Advanced Trigonometry
Sine Rule
The sine rule is also known as the “law of sines”. It enables us to find a missing side or
angle in a non-right-angled triangle.
Triangles are labelled in the following way:
The formula for the sine rule is:
𝒂
𝒃
=
𝒔𝒊𝒏𝑨 𝒔𝒊𝒏𝑩
Using the Sine Rule to find a missing side
eg.
𝑎
𝑏
=
𝑠𝑖𝑛𝐴 𝑠𝑖𝑛𝐵
𝑥
3.2
=
𝑠𝑖𝑛100 𝑠𝑖𝑛29
𝑥=
3.2𝑠𝑖𝑛100
𝑠𝑖𝑛29
𝑥 = 6.50m
Using the Sine Rule to find a missing angle
eg.
𝑎
𝑏
=
𝑠𝑖𝑛𝐴 𝑠𝑖𝑛𝐵
8
10
=
sin𝑚 sin75
sin𝑚 =
8𝑠𝑖𝑛75
10
𝑚 = 50.6°
Cosine Rule
The cosine rule is also known as the “law of cosines”. Similar to the sine rule, it also enables
us to find a missing side or angle in a non-right-angled triangle.
The formula for the cosine rule is:
𝒂𝟐 = 𝒃𝟐 + 𝒄𝟐 − 𝟐𝒃𝒄(𝒄𝒐𝒔𝑨)
Using the Cosine Rule to find a missing side
eg.
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐(𝑐𝑜𝑠𝐴)
𝑥 2 = 102 + 122 − 2(10)(12)(𝑐𝑜𝑠62)
𝑥 2 = 131.33
𝑥 = 11.5cm
Using the Cosine Rule to find a missing angle
It is useful to rearrange the formula for the cosine rule if you wish to use it to find an angle:
𝒄𝒐𝒔𝑨 =
𝒃𝟐 + 𝒄𝟐 − 𝒂𝟐
𝟐𝒃𝒄
eg.
𝑏 2 + 𝑐 2 − 𝑎2
𝑐𝑜𝑠𝐴 =
2𝑏𝑐
𝑐𝑜𝑠𝐴 =
102 + 112 − 92
2(10)(11)
𝑐𝑜𝑠𝐴 =
100 + 121 − 81
220
𝑐𝑜𝑠𝐴 =
140
220
𝐴 = 𝑐𝑜𝑠 −1 (
𝐴 = 50.5°
140
)
220