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Special angles
The document demonstrates how to construct right triangles with angles of 45, 60, and 30 degrees and sides of 1 unit in order to derive exact trigonometric ratio values. An isosceles right triangle is used to show that tan(45) = 1, sin(45) = sqrt(2)/2, and cos(45) = sqrt(2)/2. An equilateral triangle bisects an angle of 60 degrees, allowing the trig ratios of sin(60), cos(60), and tan(60) to be determined. Bisecting the top angle of 30 degrees gives the trig ratios of sin(30), cos(30), and tan(30). The key point is to remember the process
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Special angles
- 1. Exact Values ofthe Trig Ratios Thursday 26th May 2011
- 2. Construct an rightangled isosceles triangle
- 3. 45° 45° Construct an right angled isosceles triangle
- 4. 1 1 Make the sides the simplest they can be....
- 5. 45° 1 45° 1 Make the sides the simplest they can be....
- 6. 45° 1 45° 1 Find the hypotenuse....
- 7. 45° 2 1 45° 1 Find the hypotenuse....
- 8. o 1 tan 45 = = 1 1 45° o 1 2 sin 45 = = 2 2 1 o 1 2 cos 45 = = 2 2 45° 1 define the Trig ratios....
- 9. o 1 tan 45 = = 1 1 45° o 1 2 2 sin 45 = 2 = 2 1 o 1 2 cos 45 = = 2 2 45° 1 define the Trig ratios....
- 10. o 1 tan 45 = = 1 1 45° 1 45° 1 remember the process not the values :)
- 11. o 1 tan 45 = = 1 1 45° 2 1 45° 1 remember the process not the values :)
- 12. o 1 tan 45 = = 1 1 45° o 1 2 2 sin 45 = 2 = 2 1 45° 1 remember the process not the values :)
- 13. o 1 tan 45 = = 1 1 45° o 1 2 2 sin 45 = 2 = 2 1 o 1 2 cos 45 = = 2 2 45° 1 remember the process not the values :)
- 14. Construct an equilateraltriangle...
- 15. Construct an equilateraltriangle...
- 16. 60° 60° 60° Construct an equilateral triangle...
- 17. 60° 60° 60° with simple side lengths
- 18. 60° 2 2 60° 60° 2 with simple side lengths
- 19. 60° 2 2 60° 60° 2 bisect the top angle
- 20. 30° 2 2 60° 60° 1 1 bisect the top angle
- 21. 30° 2 2 60° 60° 1 1 find the perpendicular height
- 22. 30° 2 2 3 60° 60° 1 1 find the perpendicular height
- 23. 30° 2 2 60° 60° 1 1 Read off the trig ratios in the constructed right angled triangle
- 24. 30° 2 2 3 60° 60° 1 1
- 25. 30° 2 2 3 60° 60° o 3 1 1 sin 60 = 2
- 26. 30° 2 2 3 60° 60° o 3 1 1 sin 60 = 2 o 1 cos 60 = 2
- 27. 30° 2 2 3 60° 60° o 3 1 1 sin 60 = 2 o 1 cos 60 = 2 o 3 tan 60 = = 3 1
- 28. 30° 2 2 3 60° 60° o 3 1 1 o 1 sin 60 = sin 30 = 2 2 o 1 cos 60 = 2 o 3 tan 60 = = 3 1
- 29. 30° 2 2 3 60° 60° o 3 1 1 o 1 sin 60 = sin 30 = 2 2 o 1 o 3 cos 60 = cos 30 = 2 2 o 3 tan 60 = = 3 1
- 30. 30° 2 2 3 60° 60° o 3 1 1 o 1 sin 60 = sin 30 = 2 2 o 1 o 3 cos 60 = cos 30 = 2 2 o 3 1 3 tan 60 = = 3 o tan 30 = = 1 3 3
- 31. Remember the processnot the results! :) o 1 tan 45 = = 1 3 o 1 1 o sin 60 = sin 30 = 2 2 o 1 2 o 1 o 3 sin 45 = = cos 60 = cos 30 = 2 2 2 2 o 1 2 o 3 o 1 3 cos 45 = = tan 60 = = 3 tan 30 = = 2 2 1 3 3