Transcript Material Taken From: Mathematics for the international student

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) SSA
– 2 angles and 1 side ASA, AAS
The Sine Rule
a
b
c


sin A sin B sin C
Where A, B and C are angles and a, b and c
are the respective opposing sides
C
a
b
A
c
B
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:
134°
(a) the size of angle CAB;
A
(b) the length of AB.
8.15 cm
C
23º
15 cm
Diagram not to scale
B
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°
385 m
40°
410 m
Diagram not to scale
Practice
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.
diagram not to scale
A
752 m
400 m
20.0 min
30º
B
C
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.
C
32.1°
5
A
7
48°
B
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) SAS
– 3 sides
SSS
-Solving for a Side-
The Cosine Rule
a2 = b2 + c2 –2bc cosA
SSS
SAS
b2 = a2 + c2 –2ac cosB
c2 = a2 + b2 –2ab cosC
This is all one term.
Where A, B and C are angles
and a, b and c are the
respective opposing sides
C
a
b
A
c
B
Practice
Find, correct to 3 sig figs, the length of BC.
8.80 cm
-Solving for an Angle-
The Cosine Rule
b c a
cos A 
2bc
2
a c b
cos B 
2ac
2
a b c
cos C 
2ab
2
2
2
2
2
2
2
SSS
SAS
C
a
b
A
Where A, B and C are angles and a, b and c
are the respective opposing sides
c
B
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
5m
48.5°
6m
A
C
Diagram not to scale