1. Trigonometry

2. Trigonometry is a method of finding out an unknown angle or side in a right angled triangle
Both the triangles below are similar because:
The angles are the same but the sides are different

3. Trigonometry is a method of finding out an unknown angle or side in a right angled triangle
Both the triangles below are similar because:
The angles are the same but the sides are different

4. Trigonometry is a method of finding out an unknown angle or side in a right angled triangle
Both the triangles below are similar because:
The angles are the same but the sides are different

5. If we measure the height and the base:
Small triangle
Large triangle
For both triangles
5 cm
8 cm
10 cm
16 cm

6. This angle is in fact 320
So as long as the value of then this angle will always be 320

7. This is the idea behind trigonometry
If we know 2 sides then we can find the angles in the triangle
How do we know the angle is 320 ?
We can use our calculator which has been programmed to work out the angle.

8. We don`t have to know the height and the base it can be any 2 sides
Depending on which 2 sides are known then we use a different button on the calculator
Names are given to the 3 sides which all refer to the angle we are trying to find

9. Opposite, Adjacent and Hypotenuse
The names are:
Opposite, Adjacent and Hypotenuse
Opposite means on the other side from the angle we need.
Adjacent means next to the angle we need.
Hypotenuse means the side opposite the right angle
Hypotenuse
Opposite X
Adjacent

10. Identify the names of the sides of these right angled-triangles given angle k
opposite b
opposite a b
hypotenuse c c
adjacent a
hypotenuse k
opposite a k
adjacent c
adjacent b c
opposite k
hypotenuse
hypotenuse b k a
adjacent

11. Opposite, Adjacent and Hypotenuse
In each case label all the sides of the triangles as Opposite (O), Adjacent (A) and Hypotenuse (H) with relation to the angle marked as “X”. X X X x X X X X X X X

12. Using the Opposite (O), Adjacent (A) and
Hypotenuse (H) to work out the missing angle
The calculator has 3 buttons which are used to find the missing angle:
Sin – short for Sine
Cos – short for Cosine
Tan – short for Tangent

13. Deciding which button to use depends on which sides are given
SOH CAH TOA
Memory Aid
Some Old Horses Sin Opposite Hypotenuse
Can Always Hear Cos Adjacent Hypotenuse
Their Owners Approaching Tan Opposite Adjacent
Or invent one of your own

14. SOH CAH TOA S O H C A H T O A Divide it up into three groups
Place each group of three in a triangle starting in the bottom left of each triangle S O H C A H T O A

15. Trigonometric Ratios S O H
SOH C A H
CAH T O A
TOA

16. SOH CAH TOA Cos (x) = x Example 1 What have we got and need to find?
We need an angle – x.
We have the Hypotenuse and Adjacent side.
10 cm
25 cm x
Looking at the phrase, we can use C A H
Hypotenuse
Cos (x) =
Adjacent
Hypotenuse
Adjacent

17. Cos (x) =
Adjacent
Hypotenuse
10 cm
25cm x
Replace A and H by 10 and 25
Hypotenuse
Cos (x) =
= 0.4
We now need to convert this to an angle in degrees using the Cos-1 button!!!
Adjacent
x = Cos –1(0.4) = 66.42o
We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons.

18. SOH CAH TOA Tan (x) = x Example 2 What have we got and need to find?
We need an angle – x.
We have the Opposite and Adjacent side.
Looking at the phrase, we can use TOA
15 cm
Tan (x) =
Opposite
Adjacent
Opposite x
20 cm
Adjacent

19. We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons.
Tan (x) =
Opposite
Adjacent
Replace O and A by 15 and 20
Tan (x) =
= 0.75
We now need to convert this to an angle in degrees using the Tan-1 button!!!
15 cm
Opposite x
x = Tan –1(0.75) = 36.67o
20 cm
Adjacent

20. SOH CAH TOA Sin (x) = x Example 3 What have we got and need to find?
We need an angle – x.
We have the Hypotenuse and Opposite side.
Looking at the phrase, we can use S O H
Hypotenuse
12 cm
8 cm
Sin (x) =
Opposite
Hypotenuse
Opposite x

21. Sin (x) =
Opposite
Hypotenuse
We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons.
Replace O and H by 8 and 12
Sin (x) =
= 0.666
Hypotenuse
We now need to convert this to an angle in degrees using the Sin-1 button!!!
12 cm
8 cm
Opposite x
x = Sin –1(0.666) = 41.81o

22. Using Trigonometry to Find a Missing Side

23. Trigonometric Ratios S O H
SOH C A H
CAH T O A
TOA

24. Trigonometric Ratios S O H
The triangle can also be used to find either the opposite side or the hypotenuse

25. Trigonometric Ratios C A H
The triangle can also be used to find either the adjacent side or the hypotenuse

26. Trigonometric Ratios O A T
TOA
The triangle can also be used to find either the adjacent side or the opposite

27. H S O H Example 1 SOH CAH TOA What have we got and need to find?
We need the Hypotenuse H
We have an angle and the Opposite O
Hypotenuse
Looking at the phrase we can use S O H S O H
Opposite
Hypotenuse =
Opposite
Sin (angle)

28. H Opposite Hypotenuse = Sin (angle) Replace O by 3 and (angle) by 60o
Sin (60o)
60o
Use the Sin button on your calculator to find this value H
H =
Hypotenuse
H = …..
H = 3.46 m to 2 d.p.
3 m
Opposite

29. A T O A Example 2 SOH CAH TOA What have we got and need to find?
We need the Adjacent A
40o
We have an angle and the Opposite O A
Adjacent
Looking at the phrase we can use T O A T O A
3 m
Opposite

30. A Replace O by 3 and (angle) by 40o 3 40o Tan (40o) H = Adjacent H =
Use the Tan button on your calculator to find this value A
H =
Adjacent
H = 3.575…..
3 m
H = 3.58 m to 2 d.p.
Opposite

31. 8 C A H Example 3 SOH CAH TOA What have we got and need to find?
We need the Adjacent A
We have an angle and the Hypotenuse H 8
Hypotenuse
Looking at the phrase we can use C A H
70o A C A H
Adjacent

32. 8 Replace H by 8 and (angle) by 70o A = cos70 x 8 H = 0.342 x 8
Use the Cos button on your calculator to find this value 8
H = x 8
Hypotenuse
H = 2.736…..
70o A
H = 2.74 m to 2 d.p.
Adjacent