1. Section 7-1 Measurement of Angles

2. Trigonometry The word trigonometry comes two Greek words, trigon and metron, meaning “triangle measurement.”

3. Trigonometry In trigonometry, an angle often represents a rotation about a point. Thus, the angle Θ shown is the result of rotating its initial ray to its terminal ray.

4. Revolutions and Degrees A common unit for measuring very large angles is the revolution, a complete circular motion. A common unit for measuring smaller angles is the degree, of which there are 360 in one revolution

5. Degrees, Minutes, and Seconds Angles can be measured more precisely by dividing one degree into 60 minutes and by dividing one minute into 60 seconds. For example, an angle of 25 degrees, 20 minutes, and 6 seconds is written 25˚20’6”.

6. Decimal Degrees Angles can also be measured in decimal degrees. To convert between decimal degrees and degrees, minutes, and seconds, you can reason as follows: 12.3˚=12˚+ 0.3(60)’ = 12˚18’ 25˚20’6”=25˚ +

7. Radians Relatively recently in mathematical history, another unit of angle measurement, the radian, has come into use. When an arc of a circle has the same length as the radius of the circle, the measure of the central angle, is by definition 1 radian.

8. Radians Likewise, a central angle has a measure of 1.5 radians when the length of the intercepted arc is 1.5 times the radius.

9. Radian Measures In general, the radian measure of the central angle is the number of radius units in the length of arc AB. This accounts for the name radian. In the diagram at the right, the measure (Greek Theta) of the central angle is:

10. Arc Length Arc length = s = r

11. Radians Let us use this equation to see how many radians correspond to 1 revolution. Since the arc length of 1 revolution is the circumference of the circle, 2Πr, we have. Thus, 1 revolution measured in radians is 2Π and measured in degrees is 360. We have 2Π radians = 360 degrees or Π radians = 180 degrees.

12. Conversion Formulas This gives us the following conversion formulas: 1 radian = degrees ≈ 57.2958 degrees

13. Conversion Formulas This gives us the following conversion formulas: 1 degree = radians ≈0.0174533 radians

14. Radians Angle measures that can be expressed evenly in degrees cannot be expressed evenly in radians, and vice versa. That is why angles measured in radians are often given as fractional multiples of Π.

15. Degrees vs. Radians Angles whose measures are multiples of appear often in trigonometry. The diagrams below will help you keep the degree conversions for these special angles in mind. Note that a degree measure, such as 45˚, is usually written with a degree symbol, while a radian measure such as is usually written without any symbol.

16. 45° Multiples of 45°,

17. 60° Multiples of 60°,

18. 30° Multiples of 30°,

19. Standard Position When an angle is shown in a coordinate plane, it usually appears in standard position, with its vertex at the origin and its initial ray along the positive x-axis. We consider a counterclockwise rotation to be positive and a clockwise rotation to be negative. By positive and negative angles we mean angles with positive and negative measures.

20. Positive Angle An angle of 380˚

21. Negative Angle An angle of

22. Quadrantal Angles If the terminal ray of an angle in standard position lies in the first quadrant, as shown at the left above, the angle is said to be a first- quadrant angle. Second-, third-, and fourth- quadrant angles are similarly defined. If the terminal ray of an angle in standard position lies along an axis, as shown at the right above, the angle is called a quadrantal angle. The measure of a quadrantal angle is always a multiple of 90˚ or

23. Coterminal Angles Two angles in standard position are called coterminal angles if they have the same terminal ray. For any given angle there are infinitely many coterminal angles.

24. Example Convert the degree measures to radians. 270°-23.6°

25. Example Convert the radian measures to degrees 12.3

26. Example Find two angles, one positive and one negative, that are coterminal with each given angle. A. 60° B. -210° C. D.

27. Example Find two angles, one positive and one negative, that are coterminal with each given angle. 24°15’-23°37’

28. Additional Example: 1. Give the degree measure and the radian measure of the angle formed by the hour hand and the minute hand of a clock at 2:30 a.m.

29. Additional Example: 2. A gear revolves at 40 rpm. Find the number of degrees per minute through which the gear turns and the approximate number of radians per minute.