1. Created by Mr. Lafferty Maths Dept.
Trigonometry
S5 Int2
Sine Rule Finding a length
Sine Rule Finding an Angle
Cosine Rule Finding a Length
Cosine Rule Finding an Angle
Area of ANY Triangle
Mixed Problems
29-Apr-18
Created by Mr. Lafferty Maths Dept.

2. Created by Mr. Lafferty Maths Dept.
Starter Questions
S5 Int2
29-Apr-18
Created by Mr. Lafferty Maths Dept.

3. Created by Mr. Lafferty Maths Dept.
Sine Rule
S5 Int2
Learning Intention
Success Criteria
1. To show how to use the sine rule to solve REAL LIFE problems involving finding the length of a side of a triangle .
Know how to use the sine rule to solve REAL LIFE problems involving lengths.
29-Apr-18
Created by Mr. Lafferty Maths Dept.

4. Sine Rule B a c C b A www.mathsrevision.com
S5 Int2
Works for any Triangle
The Sine Rule can be used with ANY triangle
as long as we have been given enough information. B a c C b A
29-Apr-18
Created by Mr Lafferty Maths Dept

5. The Sine Rule Deriving the rule C b a B A c This can be extended to
Consider a general triangle ABC.
Deriving the rule P
Draw CP perpendicular to BA
This can be extended to
or equivalently

6. Calculating Sides Using The Sine Rule
S5 Int2
Example 1 : Find the length of a in this triangle. B
10m
34o
41o a C A
Match up corresponding sides and angles:
Now cross multiply.
Solve for a.

7. Calculating Sides Using The Sine Rule
S5 Int2
Example 2 : Find the length of d in this triangle. D
10m
133o
37o d E C
Match up corresponding sides and angles:
Now cross multiply.
Solve for d.
= 12.14m

8. Created by Mr Lafferty Maths Dept
What goes in the Box ?
S5 Int2
Find the unknown side in each of the triangles below:
(1)
12cm
72o
32o A
(2)
93o B
47o
16mm
A = 6.7cm
B = 21.8mm
29-Apr-18
Created by Mr Lafferty Maths Dept

9. Created by Mr. Lafferty Maths Dept.
Sine Rule
S5 Int2
Now try MIA Ex 2.1
Ch12 (page 247)
29-Apr-18
Created by Mr. Lafferty Maths Dept.

10. Created by Mr. Lafferty Maths Dept.
Starter Questions
S5 Int2
29-Apr-18
Created by Mr. Lafferty Maths Dept.

11. Created by Mr. Lafferty Maths Dept.
Sine Rule
S5 Int2
Learning Intention
Success Criteria
1. To show how to use the sine rule to solve problems involving finding an angle of a triangle .
Know how to use the sine rule to solve problems involving angles.
29-Apr-18
Created by Mr. Lafferty Maths Dept.

12. Calculating Angles Using The Sine Rule
S5 Int2
Example 1 :
Find the angle ao ao
45m
23o
38m
Match up corresponding sides and angles:
Now cross multiply:
Solve for sin ao
= 0.463
Use sin to find ao

13. Calculating Angles Using The Sine Rule
S5 Int2
143o
75m
38m bo
Example 2 :
Find the angle bo
Match up corresponding sides and angles:
Now cross multiply:
Solve for sin bo
= 0.305
Use sin to find bo

14. What Goes In The Box ? www.mathsrevision.com
Calculate the unknown angle in the following:
(2)
14.7cm bo
14o
12.9cm
(1)
14.5m
8.9m ao
100o
ao = 37.2o
bo = 16o

15. Created by Mr. Lafferty Maths Dept.
Sine Rule
S5 Int2
Now try MIA Ex3.1
Ch12 (page 249)
29-Apr-18
Created by Mr. Lafferty Maths Dept.

16. Created by Mr. Lafferty Maths Dept.
Starter Questions
S5 Int2
29-Apr-18
Created by Mr. Lafferty Maths Dept.

17. Created by Mr. Lafferty Maths Dept.
Cosine Rule
S5 Int2
Learning Intention
Success Criteria
1. To show when to use the cosine rule to solve problems involving finding the length of a side of a triangle .
Know when to use the cosine rule to solve problems.
2. Solve problems that involve finding the length of a side.
29-Apr-18
Created by Mr. Lafferty Maths Dept.

18. Cosine Rule B a c C b A www.mathsrevision.com
S5 Int2
Works for any Triangle
The Cosine Rule can be used with ANY triangle
as long as we have been given enough information. B a c C b A
29-Apr-18
Created by Mr Lafferty Maths Dept

