1. Let’s Investigate
The Tangent Ratio
The Tangent Angle
The Sine Ratio
The Sine Angle
The Cosine Ratio
The Cosine Angle
Mixed Problems
Extension

2. Starter Questions

3. Trigonometry
Let’s Investigate!

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

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

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

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

8. The ancient Greeks discovered this and repeated this for
possible angles.
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!

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

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

11. Opp
How high is the tower?
60°
12 m

12. Copy this!
Opposite
hypotenuse
60°
12 m
Adjacent

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

14. ?
20.8m
So the tower’s 20.8 m high!
Don’t worry, you’ll be trying plenty of examples!!

15. Starter Questions
3cm

16. Opp
Tan x° =
Adj
Opposite x°
Adjacent

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

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

19. Starter Questions

20. Using Tan to calculate angles

21. ? SOH CAH TOA Opp Tan x° = Adj 18 Tan x° = 12 Tan x° = 1.5 Example Opp
Hyp
18m
Opp ?
Tan x° =
Adj x°
12m 18
Tan x° =
Adj 12
Tan x°
= 1.5

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

23. = 1.5 Tan x° Press Enter 1.5 = x = Tan ⁻¹1.5 = 56.3° (1 d.p.) Tan ⁻¹

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

25. Starter Questions

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

27. Opp Sin x° = Hyp O Sin 34° = 11 = O 11 x Sin 34° O = 11 x Sin 34°
Example
Hyp
11cm O
Opp
Opp
Sin x° =
34°
Hyp O
Sin 34° =
Change side, change sign! 11
= O
11 x Sin 34°
O =
11 x Sin 34°
= 6.2cm (1 d.p.)

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

29. Starter Questions
57o

30. Using Sin to calculate angles

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

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

33. = 0.667 (3 d.p.) Sin x° Press Enter 0.667 = x = Sin ⁻¹0.667
2nd
Enter
0.667 =
x =
Sin ⁻¹0.667
= 41.8° (1 d.p.)

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

35. Starter Questions

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

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

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

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

40. Using Cos to calculate angles

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

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

43. Starter Questions

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

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

46. O S H A C H O T A CAH TOA SOH Sin x° = Opp Hyp Cos x° = Adj Tan x° =
Copy this!
Sin x° =
Opp
Hyp
Cos x° =
Adj
Tan x° = O
S H A
C H O
T A
CAH
TOA
SOH

47. Mixed Examples www.mathsrevision.com Cos 20° Tan 27° Sin 36° Sin 60°

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

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

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

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

52. Starter Questions
Level E

53. Extension

54. ? SOH CAH TOA Opp Sin x° = Hyp 23 Sin 30° = b Example 1 Hyp b 23cm Opp

55. 23 Sin 30° = b 23 b= Sin 30° (This means b = 23 ÷ Sin 30º) b= 46 cm
Change sides, change signs! b 23 b=
Sin 30°
(This means b = 23 ÷ Sin 30º) b=
46 cm

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

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