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:So 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 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.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.a/sinA = b/sinB
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.a/sinA = b/sinB
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.a2 = 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.Use the second version of the cosine rule
cos A = (b2 + c2 – a2) / 2bc
cos P = (64 + 25 – 49) / (2 x 8 x 5)
cos P = 0.5
P = cos−1(0.5)
P = 60°