1. 1.1
Trigonometry

2. Vocabulary: Angle – created by rotating a ray about its endpoint.
Initial Side – the starting position of the ray.
Terminal Side – the position of the ray after rotation.
Vertex – the endpoint of the ray.

3. Terminal side Vertex Initial side Initial side Vertex Terminal side
This arrow means that the rotation was in a counterclockwise direction.
Vertex
Initial side
This arrow means that the rotation was in a clockwise direction.
Initial side
Vertex
Terminal side

4. Positive Angles – angles generated by a counterclockwise rotation.
Negative Angles – angles generated by a clockwise rotation.
We label angles in trigonometry by using the Greek alphabet.
 - Greek letter alpha
 - Greek letter beta
 - Greek letter phi
 - Greek letter theta

5. Terminal side Vertex Initial side Initial side Vertex Terminal side
This represents a positive angle
Vertex
Initial side
This represents a negative angle
Initial side
Vertex
Terminal side

6. Positive angle in standard position
Standard Position – an angle is in standard position when its initial side rests on the positive half of the x-axis.
Positive angle in standard position

7. Practice sketching graphs in standard position: (degrees only)

8. There are two ways to measure angles…
Degrees
Radians

9. Degrees: Radians: There are 360 in a complete circle.
1 is 1/360th of a rotation.
Radians:
There are 2 radians in a complete circle.
1 radian is the size of the central angle when the
radius of the circle is the same size as the arc of
the central angle.

10. Length of the arc is equal to the length of the radius.
1 Radian
radius

11. Practice sketching graphs in standard positions with radians:

12. Coterminal angles – two angles that share a common vertex, a common initial side and a common terminal side.
Examples of Coterminal Angles  
 and  are coterminal angles because they share the same initial side and same terminal side.
Coterminal angles could go in opposite directions.

13.   Examples of Coterminal Angles
 and  are coterminal angles because they share the same initial side and same terminal side.
Coterminal angles could go in the same direction with multiple rotations.

14. Finding coterminal angles of angles measured in degrees:
Since a complete circle has a total of 360, you can find coterminal angles by adding or subtracting 360 from the angle that is provided.

15. Example:  = 25 positive coterminal angle: 25 + 360 = 385
Find two coterminal angles (one positive and one negative) for the following angles.
 = 25
positive coterminal angle:
= 385
negative coterminal angle:
25 – 360 = - 335

16. Example:  = 725 positive coterminal angle:
Find two coterminal angles (one positive and one negative) for the following angles.
 = 725
positive coterminal angle:
= 1085 (add a rotation) or
725 – 360 = 365 (subtract a rotation)
725 – 360 – 360 = 5 (subtract 2 rotations)
negative coterminal angle:
725 – 360 – 360 – 360 = - 355
(must subtract 3 rotations)

17. Example:  = -90 positive coterminal angle: -90 + 360 = 270
Find two coterminal angles (one positive and one negative) for the following angles.
 = -90
positive coterminal angle:
= 270
negative coterminal angle:
- 90 – 360 = - 450

18. Finding coterminal angles of angles measured in radians:
Since a complete circle has a total of 2 radians you can find coterminal angles by adding or subtracting 2 from the angle that is provided.

19. Example:  = /7 positive coterminal angle:
Find two coterminal angles (one positive and one negative) for the following angles.
 = /7
positive coterminal angle:
/7 + 2 = /7 + 14/7 = 15/7 rad
negative coterminal angle:
/7 - 2 = /7 - 14/7 = -13/7 rad

20. Example:  = -4/9 positive coterminal angle:
Find two coterminal angles (one positive and one negative) for the following angles.
 = -4/9
positive coterminal angle:
-4/9 +2 = -4/9 + 18/9 =14/9 rad
negative coterminal angle:
-4/9 -2 =-4/9 - 18/9 =-22/9 rad

21. Complementary angles – two positive angles whose sum is 90 or two positive angles whose sum is /2.
To find the complement of a given angle you subtract the given angle from 90 (if the angle provided is in degrees) or from /2 (if the angle provided is in radians).

22. Example:  = 29  = 107  = /5 complement = 90 – 29 = 61.
Example:
Find the complement of the following angles if one exists.
 = 29
complement = 90 – 29 = 61
 = 107
complement = 90 – 107 = none
(No complement because it is negative)
 = /5
complement = /2 - /5 = 5/10 - 2/10 = 3/10

23. Supplementary angles – two positive angles whose sum is 180 or two positive angles whose sum is .
To find the supplement of a given angle you subtract the given angle from 180 (if the angle provided is in degrees) or from  (if the angle provided is in radians).

24. Example:  = 29  = 107  = /5 supplement = 180 – 29 = 151
Find the supplement of the following angles if one exists.
 = 29
supplement = 180 – 29 = 151
 = 107
supplement = 180 – 107 = 73
 = /5
supplement = - /5 = 5/5 - /5 = 4/5

25. We have to become comfortable working with both forms of measuring angles.
Therefore, MEMORIZE the following:
Degrees
Radians 0
0 radians
90
/2 radians
30
/6 radians
180
 radians
45
/4 radians
270
3/2 radians
60
/3 radians
360
2 radians
We will memorize more, very, very soon.

26. Example: Multiply the given degrees by  radians/180
Manually Converting from Degrees to Radians:
Multiply the given degrees by  radians/180
Example:
Convert the following degrees to radians
135 degrees  radians =
degrees
135
135 radians =
180
3 radians 4

27. Example: Multiply the given degrees by  radians/180
Convert the following degrees to radians
540 degrees  radians =
degrees
540
540 radians =
180
3 radians 1

28. Example: Multiply the given radians by 180/ radians
Manually Converting from Radians to Degrees:
Multiply the given radians by 180/ radians
Example:
Convert the following radians to degrees.
- radians degrees =
 radians
-/3 radians
-180 degrees = 3
-60

29. Example: Multiply the given radians by 180/ radians
Convert the following radians to degrees.
9 radians degrees =
 radians
9/2 radians
1620 degrees = 2
810

30. Example: Multiply the given radians by 180/ radians
Convert the following radians to degrees.
2 radians degrees =
 radians 2
(if you don’t see the degree symbol, then the angle measure is automatically believed to be a radian.)
degrees = 2
114.59

31. Tomorrow, we can look at your individual calculators and show you how to do these conversions via those calculators.

32. The following formula is used to determine arc length: s = r 
Finding Arc Length:
The following formula is used to determine arc length:
s = r 
arc length
radius
Measure of the central angle in radians.
must have the same units of measure

33. Examples s = r  s = (14)(3) s = 42 inches s = ? 3 radians
r= 14 inches
3 radians
s = ?
s = r 
s = (14)(3)
s = 42 inches
Picture not drawn to scale.

34. Examples s = r  9 = (r)(/6) r = 54/  cm  17.19 cm s =9 cm 30
You must convert 30 to radians.
s = r 
9 = (r)(/6)
r = 54/  cm  cm
Picture not drawn to scale.