19. *Since Cos A = x/c  x = cCosA
The Cosine Rule
The Cosine Rule generalises Pythagoras’ Theorem and takes care of the 3 possible cases for Angle A.
a2 > b2 + c2
a2 < b2 + c2
a2 = b2 + c2 A 1 2 3 A B C a b c
Consider a general triangle ABC. We require a in terms of b, c and A.
Deriving the rule
BP2 = a2 – (b – x)2
Also: BP2 = c2 – x2
a2 – (b – x)2 = c2 – x2
a2 – (b2 – 2bx + x2) = c2 – x2
a2 – b2 + 2bx – x2 = c2 – x2
a2 = b2 + c2 – 2bx*
a2 = b2 + c2 – 2bcCosA P x
b - x b
Draw BP perpendicular to AC
*Since Cos A = x/c  x = cCosA
When A = 90o, CosA = 0 and reduces to a2 = b2 + c2 1
Pythagoras
When A > 90o, CosA is positive,  a2 > b2 + c2 2
Pythagoras + a bit
When A < 90o, CosA is negative,  a2 > b2 + c2 3
Pythagoras - a bit

20. Finding an unknown side.B C a b c
The Cosine Rule
The Cosine rule can be used to find:
1. An unknown side when two sides of the triangle and the included angle are given.
2. An unknown angle when 3 sides are given.
Finding an unknown side.
a2 = b2 + c2 – 2bcCosA
Applying the same method as earlier to the other sides produce similar formulae for b and c. namely:
b2 = a2 + c2 – 2acCosB
c2 = a2 + b2 – 2abCosC

21. Cosine Rule www.mathsrevision.com OR
S5 Int2
Works for any Triangle
How to determine when to use the Cosine Rule.
Two questions
1. Do you know ALL the lengths.
SAS OR
2. Do you know 2 sides and the angle in between.
If YES to any of the questions then Cosine Rule
Otherwise use the Sine Rule
29-Apr-18
Created by Mr Lafferty Maths Dept

22. Using The Cosine Rule www.mathsrevision.com
S5 Int2
Works for any Triangle
Example 1 : Find the unknown side in the triangle below: L 5m
12m
43o
Identify sides a,b,c and angle Ao
a = L
b = 5
c = 12
Ao =
43o
a2 = b2 + c2
-2bccosAo
Write down the Cosine Rule.
Substitute values to find a2.
a2 = 52 +
122
- 2 x 5 x 12 cos 43o
a2 = -
(120 x
0.731 )
a2 =
81.28
Square root to find “a”.
a = 9.02m

23. Notice the two negative signs.
Using The Cosine Rule
S5 Int2
Works for any Triangle
137o
17.5 m
12.2 m M
Example 2 :
Find the length of side M.
a = M
b = 12.2
C = 17.5
Ao = 137o
Identify the sides and angle.
a2 = b2 + c2
-2bccosAo
Write down Cosine Rule
a2 = – ( 2 x 12.2 x 17.5 x cos 137o )
a2 = – ( 427 x – )
Notice the two negative signs.
a2 =
a2 =
a = 27.7m

24. What Goes In The Box ? www.mathsrevision.com
Find the length of the unknown side in the triangles:
(1)
78o
43cm
31cm L
L = 47.5cm
(2) 8m
5.2m
38o M
M =5.05m

25. Created by Mr. Lafferty Maths Dept.
Cosine Rule
S5 Int2
Now try MIA Ex4.1
Ch12 (page 254)
29-Apr-18
Created by Mr. Lafferty Maths Dept.

26. Created by Mr. Lafferty Maths Dept.
Starter Questions
S5 Int2
54o
29-Apr-18
Created by Mr. Lafferty Maths Dept.

27. Created by Mr. Lafferty Maths Dept.
Cosine Rule
S5 Int2
Learning Intention
Success Criteria
1. To show when to use the cosine rule to solve REAL LIFE problems involving finding an angle of a triangle .
Know when to use the cosine rule to solve REAL LIFE problems.
2. Solve REAL LIFE problems that involve finding an angle of a triangle.
29-Apr-18
Created by Mr. Lafferty Maths Dept.

28. Cosine Rule B a c C b A www.mathsrevision.com
S5 Int2
Works for any Triangle
The Cosine Rule can be used with ANY triangle
as long as we have been given enough information. B a c C b A
29-Apr-18
Created by Mr Lafferty Maths Dept

29. Finding Angles Using The Cosine Rule
S5 Int2
Works for any Triangle
Consider the Cosine Rule again:
a2 = b2 + c2
-2bccosAo
We are going to change the subject of the formula to cos Ao
b2 + c2 – 2bc cos Ao = a2
Turn the formula around:
-2bc cos Ao = a2 – b2 – c2
Take b2 and c2 across.
Divide by – 2 bc.
Divide top and bottom by -1
You now have a formula for finding an angle if you know all three sides of the triangle.

30. Finding Angles Using The Cosine Rule
S5 Int2
Works for any Triangle xo
16cm
9cm
11cm
Example 1 : Calculate the
unknown angle xo .
Write down the formula for cos Ao
Ao = xo
a = 11
b = 9
c = 16
Label and identify Ao and a , b and c.
Substitute values into the formula.
Cos Ao =
0.75
Calculate cos Ao .
Ao = 41.4o
Use cos to find Ao

31. Finding Angles Using The Cosine Rule
S5 Int2
Works for any Triangle
26cm
15cm
13cm yo
Example 2: Find the unknown
Angle in the triangle:
Write down the formula.
Ao = yo
a = 26
b = 15
c = 13
Identify the sides and angle.
Find the value of cosAo
The negative tells you the angle is obtuse.
cosAo =
Ao =
136.3o

32. What Goes In The Box ? ao www.mathsrevision.com bo
Calculate the unknown angles in the triangles below:
(1)
10m 7m 5m ao
(2)
12.7cm
7.9cm
8.3cm bo
ao =111.8o
bo = 37.3o

33. Created by Mr. Lafferty Maths Dept.
Cosine Rule
S5 Int2
Now try MIA Ex 5.1
Ch12 (page 256)
29-Apr-18
Created by Mr. Lafferty Maths Dept.

