Chapter 4 Angles and their measures - PowerPoint PPT Presentation

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Chapter 4 Angles and their measures

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  1. Chapter 4 Angles and their measures

  2. Degrees vs Radians What do you know about degrees? What do you know about Radians?

  3. Degrees Degree is represented by the symbol . It is a unit of angular measure equal to 1/180th of a straight angle. In DMS (Degrees-minutes-second) system of angular measure, each degree is subdivided into 60 minutes and each minute is subdivided into 60 seconds.

  4. Example: A) Convert B) Convert

  5. How to do it for A: • A) First we convert • Then we have to convert .5 minute into seconds So the answer is

  6. How to do it for B: Each minute is 1/60th of a degree and each second is 1/3600th of a degree So it is

  7. Navigation In navigation, the course or bearing of an object is sometimes given as the angle of the line of travel measured clockwise from due north.

  8. What is Radian? A central angle of a circle has measure 1 radian if it intercepts an arc with the same length as the radius.

  9. Degree-Radian Conversion

  10. Example Working with Radian Measure How many radians are in 60 degrees?

  11. Answer

  12. Group Work: How many radians are in 90 degrees? How many degrees are in

  13. Arc Length Formula (Radian Measure) You may seen this as

  14. Arc Length Formula (Degree Measure) You may seen this as

  15. Note: So basically, when you find the arc length of a circle, you are finding the radian!!

  16. Example Perimeter of a Pizza Slice

  17. Example Perimeter of a Pizza Slice

  18. Angular and Linear Motion Angular speed is measured in units like revolutions per minute. Linear speed is measured in units like miles per hour.

  19. Example: You have a car, it’s wheels are 36 inches in diameter. If the wheels are rotating at 630 rpm, find the truck’s speed in miles per hour

  20. Answer 67.47 mi/hr Note remember 1 radian = the radius

  21. Nautical Mile A nautical mile (naut mi) is the length of 1 minute of arc along Earth’s equator.

  22. Statute mile Statue mile is the “land mile”

  23. Distance Conversions

  24. Example From Boston to San Franciso is 2698 stat mi, convert it to nautical mile.

  25. Answer

  26. Homework Practice P 356 #1-39 every other odd

  27. Trigonometric Functions, special right triangles and the unit circle

  28. Day 1 Exploration activity! You are to cut out the unit circle I provided onto the notebook. You are to trace as many special right triangle onto the unit circle as possible, but here is the rule. The hypotenuse of the triangle must be the radius of the circle and one leg on the axis. Hint: You should have 3 per quadrant After tracing it all, find the coordinates of the points that lies right on the circle and find the cumulative degrees of each point on the circle. Then answer the following questions in your group

  29. Result from the activity should be like this:

  30. Day 1 Why is it called a unit circle? How does the special right triangles and unit circle relate? What does the special right triangles give you relating to the circle? Is it possible to convert the degrees into radians? How do you do it?

  31. Unit Circle The unit circle is a circle of radius 1 centered at the origin.

  32. Review SOHCAHTOA

  33. Practice *Teacher make up different problems regarding SOHCAHTOA

  34. Note: Please remember the unit circle and the coordinates with its radian/degrees In an Unit Circle, the radius r is 1 is the degree or radians Also please note that (X,Y) is a rectangular coordinate In the unit circle with regarding to Trig: (X,Y) = ) X is Y is

  35. Practice Using Unit Circle *Teacher make up different practices regarding Unit Circle

  36. Trigonometry Extension sin csc cos t

  37. Practice Teacher make up different practices regarding the 6 trigonometry functions

  38. Initial Side, Terminal Side

  39. Positive Angle, Negative Angle

  40. Coterminal Angles Two angles in an extended angle-measurement system can have the same initial side and the same terminal side, yet have different measures. Such angles are called coterminal angles.

  41. Example Finding Coterminal Angles

  42. Example Finding Coterminal Angles

  43. Example Finding Coterminal Angles

  44. Example Evaluating Trig Functions Determined by a Point in QI

  45. Example Evaluating Trig Functions Determined by a Point in QI

  46. Example Evaluating More Trig Functions

  47. Example Using one Trig Ration to Find the Others

  48. Homework Practice P 366 #1-55 EOO P 381 #1-48 EOE

  49. Solving Trig Functions

  50. Function and its inverse