodic Nature of Circular Functions
= sin t = y
and y = when t =
60o
.
Answer:

56. Use the Periodic Nature of Circular Functions
C. Find the exact value of .
Rewrite as the sum of a
number and 2 and an integer
multiple of π.
map to the
same point (x, y) =
on the unit circle.

57. Use the Periodic Nature of Circular Functions
Answer:

58. A. 1
B. –1
C.
D.
Find the exact value of cos