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 c b C B 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 sin a A = sin b B = sin c C For unknown sides a sin A = b sin B = c sin C

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 75  9 cm

Find the size of angle R

R

sinA a sin 4 R

=

sinB b

=

sin 9 75 sin R

=

sin 9 75 x 4 sinR  0.4293

R = 25.4



Example 2:

Q 65  75  12 cm P y

Find the size of PR

R

a

sin

A

=

b

sin

B y

sin 65 = 12 sin 75

y

= 12 sin 75 x sin65

y



11.3

cm (to 1d.p.)

The cosine rule

a 2 = b 2 + c 2 - 2bc cos A b 2 = a 2 + c 2 - 2ac cos B c 2 = a 2 + b 2 - 2ab cos C

These formulae can be rearranged to give:

Cos A = Cos B = Cos C = b 2



c 2 2bc



a 2

a

2 a 2



c

2

2ac



b

2



b 2 2ab



c 2

The cosine rule

a 2 = b 2 + c 2 - 2bc cos A Cos A = b 2



c 2 2bc



a 2