1. Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce Haese and Haese Publications, AND Mathematical Studies Standard Level Peter Blythe, Jim Fensom, Jane Forrest and Paula Waldman de Tokman Oxford University Press, 2012

2. The Sine Rule and Cosine Rules
The sine and cosine rules are formulae that will help you find unknown sides and angles in a triangle.
These rules let you apply trigonometry to triangles that are not right-angled.

3. The Sine Rule
Use the sine rule if you are given this information about the triangle. Either:
2 sides and a non-included angle (an angle opposite)
2 angles and 1 side
SSA
ASA, AAS

4. The Sine Rule SSA, ASA, AAS Where A, B and C are angles and a, b and c
are the respective opposing sides A C B a b c
SSA, ASA, AAS

5. Practice
Find the length of AC.
16.2 cm

6. Practice
Find the length of AB.
12.0 m

7. Practice
In the diagram, triangle ABC is isosceles. AB = AC, CB = 15 cm and angle ACB is 23°.
Find:
(a) the size of angle CAB;
(b) the length of AB.
134° A C B
23º
15 cm
8.15 cm
Diagram not to scale

8. Practice
A farmer wants to construct a new fence across a field. The plan is shown below. The new fence is indicated by a dotted line. Calculate the length of the fence.
75°
40°
410 m
385 m
Diagram not to scale

9. Practice (a) Find the length of AC using the above information.
The figure shows a triangular area in a park surrounded by the paths AB, BC and CA, where AB = 400 m and ABC = 70
(a) Find the length of AC using the above information.
Diana goes along these three paths in the park at an average speed of 1.8 m/s.
(b) Given that BC = 788m, calculate how many minutes she takes to walk once around the park.
752 m
20.0 min

10. In triangle ABC, AC = 5, BC = 7, A = 48°, as shown in the diagram
Practice
In triangle ABC, AC = 5, BC = 7, A = 48°, as shown in the diagram
Find the measure of angle ABC giving your answer correct to the nearest degree. A B C 5 7
48°
32.1°
diagram not to scale

11. The Cosine Rule
Use the cosine rule if you are given this information about the triangle. Either:
2 sides and the included angle (an angle between)
3 sides
SAS
SSS

12. SSS SAS -Solving for a Side- The Cosine Rule a2 = b2 + c2 –2bc cosA
a2 + c2
–2ac cosB
c2 =
a2 + b2
–2ab cosC
This is all one term. A C B a b c
Where A, B and C are angles and a, b and c are the respective opposing sides

13. Find, correct to 3 sig figs, the length of BC.
Practice
Find, correct to 3 sig figs, the length of BC.
8.80 cm

14. SSS SAS The Cosine Rule -Solving for an Angle-B a b c
Where A, B and C are angles and a, b and c
are the respective opposing sides

15. Practice
In triangle ABC, if AB = 7 cm, BC = 8 cm and CA = 5 cm, find the measure of angle BCA.
60°

16. Practice
In triangle ABC, if AC = 8.6 m, AB = 6.3 m and angle A = 50, find the length of BC.
6.63 m

17. Practice
A gardener pegs out a rope, 19 meters long, to form a triangular flower bed as shown in this diagram.
Calculate:
(a) the size of the angle BAC;
(b) the area of the flower bed. B
A C
5 m
6 m
48.5°
Diagram not to scale