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Splash Screen

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Splash Screen Example 4 A. B. C. D. Find the exact value of cos . Example 5 Use One Trigonometric Value to Find Others To find the other function values, you ... – PowerPoint PPT presentation

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Title: Splash Screen


1
Splash Screen
2
Lesson Menu
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
3
5Minute Check 1
Write 62.937 in DMS form.
A. 6254'13" B. 6322'2" C. 6254'2" D. 6256'13.2
"
4
5Minute Check 2
Write 9642'16'' in decimal degree form to the
nearest thousandth.
A. 96.704o B. 96.422o C. 96.348o D. 96.259o
5
5Minute Check 3
Write 135º in radians as a multiple of p.
6
5Minute Check 4
A. 240o B. 60o C. 120o D. 240o
7
5Minute Check 5
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
8
Then/Now
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.

9
Vocabulary
  • quadrantal angle
  • reference angle
  • unit circle
  • circular function
  • periodic function
  • period

10
Key Concept 1
11
Example 1
Evaluate Trigonometric Functions Given a Point
Let (4, 3) be a point on the terminal side of an
angle ? in standard position. Find the exact
values of the six trigonometric functions of ?.
Use x 4, y 3, and r 5 to write the six
trigonometric ratios.
12
Example 1
Evaluate Trigonometric Functions Given a Point
Answer
13
Example 1
Let (3, 6) be a point on the terminal side of an
angle ? in standard position. Find the exact
values of the six trigonometric functions of ?.
14
Key Concept 2
15
Example 2
Evaluate Trigonometric Functions of Quadrantal
Angles
A. Find the exact value of cos p. If not defined,
write undefined.
The terminal side of p 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.
16
Example 2
Evaluate Trigonometric Functions of Quadrantal
Angles
Answer 1
17
Example 2
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.
18
Example 2
Evaluate Trigonometric Functions of Quadrantal
Angles
Answer undefined
19
Example 2
Evaluate Trigonometric Functions of Quadrantal
Angles
20
Example 2
Evaluate Trigonometric Functions of Quadrantal
Angles
Cotangent function
Answer 0
21
Example 2
A. 1 B. 0 C. 1 D. undefined
22
Key Concept 3
23
Example 3
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
24
Example 3
Find Reference Angles
25
Example 3
Find the reference angle for a 520o angle.
A. 20 B. 70 C. 160 D. 200
26
Key Concept 4
27
Example 4
Use Reference Angles to Find Trigonometric Values
28
Example 4
Use Reference Angles to Find Trigonometric Values
29
Example 4
Use Reference Angles to Find Trigonometric Values
B. Find the exact value of tan 150º.
Because the terminal side of ? lies in Quadrant
II, the reference angle ?' is 180o 150o or 30o.
30
Example 4
Use Reference Angles to Find Trigonometric Values
tan 150 tan 30 In Quadrant II, tan ? is
negative.
31
Example 4
Use Reference Angles to Find Trigonometric Values
32
Example 4
Use Reference Angles to Find Trigonometric Values
33
Example 4
Use Reference Angles to Find Trigonometric Values
CHECK You can check your answer by using a
graphing calculator.
34
Example 4
35
Example 5
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 ?. You know that sec ? is positive and sin ?
is positive, so ? must lie in Quadrant I. This
means that both x and y are positive.
36
Example 5
Use One Trigonometric Value to Find Others
37
Example 5
Use One Trigonometric Value to Find Others
38
Example 5
Use One Trigonometric Value to Find Others
39
Example 5
Let csc ? 3, tan ? lt 0. Find the exact values
o the five remaining trigonometric functions of ?.
40
Example 6
Find Coordinates Given a Radius and an Angle
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.
41
Example 6
Find Coordinates Given a Radius and an Angle
42
Example 6
Find Coordinates Given a Radius and an Angle
43
Example 6
Find Coordinates Given a Radius and an Angle
44
Example 6
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
Key Concept 7
46
Example 7
Find Trigonometric Values Using the Unit Circle
47
Example 7
Find Trigonometric Values Using the Unit Circle
cos t x Definition of cos t
48
Example 7
Find Trigonometric Values Using the Unit Circle
49
Example 7
Find Trigonometric Values Using the Unit Circle
50
Example 7
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
51
Example 7
52
Key Concept 8
53
Example 8
Use the Periodic Nature of Circular Functions
54
Example 8
Use the Periodic Nature of Circular Functions
55
Example 8
Use the Periodic Nature of Circular Functions
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.
56
Example 8
Use the Periodic Nature of Circular Functions
57
Example 8
Use the Periodic Nature of Circular Functions
58
Example 8
Use the Periodic Nature of Circular Functions
59
Example 8
60
End of the Lesson
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