2. www.mathsrevision.com Trigonometry Let’s Investigate Extension The Tangent Ratio The Tangent Angle The Sine Ratio The Sine Angle The Cosine Ratio The Cosine Angle Mixed Problems

3. www.mathsrevision.com Starter Questions Starter Questions www.mathsrevision.com

4. www.mathsrevision.com Let’s Investigate! Trigonometry

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

6. www.mathsrevision.com Trigonometry 30° Adjacent Opposite hypotenuse Opposite Adjacent = 0.6 Mathemagic!

7. www.mathsrevision.com Trigonometry 45° Adjacent Opposite hypotenuse Opposite Adjacent = 1 Try another!

8. www.mathsrevision.com Trigonometry 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. www.mathsrevision.com Trigonometry 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 possible angles.

10. www.mathsrevision.com Trigonometry 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. www.mathsrevision.com Trigonometry What’s the point of all this???Don’t worry, you’re about to find out!

12. www.mathsrevision.com Trigonometry 12 m How high is the tower? Opp 60°

13. www.mathsrevision.com Trigonometry 60° 12 m Adjacent Opposite hypotenuse Copy this!

14. www.mathsrevision.com Trigonometry Tan x° = Opp Adj Tan 60° = Opp 12 = Opp12 x Tan 60° Opp =12 x Tan 60°= 20.8m (1 d.p.) Change side, change sign! Copy this!

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

16. www.mathsrevision.com Starter Questions Starter Questions www.mathsrevision.com 3cm

17. www.mathsrevision.com Trigonometry Adj x°x°x°x° Tan x° = O p p o s i t e Opp Adjacent

18. www.mathsrevision.com Trigonometry 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!

19. www.mathsrevision.com Trigonometry Now try Exercise 1. ( HSDU Support Materials)

20. www.mathsrevision.com Starter Questions Starter Questions www.mathsrevision.com

21. www.mathsrevision.com Using Tan to calculate angles

22. www.mathsrevision.com Trigonometry Example x°x°x°x° Tan x° = Op p Adj Hyp S O H C A H T O A 12m Tan x° = 18 12 = 1.5Tan x° Adj 18m ?

23. www.mathsrevision.com Trigonometry = 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

24. www.mathsrevision.com Trigonometry x = Tan ⁻ ¹ 1.5 = 56.3° (1 d.p.) = 1.5Tan x° 2 nd Tan Tan ⁻ ¹ Press Enter = 1.5

25. www.mathsrevision.com Trigonometry Now try Exercise 2. ( HSDU Support Materials)

26. www.mathsrevision.com Starter Questions Starter Questions www.mathsrevision.com

27. Trigonometry The Sine Ratio x°x°x°x° Sin x° = O p p o s i t e Opp Hyp h y p o t e n u s e

28. www.mathsrevision.com Trigonometry Example 34° Sin x° = Op p Hyp O 11cm Sin 34° = O 11 = O 11 x Sin 34° O =11 x Sin 34°= 6.2cm (1 d.p.) Change side, change sign!

29. www.mathsrevision.com Trigonometry Now try Exercise 3. ( HSDU Support Materials)

30. www.mathsrevision.com Starter Questions Starter Questions www.mathsrevision.com 57 o

31. www.mathsrevision.com Using Sin to calculate angles

32. www.mathsrevision.com Trigonometry 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° ?

33. www.mathsrevision.com Trigonometry =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

34. www.mathsrevision.com Trigonometry 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

35. www.mathsrevision.com Trigonometry Now try Exercise 4. ( HSDU Support Materials)

36. www.mathsrevision.com Starter Questions Starter Questions www.mathsrevision.com

37. Trigonometry The Cosine Ratio Cos x° = Adjacent Adj x°x°x°x° Hyp h y p o t e n u s e

38. www.mathsrevision.com Trigonometry 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!

39. www.mathsrevision.com Trigonometry Now try Exercise 5. ( HSDU Support Materials)

40. www.mathsrevision.com Starter Questions www.mathsrevision.com Q1.Calculate Q2.Round to 1 decimal place 2.354. Q3. How many minutes in 3hours Q4.The answer to the question is 180. What is the question.

41. www.mathsrevision.com Using Cos to calculate angles

42. www.mathsrevision.com Trigonometry Example x°x°x°x° Cos x° = Op p Adj Hyp S O H C A H T O A 45cm Cos x° = 34 45 = 0.756 (3 d.p.)Cos x° x = Cos ⁻ ¹0.756 =40.9° (1 d.p.) Adj 34cm

43. www.mathsrevision.com Trigonometry Now try Exercise 6. ( HSDU Support Materials)

44. www.mathsrevision.com Starter Questions Starter Questions www.mathsrevision.com

45. www.mathsrevision.com The Three Ratios Cosine Sine Tangent Sine Tangent Cosine Sine www.mathsrevision.com

46. Trigonometry The Three Ratios Sin x° = Opp Hyp Cos x° = Adj Hyp Tan x° = Opp Adj

47. www.mathsrevision.com Trigonometry Sin x° = Opp Hyp Cos x° = Adj Hyp Tan x° = Opp Adj CAHCAHTOATOASOHSOH A C H O T A O S H Copy this!

48. www.mathsrevision.com Mixed Examples Cos 12° Sin 60° Tan 27° Sin 30° Sin 35° Tan 40° Cos 20° Cos 79° Sin 36° www.mathsrevision.com

49. Trigonometry Example 1 40° Sin x° = Op p Hyp S O H C A H T O A O 15m Sin 40° = O 15 = O15 x Sin 40° O =15 x Sin 40°= 9.6m (1 d.p.) Change side, change sign!

50. www.mathsrevision.com Trigonometry Example 2 35° Cos x° = Op p Adj Hyp S O H C A H T O A 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!

51. www.mathsrevision.com Trigonometry Example 3 60° Tan x° = Op p Adj Hyp S O H C A H T O A 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!

52. www.mathsrevision.com Trigonometry Now try Exercise 7. ( HSDU Support Materials)

53. www.mathsrevision.com Starter Questions Starter Questions www.mathsrevision.com Level E

54. www.mathsrevision.com Extension www.mathsrevision.com

55. Trigonometry Example 1 30° Sin x° = Op p Hyp S O H C A H T O A 23cm b Sin 30° = 23 b ?

56. www.mathsrevision.com Trigonometry Sin 30° = 23 b Change sides, change signs! Sin 30° 23 b= (This means b = 23 ÷ Sin 30º) b=46 cm

57. www.mathsrevision.com Trigonometry Example 2 50° Cos x° = Op p Adj Hyp S O H C A H T O A 7m p Cos 50° = 7 p p= 10.9m (1 d.p.) Adj Change sides, change signs! Cos 50° 7

58. www.mathsrevision.com Trigonometry Example 3 55° Tan x° = Op p Adj Hyp S O H C A H T O A 9m d Adj Tan 55° = 9 d d= 6.3m (1 d.p.) Change sides, change signs! Tan 55° 9