1. Trigonometry-4 Ratios of the Sides of Triangles. The meaning of sin, cos and tan sin is short for sine cos is short for cosine tan is short for tangent

2. Trigonometry Triangles – Ratios of Sides Construct a Right Angled Triangle Ratios in a Right Angled Triangle

3. Triangles – Ratios of Sides  In triangle ABC, AB = 5 units, BC = 3 units and AC = 4 units.  A ratio is one number divided by another number. BC A B C 3 4 5  The ratio of side BC to side AC is written as: AC4 3 = 0,75 =

4. Triangles – Ratios of Sides  In triangle ABC you are given the length of the sides.  A ratio is one number divided by another number. BC A B C 3 4 5  The ratio of side BC to side AB is written as: AB5 3 = 0,6 =

5. Triangles – Ratios of Sides  In triangle ABC you are given the length of the sides.  A ratio is one number divided by another number. AC A B C 3 4 5  The ratio of side AC to side AB is written as: AB5 4 = 0,8 =

6. Construct a Right Angled Triangle The tools you need: ruler pencil protractor set square

7. Construct a Right Angled Triangle with an angle of 30  1: Draw a line

8. Construct a Right Angled Triangle 1: Draw a line

9. Construct a Right Angled Triangle 2: Make a point on the line

10. Construct a Right Angled Triangle 3: Move the protractor onto the point

11. Construct a Right Angled Triangle 4: Mark your angle with a point

12. Construct a Right Angled Triangle 4: Mark your angle with a point

13. Construct a Right Angled Triangle 5: Draw a line through the two points

14. Construct a Right Angled Triangle 5: Draw a line through the two points

15. Construct a Right Angled Triangle 6: Now draw a right angle to complete the triangle.

16. Construct a Right Angled Triangle 6: Now draw a right angle to complete the triangle.

17. Construct a Right Angled Triangle 30  7: Write in the size of the angle.

18. Ratios in a Right Angled Triangle Label the hypotenuse H, the opposite side O and the adjacent side A. 30  H O A

19. Ratios in a Right Angled Triangle Measure the length of H, O and A. 30  H O A Do you think everyone will get the same answers? No – all the triangles are different. The only thing that is the same is the angle.

20. Ratios in a Right Angled Triangle Measure the length of H, O and A. 30  H O A A = H = O = 5,0 Remember – your answers won’t be the same.

21. Ratios in a Right Angled Triangle Measure the length of H, O and A. 30  H O A A = H = O = 5,0 5,7 Remember – your answers won’t be the same.

22. Ratios in a Right Angled Triangle Measure the length of H, O and A. 30  H O A A = H = O = 5,0 5,7 2,8 Remember – your answers won’t be the same.

23. Ratios in a Right Angled Triangle 30  H O A A = 5,0 H = 5,7 O = 2,8 O A = 2,8 5,0 Work out the ratio: O A =0.56 O A =

24. Ratios in a Right Angled Triangle 30  H O A A = 5,0 H = 5.7 O = 2,8 O H = 2,8 5,7 Work out the ratio: O H =0.49 O H = O A =0.56

25. Ratios in a Right Angled Triangle 30  H O A A = 5,0 H = 5.7 O = 2,8 A H = 5.0 5,7 Work out the ratio: A H =0.87 A H = O H =0.49 O A =0.56

26. Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O A =0.56 O H =0.49 A H =0.87 Do you think our diagrams are very accurate? Probably not! With our tools we can’t be exact! What should we do to our answers?

27. Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O A =0.56 O H =0.49 A H =0.87 The second decimal place won’t be accurate. Let’s round the answers to one decimal place.

28. Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O A =0.56 O H =0.49 A H =0.87 How do we round the answers to one decimal place? If the second decimal is 4 or less, then the first decimal stays the same. If the second decimal is 5 or more, then the first decimal becomes bigger by 1.

29. O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.49 A H =0.87 How do we round the answers to one decimal place? If the second decimal is 5 or more, then the first decimal becomes bigger by 1. If the second decimal is 4 or less, then the first decimal stays the same.

30. O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.5 A H =0.87 How do we round the answers to one decimal place? If the second decimal is 5 or more, then the first decimal becomes bigger by 1. If the second decimal is 4 or less, then the first decimal stays the same.

31. O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.5 A H =0.9 How do we round the answers to one decimal place? If the second decimal is 5 or more, then the first decimal becomes bigger by 1. If the second decimal is 4 or less, then the first decimal stays the same.

32. O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.5 A H =0.9 Does everyone in the class get the same answers? YES ! Amazing! From all the different sized triangles you drew we get the same answers for the ratios.

33. O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.5 A H =0.9 On a scientific calculator do the following: Enter 30. Press the ‘tan’ button. Round the answer to 1 decimal. Answer is same as:

34. O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.5 A H =0.9 On a scientific calculator do the following: Enter 30. Press the ‘sin’ button. Round the answer to 1 decimal. Answer is same as:

35. O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.5 A H =0.9 On a scientific calculator do the following: Enter 30. Press the ‘cos’ button. Round the answer to 1 decimal. Answer is same as:

36. O A =0.6 Ratios in a Right Angled Triangle A = 5,0 H = 5.7 O = 2,8 Our Answers O H =0.5 A H =0.9 Answers on a scientific calculator: Gives same answer as ‘tan’ button. Gives same answer as ‘sin’ button. Gives same answer as ‘cos’ button.

37. Ratios in a Right Angled Triangle Instead of an angle of 30  in the triangle, you can put in any angle and get the same results. Use the ‘tan’ button. Use the ‘sin’ button. Use the ‘cos’ button. O A Ratio: O H A H

38. Ratios in a Right Angled Triangle  Instead of drawing triangles and measuring sides to get the ratios, we can just use a scientific calculator.  In any right angled triangle: O  A H O A Tan  = O H Sin  = A H Cos  =

39. Ratios in a Right Angled Triangle Tan 20  = 0,36 O A O H A H O 20  A H O 40  A H O 60  A H Tan 40  = 0,83Tan 60  = 1,73 sin 20  = 0,34sin 40  = 0,64sin 60  = 0,86 cos 20  = 0,93cos 40  = 0,76cos 60  = 0,5 These are true for any triangle with these angles.

40. Ratios in a Right Angled Triangle  We have to remember these trigonometric ratios.  In any right angled triangle: O  A H O A Tan  = O H Sin  = A H Cos  = Some Old Hens Cackle And Howl Till Old Age

41. Ratios in a Right Angled Triangle O  A H O A Tan  = O H Sin  = A H Cos  = Some Old HensCackle And HowlTill Old Age

42. Well Done!