1. What you will learn A short history of Trigonometry
How to convert decimal degree measures to degrees, minutes and seconds, and vice versa
How to find the number of degrees in a given rotation
How to identify angles that are coterminal with another angle
How to find a reference angle

2. A Brief History of Trigonometry
Trigonometry was originally created by the Greeks to aid in the study of astronomy. Hipparchus of Bithynia ( B.C.) tabulated trigonometric ratios, to enable the calculation of a planet's position. Ptolemy continued some of this early work.
The Chinese, in the medieval times, studied astronomy, and hence, trigonometry. They introduced the tangent function. However, most of their work was in the field of astronomy, and many of their trigonometric advancements were not continued.
Objective: 5-1 Angles and Degree Measure

3. History (continued)
The Indians were the next to advance the study of trigonometry. They developed their own sine tables, using the Greek half-angle formula. Later, the cosine table was also constructed. Techniques of approximation to a relatively high accuracy were also introduced.
The Indian works were translated and read by the Islamic mathematicians, who also worked on trigonometry. Similar to the Greeks and Indians, they related trigonometry and astronomy. The Indian sine was used, as well as the chord. The cosine was also formally introduced, by Abu Abdallah Muhammad ibn Jabir al-Battani.
Objective: 5-1 Angles and Degree Measure

4. History (continued)
Trigonometry reached Europe in the medieval times. Richard of Wallingford wrote a text on trigonometry, Quadripartium. In the 16th century, trigonometry was incorporated into geography and navigation. Knowledge of trigonometry was used to construct maps, determining the position of a land mass in relation to the longitudes and latitudes.
Johannes Muller wrote a text On Triangles. He studied plane trigonometry, including results for solving triangles.
Objective: 5-1 Angles and Degree Measure

5. Some Fun Definitions Terminal Side Initial Side Vertex
Objective: 5-1 Angles and Degree Measure

6. More Angle Information
An angle measure provides information about the direction of rotation of a terminal side and the amount of rotation.
- If the rotation is in a counterclockwise direction, the angle formed is a positive angle.
- If the rotation is clockwise, the angle is negative.
Objective: 5-1 Angles and Degree Measure

7. Angles in the Coordinate Plane
All of these angles are in standard position. The initial side is on the x-axis. y y
+120o x x
-120o y
+90o x
Objective: 5-1 Angles and Degree Measure

8. Angle Measures We use degrees to measure angles.
Degrees can then be divided into minutes and seconds.
Each minute is 1/60 of a degree.
Each second is 1/60 of a minute or 1/3600 of a degree.
Objective: 5-1 Angles and Degree Measure

9. Converting Decimals to Degrees, Minutes, Seconds
Example:
Change north latitude o to degrees, minutes and seconds.
Objective: 5-1 Angles and Degree Measure

10. You Try Convert 329.125o to degrees, minutes, and seconds.
Objective: 5-1 Angles and Degree Measure

11. Converting Degrees, Minutes, Seconds to Decimal
Write north latitude 39o5’34” to a decimal rounded to the nearest thousandth.
Objective: 5-1 Angles and Degree Measure

12. You Try
If the bearing from the plane to tower B is 35o12’7”, write the bearing as a decimal rounded to the nearest thousandth.
Objective: 5-1 Angles and Degree Measure

13. Quadrantal Angles
If the terminal side of an angle that is in standard position coincides with one of the axes, the angle is called a quadrantal angle. y
+90o x
Objective: 5-1 Angles and Degree Measure

14. Rotations A full rotation is 360 degrees.
A measure of more than 360 degrees represents more than one rotation.
Objective: 5-1 Angles and Degree Measure

15. Example Give the angle measure represented by each rotation.
rotations clockwise:
rotations counterclockwise:
Objective: 5-1 Angles and Degree Measure

16. You Try Give the angle measure represented by each rotation.
rotations clockwise.
rotations counterclockwise.
Objective: 5-1 Angles and Degree Measure

17. Coterminal Angles
Two angles in standard position are called coterminal angles if they have the same terminal side. Since angles differing in degree measure by multiples of 360 degrees are equivalent. y x
Objective: 5-1 Angles and Degree Measure

18. Math Definition
If is the degree measure of an angle, then all angles measuring k, where k is an integer; are coterminal with .
Example: Any angle coterminal with 75o can be written 75o + 360ko, where k is the number of rotations around the circle.
Objective: 5-1 Angles and Degree Measure

19. Example
Identify all angles that are coterminal with each angle. Then find one positive angle and one negative angle coterminal with the angle.
1. 45o o y x
Objective: 5-1 Angles and Degree Measure

20. You Try
Identify all angles that are coterminal with each angle. Then find one positive angle and one negative angle that are coterminal with the angle.
1. 86o o
Objective: 5-1 Angles and Degree Measure

21. More Fun
If each angle is in standard position, determine a coterminal angle that is between 0o and 360o. State the quadrant in which the terminal side lies. o o
Objective: 5-1 Angles and Degree Measure

22. You Try
If each angle is in standard position, determine a coterminal angle that is between 0o and 360o. State the quadrant in which the terminal side lies. o o
Objective: 5-1 Angles and Degree Measure

23. VERY IMPORTANT – Reference Angles
If is a nonquadrantal angle in standard position, its reference angle is defined as the acute angle formed by the terminal side of the given angle and the x-axis.
Objective: 5-1 Angles and Degree Measure

24. The Math Definition
For an angle , 0o< <360o its reference angle is defined by:
when the terminal side is in Quadrant I
o when the terminal side is in Quadrant II
o when the terminal side is in Quadrant III
o when the terminal side is in Quadrant IV.
Objective: 5-1 Angles and Degree Measure

25. Example Find the measure of the reference angle for each angle:
Objective: 5-1 Angles and Degree Measure

26. You Try Find the measure of the reference angle for each angle.
Objective: 5-1 Angles and Degree Measure

27. Homework
Page 281, 18, 20, 24, 26, even
Objective: 5-1 Angles and Degree Measure