1. Angles and Radian Measure

2. 4.1 – Angles and Radian Measure An angle is formed by rotating a ray around its endpoint. The original position of the ray is called the initial side of the angle, and the ending position of the ray is called the terminal side. The endpoint of the ray is called the vertex of the angle. vertex terminal side initial side

3. 4.1 – Angles and Radian Measure An angle is said to be in standard position if its vertex is at the origin, and its initial side is on the positive x-axis. An angle is positive if it is rotated counterclockwise, negative if it is rotated clockwise. positive negative

4. 4.1 – Angles and Radian Measure Degree Measure 90º (right angle) 180º (straight angle) 270º 360º

5. 4.1 – Angles and Radian Measure Degree Measure 90º 180º 360º 270º acute (between 0º and 90º) obtuse (between 90º and 180º)

6. Converting from Degrees-Minutes-Seconds (DMS) to Decimal Degrees (DD) “DMS” stands for Degrees-Minutes-Seconds. The conversions are based on subdivisions of a whole degree where… 1 degree = 60 minutes 1 minute = 60 seconds It follows that 1 degree = 3600 seconds

7. From DMS to DD

8. From DD to DMS 1. Multiply the decimal part of the degrees (to the right of the decimal point) by 60. These are your minutes. 2. Now multiply the decimal part of your minutes (to the right of the decimal point) by 60. These are your seconds. *There is a DMS button on your calculator.*

9. Examples 35 degrees 23 minutes 57 seconds to DD (round to four decimal places) 97.554 degrees to DMS

10. 4.1 – Angles and Radian Measure If θ is an angle with its vertex at the center of a circle of radius r, and θ intercepts an arc of length s on that circle, then θ = radians. s θ r

11. 4.1 – Angles and Radian Measure If s is the length of an arc intercepted by a central angle θ (in radians) in a circle of radius r, then s = rθ. s θ r

12. 4.1 – Angles and Radian Measure If θ intercepts an arc of length equal to the radius of the circle, θ has a measure of 1 radian. r θ r

13. 4.1 – Angles and Radian Measure One complete rotation s θ r s = circumference = 2πr θ = 2πrr2πrr = 2π radians

14. 4.1 – Angles and Radian Measure Radian Measure (right angle) π (straight angle) 2π2π 180 =

15. 4.1 – Angles and Radian Measure To convert from degrees to radians, multiply by. To convert from radians to degrees, multiply by. 60º = 5 radians 60 degrees 3 = 150º 30 = 286.48º

16. Examples Convert the angle in degrees to radians. Convert the angle in radians to degrees.

17. 4.1 – Angles and Radian Measure Two angles are said to be coterminal if they have the same terminal side. 40º Angle = 40º Coterminal angle:40º − 360º = −320º 40º + 360º = 400º 40º + 720º = 760º

18. 4.1 – Angles and Radian Measure Two angles are said to be coterminal if they have the same terminal side. Angle = Coterminal angle:

19. 4.1 – Angles and Radian Measure A clock has a minute hand which is 8 inches long. If the minute hand moves from 12:00 to 3:00, how far does the tip of the minute hand move? s = rθ = (8)(90)= 720 inches = 4π= 12.57 inches

20. 4.1 – Angles and Radian Measure Two angles whose sum is 90º are called complementary. Two angles whose sum is 180º are called supplementary. 40º Angle = 40º Complement:90º − 40º = 50º Supplement:180º − 40º = 140º