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, 2004 AND Mathematical Studies Standard Level Peter Blythe, Jim Fensom, Jane Forrest and Paula Waldman de Tokman Oxford University Press, 2012

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.

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

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

Practice Find the length of AC. 16.2 cm

Practice Find the length of AB. 12.0 m

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

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

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

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

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

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

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

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

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

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

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