The document discusses the sine rule and its applications in solving problems involving triangles. It introduces the sine rule formula relating the ratios of sides and opposite angles. Examples are provided to demonstrate using the sine rule to calculate unknown side lengths and angles. It also covers calculating the area of a triangle using various formulas relating the area to combinations of sides and angles.

1. Sine Rule
A
c b
B a C

2. Sine Rule
A
c b
h
B a C

3. Sine Rule
A h
 sin B
c
c h  c sin B
h b
B a C

4. Sine Rule
A h h
 sin B  sin C
c b
c h  c sin B h  b sin C
h b
B a C

5. Sine Rule
A h h
 sin B  sin C
c b
c h  c sin B h  b sin C
h b
 c sin B  b sin C
B C c b
a 
sin C sin B

6. Sine Rule
A h h
 sin B  sin C
c b
c h  c sin B h  b sin C
h b
 c sin B  b sin C
B C c b
a 
sin C sin B
In any ABC
a b c
 
sin A sin B sin C

7. e.g.  i  H
57 46
37.2
Q 43 26

h
L

8. e.g.  i  H
h q

sin H sin Q
57 46 h 37.2

37.2 sin 57 46 sin 43 26
Q 43 26

h
L

9. e.g.  i  H
h q

sin H sin Q
57 46 h 37.2

37.2 sin 57 46 sin 43 26
Q 43 26

37.2sin 57 46
h
h sin 43 26
L
h  45.8 units (to 1 dp)

10. e.g.  i  H
h q

sin H sin Q
57 46 h 37.2

37.2 sin 57 46 sin 43 26
Q 43 26

37.2sin 57 46
h
h sin 43 26
L
h  45.8 units (to 1 dp)
(ii )
F
16.21
12.36
Y
10632
Z

11. e.g.  i  H
h q

sin H sin Q
57 46 h 37.2

37.2 sin 57 46 sin 43 26
Q 43 26

37.2sin 57 46
h
h sin 43 26
L
h  45.8 units (to 1 dp)
(ii )
sin Y sin Z

F y z
16.21 sin Y sin10632

12.36 12.36 16.21
Y
10632
Z

12. e.g.  i  H
h q

sin H sin Q
57 46 h 37.2

37.2 sin 57 46 sin 43 26
Q 43 26

37.2sin 57 46
h
h sin 43 26
L
h  45.8 units (to 1 dp)
(ii )
sin Y sin Z

F y z
16.21 sin Y sin10632

12.36 12.36 16.21
Y 12.36sin10632
10632 sin Y 
16.21
Z
Y  4658

13. Note: does your answer make sense?

14. Note: does your answer make sense?
Check whether your answer might be obtuse, remember;

15. Note: does your answer make sense?
Check whether your answer might be obtuse, remember;
 angle sum  = 180

16. Note: does your answer make sense?
Check whether your answer might be obtuse, remember;
 angle sum  = 180
 largest angle is opposite the largest side

17. Note: does your answer make sense?
Check whether your answer might be obtuse, remember;
 angle sum  = 180
 largest angle is opposite the largest side
A B
C

18. Note: does your answer make sense?
Check whether your answer might be obtuse, remember;
 angle sum  = 180
 largest angle is opposite the largest side
A B
d C
circumcircle

19. Note: does your answer make sense?
Check whether your answer might be obtuse, remember;
 angle sum  = 180
 largest angle is opposite the largest side
A B
d C
a b c
   diameter
sin A sin B sin C
circumcircle

20. Area of a Triangle
A
c b
B a C

21. Area of a Triangle
A
c b
h
B a C

22. Area of a Triangle
A 1
Area  ah
2
c b
h
B a C

23. Area of a Triangle
A h
1  sin C
Area  ah b
2
c b h  b sin C
h
B a C

24. Area of a Triangle
A h
1  sin C
Area  ah b
2
1 h  b sin C
c
h b  Area  ab sin C
2
B a C

25. Area of a Triangle
A h
1  sin C
Area  ah b
2
1 h  b sin C
c
h b  Area  ab sin C
2
B C In any ABC
a
1
Area  ab sin C
2
1
 bc sin A
2
1
 ac sin B
2

26. Area of a Triangle
A h
1  sin C
Area  ah b
2
1 h  b sin C
c
h b  Area  ab sin C
2
B C In any ABC
a
e.g. M 1
Area  ab sin C
9.21 2
1
F 60 15
  bc sin A
2
6.37 1
D  ac sin B
2

27. Area of a Triangle
A h
1  sin C
Area  ah b
2
1 h  b sin C
c
h b  Area  ab sin C
2
B C In any ABC
a
e.g. M 1
1 Area  ab sin C
9.21 Area  dm sin F 2
2
1 1
F 60 15

  9.21 6.37  sin 6015  bc sin A
2 2
6.37 1
D  ac sin B
2

28. Area of a Triangle
A h
1  sin C
Area  ah b
2
1 h  b sin C
c
h b  Area  ab sin C
2
B C In any ABC
a
e.g. M 1
1 Area  ab sin C
9.21 Area  dm sin F 2
2
1 1
F 60 15

  9.21 6.37  sin 6015  bc sin A
2 2
6.37  25.47 units 2 (to 2 dp) 1
D  ac sin B
2

29. Area of a Triangle A h
1  sin C
Area  ah b
2
1 h  b sin C
c
h b  Area  ab sin C
2
B C In any ABC
a
e.g. M 1
1 Area  ab sin C
9.21 Area  dm sin F 2
2
1 1
F 60 15

  9.21 6.37  sin 6015  bc sin A
2 2
6.37  25.47 units 2 (to 2 dp) 1
D  ac sin B
2
Exercise 4H; 1a, 2b, 3a, 4, 8, 9, 10, 12, 14, 16, 18, 20, 22*