The Sine and Cosine Rules

SOH CAH TOA applies to right angled triangles….but when you have other triangles the sine and cosine rules come into play. 

In the following formulae the triangle is labelled like this:sine and cosine notationSo angle A is opposite side a etc – notice that the angle has a capital and the side has a lower case letter.

Here’s the three formulae:

The Sine Rulethe sine ruleThe Cosine Rule

The first one is the normal form….but if you rearrange it you get the second one which is good for finding an angle.the cosine ruleArea Of A Triangletrig area of a triangleWell this all looks really complicated right? However, there are only four types of question you can get using the sine and cosine rules…so let’s have a look at them now.


You Are Given Two Angles And One Side – Sine Rule

Example: Find side x.two angles one sidea/sinA = b/sinB

So:

7/sin60 = x/sin80

Therefore x = sin80(7/sin60) = 7.96 (2dp)

2 Sides And An Angle Not Enclosed By Them – Sine Rule

Example: Find angle m.2 sides and non enclosed anglea/sinA = b/sinB

So:

10/sin75 = 8/sinm

So rearrange the formula.

Sinm = (sin75 x 8) / 10 = 0.77274…

m = Sin−1 (0.77274…) = 50.6° (1dp)

NOTE: MAKE SURE YOU KNOW HOW TO DO SINE, AND INVERSE SINE AND ALL THE OTHER TRIG FUNCTIONS ON YOUR CALCULATOR BEFORE YOU GO INTO THE EXAM. CHECK WITH YOUR TEACHER!! THIS IS VERY IMPORTANT

ALSO MAKE SURE YOU CALCULATOR IS IN DEGREES NOT RADIANS

Two Sides Plus The Enclosed Angle – Cosine Rule

Example: Find side x.two sides and enclosed anglea2 = b2 + c2 – 2bc cos(A)

Put in the numbers

x2 = (22×22) + (28×28) – 2(22×28) x cos97

x2 = 484 + 784 – 1232 x cos97

x2 = 1418.143…

So x equals the square root of 1418.143…

So x = 37.7 (1dp)

All Three Sides But No Angles – Cosine Rule

Example: Find angle P.all three sidesUse the second version of the cosine rule

cos A = (b2 + c2 – a2) / 2bc

So:

cos P = (64 + 25 – 49) / (2 x 8 x 5)

cos P = 0.5

P = cos−1(0.5)

P = 60°