2. Presents

3. Let’s Investigate Extension The Tangent ratio The Sine ratio The Cosine ratio The three ratios

4. Let’s Investigate!

5. Trigonometry means “triangle” and “measurement”. Adjacent Opposite x°x°x°x° hypotenuse We will be using right-angled triangles.

6. 30° Adjacent Opposite hypotenuse Opposite Adjacent = 0.6 Mathemagic!

7. 45° Adjacent Opposite hypotenuse Opposite Adjacent = 1 Try another!

8. For an angle of 30°, Opposite Adjacent = 0.6 We write tan 30° = 0.6 Opposite Adjacent is called the tangent of an angle.

9. Tan 25°0.466 Tan 26°0.488 Tan 27°0.510 Tan 28°0.532 Tan 29°0.554 Tan 30°0.577 Tan 31°0.601 Tan 32°0.625 Tan 33°0.649 Tan 34°0.675 Tan 30° = 0.577 Accurate to 3 decimal places! The ancient Greeks discovered this and repeated this for all possible angles.

10. Now-a-days we can use calculators instead of tables to find the Tan of an angle. Tan On your calculator press Notice that your calculator is incredibly accurate!! Followed by 30, and press = Accurate to 9 decimal places!

11. What’s the point of all this???Don’t worry, you’re about to find out!

12. 12 m How high is the tower? h 60°

13. 12 m Adjacent Opposite hypotenuse h Copy this!

14. Tan x° = Opp Adj Tan 60° = h 12 = h12 x Tan 60° h =12 x Tan 60°= 20.8m (1 d.p.) Change side, change sign! Copy this!

15. So the tower’s 20.8 m high! Don’t worry, you’ll be trying plenty of examples!! 20.8m ?

16. The Tangent Ratio x°x°x°x° Tan x° = Opposite Opp Adjacent Adj

17. Example 65° Tan x° = Op p Adj Hyp c 8m Tan 65° = c 8 = c8 x Tan 65° c =8 x Tan 65°= 17.2m (1 d.p.) Adj Change side, change sign!

18. Now try Exercise 1. ( HSDU Support Materials)

19. Using Tan to calculate angles

20. Example x°x°x°x° Tan x° = Op p Adj Hyp SOH CAH TOA 12m Tan x° = 18 12 = 1.5Tan x° Adj 18m ?

21. = 1.5Tan x° How do we find x°? We need to use Tan ⁻ ¹ on the calculator. 2 nd Tan ⁻ ¹is written above Tan Tan ⁻ ¹ To get this press Tan Followed by

22. x = Tan ⁻ ¹ 1.5 = 56.3° (1 d.p.) = 1.5Tan x° 2 nd Tan Tan ⁻ ¹ Press Enter = 1.5

23. Now try Exercise 2. ( HSDU Support Materials)

24. The Sine Ratio x°x°x°x° Sin x° = Opposite Opp Hyp hypotenuse

25. Example 34° Sin x° = Op p Hyp h 11cm Sin 34° = h 11 = h 11 x Sin 34° h =11 x Sin 34°= 6.2cm (1 d.p.) Change side, change sign!

26. Now try Exercise 3. ( HSDU Support Materials)

27. Using Sin to calculate angles

28. Example x°x°x°x° Sin x° = Op p Hyp SOH CAH TOA 6m 9m Sin x° = 6 9 = 0.667 (3 d.p.)Sin x° ?

29. =0.667 (3 d.p.)Sin x° How do we find x°? We need to use Sin ⁻ ¹ on the calculator. 2 nd Sin ⁻ ¹is written above Sin Sin ⁻ ¹ To get this press Sin Followed by

30. x = Sin ⁻ ¹ 0.667 = 41.8° (1 d.p.) = 0.667 (3 d.p.)Sin x° 2 nd Sin Sin ⁻ ¹ Press Enter = 0.667

31. Now try Exercise 4. ( HSDU Support Materials)

32. The Cosine Ratio Cos x° = Adjacent Adj x°x°x°x° Hyp hypotenuse

33. Example 40° Cos x° = Op p Adj Hyp b 35mm Cos 40° = b 35 = b35 x Cos 40° b =35 x Cos 40°= 26.8mm (1 d.p.) Adj Change side, change sign!

34. Now try Exercise 5. ( HSDU Support Materials)

35. Using Cos to calculate angles

36. Example x°x°x°x° Cos x° = Op p Adj Hyp SOH CAH TOA 45cm Cos x° = 34 45 = 0.756 (3 d.p.)Cos x° x = Cos ⁻ ¹0.756 =40.9° (1 d.p.) Adj 34cm

37. Now try Exercise 6. ( HSDU Support Materials)

38. The Three Ratios Cosine Sine Tangent Sine Tangent Cosine Sine

39. The Ratios Sin x° = Opp Hyp Cos x° = Adj Hyp Tan x° = Opp Adj

40. The Ratios Sin x° = Opp Hyp Cos x° = Adj Hyp Tan x° = Opp Adj CAHTOASOH A C H O T A O S H Copy this!

41. Mixed Examples Cos 12° Sin 60° Tan 27° Sin 30° Sin 35° Tan 40° Cos 20° Cos 79° Sin 36°

42. Example 1 40° Sin x° = Op p Hyp SOH CAH TOA h 15m Sin 40° = h 15 = h 15 x Sin 40° h =15 x Sin 40°= 9.6m (1 d.p.) Change side, change sign!

43. Example 2 35° Cos x° = Op p Adj Hyp SOH CAH TOA b 23cm Cos 35° = b 23 = b23 x Cos 35° b =23 x Cos 35°= 18.8cm (1 d.p.) Adj Change side, change sign!

44. Example 3 60° Tan x° = Op p Adj Hyp SOH CAH TOA c 15m Tan 60° = c 15 = c15 x Tan 60° c =15 x Tan 60°= 26.0m (1 d.p.) Adj Change side, change sign!

45. Now try Exercise 7. ( HSDU Support Materials)

46. Extension

47. Example 1 30° Sin x° = Op p Hyp SOH CAH TOA 23cm b Sin 30° = 23 b ?

48. Sin 30° = 23 b Change sides, change signs! Sin 30° 23 b= (This means b = 23 ÷ Sin 30º) b=46 cm

49. Example 2 50° Cos x° = Op p Adj Hyp SOH CAH TOA 7m p Cos 50° = 7 p p= 10.9m (1 d.p.) Adj Change sides, change signs! Cos 50° 7

50. Example 3 55° Tan x° = Op p Adj Hyp SOH CAH TOA 9m d Adj Tan 55° = 9 d d= 6.3m (1 d.p.) Change sides, change signs! Tan 55° 9

51. © K Hughes 2001