34. Created by Mr. Lafferty Maths Dept.
Starter Questions
S5 Int2
29-Apr-18
Created by Mr. Lafferty Maths Dept.

35. Created by Mr. Lafferty Maths Dept.
Area of ANY Triangle
S5 Int2
Learning Intention
Success Criteria
1. To explain how to use the Area formula for ANY triangle.
Know the formula for the area of any triangle.
2. Use formula to find area of any triangle given two length and angle in between.
29-Apr-18
Created by Mr. Lafferty Maths Dept.

36. General Formula for Area of ANY TriangleBo Co a b c h
Consider the triangle below:
Area = ½ x base x height
What does the sine of Ao equal
Change the subject to h.
h = b sinAo
Substitute into the area formula

37. Area of ANY Triangle B B a c C C b A A www.mathsrevision.com
Key feature
To find the area
you need to knowing
2 sides and the angle in between (SAS)
Area of ANY Triangle
S5 Int2
The area of ANY triangle can be found
by the following formula. B B a
Another version c C C
Another version b A A
29-Apr-18
Created by Mr Lafferty Maths Dept

38. Area of ANY Triangle B B 20cm c C C 30o 25cm A A www.mathsrevision.com
S5 Int2
Example : Find the area of the triangle. B B
The version we use is
20cm c C C
30o
25cm A A
29-Apr-18
Created by Mr Lafferty Maths Dept

39. Area of ANY Triangle E 10cm 60o 8cm F D www.mathsrevision.com
S5 Int2
Example : Find the area of the triangle. E
The version we use is
10cm
60o
8cm F D
29-Apr-18
Created by Mr Lafferty Maths Dept

40. What Goes In The Box ? www.mathsrevision.com Key feature
Remember (SAS)
What Goes In The Box ?
S5 Int2
Calculate the areas of the triangles below:
(1)
23o
15cm
12.6cm
A =36.9cm2
(2)
71o
5.7m
6.2m
A =16.7m2

41. Created by Mr. Lafferty Maths Dept.
Area of ANY Triangle
S5 Int2
Now try MIA Ex6.1
Ch12 (page 258)
29-Apr-18
Created by Mr. Lafferty Maths Dept.

42. Created by Mr. Lafferty Maths Dept.
Starter Questions
S5 Int2
61o
29-Apr-18
Created by Mr. Lafferty Maths Dept.

43. Created by Mr. Lafferty Maths Dept.
Mixed problems
S5 Int2
Learning Intention
Success Criteria
1. To use our knowledge gained so far to solve various trigonometry problems.
Be able to recognise the correct trigonometric formula to use to solve a problem involving triangles.
29-Apr-18
Created by Mr. Lafferty Maths Dept.

44. The Sine Rule
Application Problems A D
The angle of elevation of the top of a building measured from point A is 25o. At point D which is 15m closer to the building, the angle of elevation is 35o Calculate the height of the building. T B
35o
25o
10o
36.5
145o
15 m
Angle TDA =
180 – 35 = 145o
Angle DTA =
180 – 170 = 10o

45. The Sine Rule A
The angle of elevation of the top of a column measured from point A, is 20o. The angle of elevation of the top of the statue is 25o. Find the height of the statue when the measurements are taken 50 m from its base B T C
180 – 110 = 70o
180 – 70 = 110o
Angle ATC =
180 – 115 = 65o
Angle BCA =
Angle ACT =
25o
65o
110o
20o
70o
53.21 m 5o
50 m

46. The Cosine Rule Application Problems L H B 57 miles 24 miles A
A fishing boat leaves a harbour (H) and travels due East for 40 miles to a marker buoy (B). At B the boat turns left and sails for 24 miles to a lighthouse (L). It then returns to harbour, a distance of 57 miles.
Make a sketch of the journey.
Find the bearing of the lighthouse from the harbour. (nearest degree) H
40 miles
24 miles B L
57 miles A

47. Not to Scale The Cosine Rule a2 = b2 + c2 – 2bcCosA P Q W
An AWACS aircraft takes off from RAF Waddington (W) on a navigation exercise. It flies 530 miles North to a point (P) as shown, It then turns left and flies to a point (Q), 670 miles away. Finally it flies back to base, a distance of 520 miles.
Find the bearing of Q from point P. P
670 miles W
530 miles
Not to Scale Q
520 miles

48. Now try MIA Ex 7.1 & 7.2 Ch12 (page 262)
Mixed Problems
S5 Int2
Now try MIA Ex 7.1 & Ch12 (page 262)
29-Apr-18
Created by Mr. Lafferty Maths Dept.