1. Mr Barton’s Maths Notes
Trigonometry
4. Sine and Cosine Rules

2. 4. Sine and Cosine Rules The Big Problem with Trigonometry
As far as mathematical things go, Pythagoras, and the trio of Sin, Cos and Tan, were pretty good… weren’t they?
However, they had one major draw back…
They only worked for right-angled triangles!
That certainly limited their use.
Well, imagine if we had some rules which worked for… wait for it… any triangle!
Well, you’ll never guess what… we do!... The Sine and Cosine Rules!
The Crucial Point about the Sine and Cosine Rules
You must know when to use each rule… what information do you need to be given?
If you can get your head around that, then it’s just plugging numbers into formulas!
Note: In all the formulas, small letters represent sides, and Capital Letters represent Angles!

3. C b a B A c x 1. The Sine Rule – Finding an unknown Side
What Information do you need to be given?
Two angles and the length of a side
What is the Formula?
Remember:
If you are given two angles, you can easily work out the 3rd by remembering that angles in a triangle add up to 1800!
Example C a b B A c x
Multiply both sides by sin 37

4. C b a B A c x 2. The Sine Rule – Finding an unknown Angle
What Information do you need to be given?
Two lengths of sides and the angle NOT INCLUDED
(i.e. not between those two sides!)
What is the Formula?
Remember:
If the angle is included, you will have to use the Cosine Rule!
Example C a b B A c x
Multiply both sides by 16

5. a2 = b2 + c2 – 2bcCosA a2 = b2 + c2 – 2bcCosA C b a B A c
3. The Cosine Rule – Finding an unknown Side
What Information do you need to be given?
Two sides of the triangle and the INCLUDED ANGLE
(i.e. the angle between the two sides!)
What is the Formula?
Remember:
You must be pretty good on your calculator to get these ones correct!
Example C a b
a2 = b2 + c2 – 2bcCosA B A c
a2 = b2 + c2 – 2bcCosA
x2 = – 2 x 5.2 x 4.5 x Cos58
x2 = – 2 x 5.2 x 4.5 x Cos58 x
x2 = …
Square root both sides
x = 4.74m (2dp)

6. C b a B A c x 4. The Cosine Rule – Finding an unknown Angle
What Information do you need to be given?
All three lengths of the triangle must be given!
What is the Formula?
Remember:
This is just a re-arrangement of the previous formula, so you only need to remember one!
Example C a b B A c x

7. Finding Sides Finding Angles Cosine Rule Sine Rule Cosine Rule 17 ? 11
A Nice Little Summary
Cosine Rule 17 ? 11 8
a2 = b2 + c2 – 2bcCosA ?
65o 14 10
Sine Rule ? ? 9 10
62o
55o
43o 16
Finding Sides
Finding Angles
Cosine Rule
Sine Rule
Need 2 sides and included angle
Need all 3 sides
Need 2 angles and any side
Need 2 sides and an angle not included

8. Good luck with your revision!