2. Use your notes from last week: Find the value of x and y.

3. Use your notes from last week: Find the value of x and y.

4. Use your notes from last week: Find the value of x and y.

5. Section 7-1 Measurement of Angles Objective: To find the measure of an angle in either degrees or radians and to find coterminal angles.

6. What we are going to learn in Sec 7.1 Vocabulary Angle measure in degrees and radians Standard position The critical values on the Unit Circle Coterminal angles

7. Section 7-1 Measurement of Angles Objective: 1.To find the measure of an angle in either degrees or radians. 2.To find coterminal angles.

8. Common Terms Initial ray is the ray that an angle starts from. Terminal ray is the ray that an angle ends on. Vertex

9. Common Terms A revolution is one complete circular motion.

10. Angles in standard position

11. Standard Position Vertex at origin The initial side of an angle in standard position is always located on the positive x-axis.

12. Angles in standard position The vertex of the angle is on (0,0). Initial ray starts on the positive x-axis The angle is measure counter clockwise. The terminal ray can be in any of the quadrants.

13. Angle describes the amount and direction of rotation 120°–210° Positive Angle: rotates counter-clockwise (CCW) Negative Angle: rotates clockwise (CW)

14. 13 Positive and negative angles When sketching angles, always use an arrow to show direction.

15. Units of angle measurement There are two ways to measure an angle: Degrees & Radians

16. Units of angle measurement There are two ways to measure an angle: Degree: 1/360 th of a circle. That is the measure one sees on a protractor and most people are familiar with. Angles can be further split into 60 minutes per degree and 60 seconds per minute.

17. Quadrantal Angle

18. Quadrantal angles

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.

20. Degrees On one of the circles provided measure 1°

21. Radian Measure Use the string provided to measure the radius. Start on the “x-axis” and use the string to measure an arc the same length on the circle. The angle created is one radian.

22. Angle θ is one radian

23. Units of angle measurement Radian: when the arc of circle has the same length as the radius of the circle. Angle  measures 1 radian.

25. approximations 1 radian ~ 57.2958 degrees 1 degree ~ 0.0174533 radians* *note: the radian measure is usually stated as a fraction of .

26. Sec 7.1 day 2 Warm up While I check your UC, work on the following: a)Display the measure of one radian on circle. Display the measure of two radians on a cirlce. b)Describe what one radian is in terms of the radius of a circle r. c)Draw a circle and identify a central angle. Describe relationship between central angle and the intercepting arc.

27. Find the measure of the central angle

29. Find the measure of the central angle COH

30. Measure of central angle: For radian measure: For degree measure: s= arc length r= radius

31. 30 Radian Measure

32. Working with Radians

33. The conversion process

34. Example 1 Convert 196° to radians

36. Calculator 2 nd APP (Angle) Use DMS to convert to Degree, minute and second. Use Angle to change 40° 20’ to a decimal value. For more information click herehere

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

38. Example Find two angles, one positive and one negative, that are coterminal with the angle 52°. Sketch all three angles

39. Solution

40. Example Find two angles, one positive and one negative, that are coterminal with the angle Sketch all three angles.

42. Coterminal Angles generalized: Degree measure: θ  360°n Radian measure: θ  2π n Where n is a counting number.

43. Helpful websites Trig flash cards http://mathmistakes.info/facts/TrigFacts/ Hot math flash cards: http://hotmath.com/learning_activities/inter activities/trig_flashcard.swfhttp://hotmath.com/learning_activities/inter activities/trig_flashcard.swf

44. Homework Sec 7.1 Written Exercises Problems # 1-8 all and # 9-29 odds UC with coordinates filled out.

47. 1 degree = 60 minutes 1° = 60 1 minute = 60 seconds 1 = 60  So … 1 degree = _________seconds 3600 Express 36  5010  as decimal degrees 36 36 +.8333 +.00277

48. OR Use your calculator!! Express 36  5010  as decimal degrees Enter 36 Press this button  ’ ’’ Press enter Enter 50 Press this button  ’ ’’ Go over to the ’ symbol -- enter Enter 10 Press this button  ’ ’’ Go over to the ’’ symbol -- enter Press enter

49. Convert 50  47’ 50’’ to decimal degree 50.7972 Convert 125  27’ 6’’ to decimal degree 125.4517 Can you go backwards and convert the decimal degree to degrees minutes seconds? Enter 125.4517 Go to DMS hit enter.

50. Express 50.525  in degrees, minutes, seconds 50º +.525(60) 50º + 36.5 50º + 36 +.5(60)  50 degrees, 36 minutes, 30 seconds