Transcript Document

We are now going to extend
trigonometry beyond right angled
triangles and use it to solve problems
involving any triangle.
1. Sine Rule
2. Cosine Rule
3. Area of a triangle
Throughout we will use the common
triangle notation of capital letters for
the vertices and corresponding,
lower case letters for the sides
opposite these vertices.
A
b
c
B
C
a
Side a is opposite to vertex A, side b
opposite vertex B and side c opposite
vertex C.
The sine rule
For unknown angles
sinA = sinB
sin
C
=
a
b
c
For unknown sides
c
b
a
= sinC
=
sinA
sinB
We can use the sine rule when we
are given:
1. Two sides and an angle opposite
to one of the two sides.
2. One side and any two angles.
Remember
Try to use the formula with the
unknown at the top of the fraction.
We can use the sine rule to find the
size of an angle or the length of a
side.
Example 1:
Q
4 cm
P
9 cm
75
R
Find the size of angle R
sinB
b
sinA
a
=
sinR
4
= sin75
9
sin
75
sinR = 9 x 4
sinR  0.4293
R = 25.4
Example 2:
Q
65
P
12 cm
75
y
Find the size of PR
R
a
sin A
y
sin 65
=
b
sin B
=
12
sin 75
y =
12
x sin65
sin 75
y  11.3 cm (to 1d.p.)
The cosine rule
a2 = b2 + c2 - 2bc cos A
2
b
=
2
a
+
2
c
- 2ac cos B
c2 = a2 + b2 - 2ab cos C
These formulae can be
rearranged to give:
2
2
2
b

c

a
Cos A =
2bc
2
2
2
a

c

b
Cos B =
2ac
2
2
2
a

b

c
Cos C =
2ab
The cosine rule
a2 = b2 + c2 - 2bc cos A
Cos A =
b2  c2  a2
2bc