1. TRIGONOMETRY Trigonometry
Comes from two Greek words trigonon and metron, meaning “triangle measurement”

2. 7.1 Measurement of Angles

3. DEGREES-unit of measure for angles 360°=1 revolution
Revolution –one complete circular motion
Minutes- 1°=60 minutes (60’)
Seconds-1’ =60 seconds(60”)
1°=3600”

4. Convert to degrees only

5. Convert to degrees only

6. Convert to DMS

7. Convert to DMS

8. Radian Measure Consider an arc of length s on a circle of radius r.
The measure of the central angle that intercepts
the arc is
 = s/r radians.  O r s

9. Definition of a Radian
One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle.
RADIAN-unit of measure for angles 2π=1 revolution

10. Converting degrees to radians
Converting radians to degrees

11. Example 240°
Example °

12. Example
Example

13. Angles
An angle is formed by two rays that have a common endpoint called the vertex. One ray is called the initial side and the other the terminal side. The arrow near the vertex shows the direction and the amount of rotation from the initial side to the terminal side. A θ B
Terminal Side
Initial Side
Vertex C

14. Angles of the Rectangular Coordinate System
An angle is in standard position if
its vertex is at the origin of a rectangular coordinate system and
its initial side lies along the positive x-axis. x y
Terminal Side
Initial Side
Vertex 
 is positive
Positive angles rotate counterclockwise. x y
Terminal Side
Initial Side
Vertex 
 is negative
Negative angles rotate clockwise.

15. Measuring Angles Using Degrees
The figures below show angles classified by their degree measurement. An acute angle measures less than 90º. A right angle, one quarter of a complete rotation, measures 90º and can be identified by a small square at the vertex. An obtuse angle measures more than 90º but less than 180º. A straight angle has measure 180º.
90º
Right angle
1/4 rotation 
Acute angle
0º <  < 90º 
Obtuse angle
90º <  < 180º
180º
Straight angle
1/2 rotation

16. Coterminal Angles
An angle of xº is coterminal (have the same terminal ray) with angles of
xº + k · 360º
where k is an integer.

17. Example
Assume the following angles are in standard position. Find a positive angle less than 360º that is coterminal with:
a. a 420º angle b. a –120º angle.
Solution We obtain the coterminal angle by adding or subtracting 360º. Our need to obtain a positive angle less than 360º determines whether we should add or subtract.
a. For a 420º angle, subtract 360º to find a positive coterminal angle.
420º – 360º = 60º
A 60º angle is coterminal with a 420º angle. These angles, shown on the next slide, have the same initial and terminal sides.

18. Example cont. x y
420º
60º x y
240º
-120º
Solution
b. For a –120º angle, add 360º to find a positive coterminal angle.
-120º + 360º = 240º
A 240º angle is coterminal with a –120º angle. These angles have the same initial and terminal sides.

19. Examples- find two angles, one positive and one negative, that are coterminal with each given angle
10°
100°
-5°
400° π
π/2
-π/3 4π
370°,-350°
460°,-260°
355°,-365°
40°,-320°
3π,-π
5π/2,-3π/2
5π/3,-7π/3
±2π

20. Conversion between Degrees and Radians
Using the basic relationship  radians = 180º,
To convert degrees to radians, multiply degrees by ( radians) / 180
To convert radians to degrees, multiply radians by 180 / ( radians)

21. Example
Convert each angle in degrees to radians
40º
75º
-160º

22. Example cont. Solution: 40º = 40*/180 = 2  /9
75º = 75*  /180 = 5  /12
-160º = -160*  /180 = -8  /9

23. Length of a Circular Arc
Let r be the radius of a circle and  the non-
negative radian measure of a central angle
of the circle. The length of the arc
intercepted by the central angle is
s = r   O s r

24. Example
A circle has a radius of 7 inches. Find the length of the arc intercepted by a central angle of 2/3
Solution:
s = (7 inches)*(2  /3) =14  /3 inches

25. Definitions of Linear and Angular Speed
If a point is in motion on a circle of radius r through an angle of  radians in time t, then its linear speed is
v = s/t
where s is the arc length given by s = r , and its angular speed is
 = /t

26. Linear Speed in Terms of Angular Speed
The linear speed, v, of a point a distance r from the center of rotation is given by v=r where  is the angular speed in radians per unit of time.

27. ASSIGNMENT
Page 261 #1-10, (even)