1. Introduction to
Trigonometry

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

3. 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. Accurate to 3 decimal places!
The ancient Greeks discovered this and repeated this for all 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. How high is the tower? h
60°
12 m

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

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

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

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

16. Example Opp Tan x° = Adj c Tan 65° = 8 8 x Tan 65° = c c = 8 x Tan 65°
Hyp
Tan x° = c
Adj c
65°
Tan 65° = 8 8m
Adj
8 x Tan 65°
= c
c =
8 x Tan 65°
= 17.2m (1 d.p.)

17. Now try
Exercise 1.

18. Using Tan to calculate angles

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

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

21. = 1.5 Tan x° Press Enter 1.5 = x = Tan ⁻¹1.5 = 56.3° (1 d.p.) Tan ⁻¹

22. Now try
Exercise 2.

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

24. Example Opp Sin x° = Hyp h Sin 34° = 11 = h 11 x Sin 34° h =
11cm h
Opp
34°
Opp
Sin x° =
Hyp h
Sin 34° = 11
= h
11 x Sin 34°
h =
11 x Sin 34°
= 6.2cm (1 d.p.)

25. Now try
Exercise 3.

26. Using Sin to calculate angles

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

28. 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

29. = 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.)

30. Now try
Exercise 4.

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

32. Example Adj Cos x° = Hyp b Cos 40° = 35 35 x Cos 40° = b b =
Opp
Hyp
35mm
Adj
Cos x° =
Hyp b
Cos 40° = 35
35 x Cos 40°
= b
b =
35 x Cos 40°
= 26.8mm (1 d.p.)

33. Now try
Exercise 5.

34. Using Cos to calculate angles

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

36. Now try
Exercise 6.

37. The Three Ratios Tangent Sine Cosine Sine Sine Tangent Cosine Sine

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

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

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

41. Example 1 SOH CAH TOA Opp Sin x° = Hyp h Sin 40° = 15 = h 15 x Sin 40°
= 9.6m (1 d.p.)

42. Example 2 SOH CAH TOA Adj Cos x° = Hyp b Cos 35° = 23 23 x Cos 35° = b
Opp
Hyp
23cm
Adj
Cos x° =
Hyp b
Cos 35° = 23
23 x Cos 35°
= b
b =
23 x Cos 35°
= 18.8cm (1 d.p.)

43. Example 3 SOH CAH TOA Opp Tan x° = Adj c Tan 60° = 15 15 x Tan 60° = c
Hyp c
60°
Opp
Tan x° =
15m
Adj
Adj c
Tan 60° = 15
15 x Tan 60°
= c
c =
15 x Tan 60°
= 26.0m (1 d.p.)

44. Now try
Exercise 7.

45. Extension

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

47. 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

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

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