Mr Barton’s Maths Notes
Trigonometry
4. Sine and Cosine Rules
www.mrbartonmaths.com
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!
1. The Sine Rule – Finding an unknown Side
What Information do you need to be given?
Two angles and the length of a side
C
What is the Formula?
a
a
b
c


SinA SinB SinC
B
b
A
Remember:
c
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
x
a
b

SinA SinB
x
7.0

Sin 37 Sin 42
x 
7.0
 Sin 37
Sin 42
x  6.3cm (1dp )
Multiply both
sides by sin 37
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?
C
b
a
SinA SinB SinC


a
b
c
B
Remember:
If the angle is included, you will have to use the Cosine Rule!
c
A
Example
SinA SinB

a
b
Sinx Sin 37
16
x

11
Sin 37
Multiply both
Sinx 
 16
sides by 16
11
Sinx  0.8753...  x  61.10 (1dp )
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?
C
a
a2 = b2 + c2 – 2bcCosA
B
Remember:
c
You must be pretty good on your calculator to get these ones correct!
Example
b
A
a2 = b2 + c2 – 2bcCosA
x2 = 5.22 + 4.52 – 2 x 5.2 x 4.5 x Cos58
x
x2 = 5.22 + 4.52 – 2 x 5.2 x 4.5 x Cos58
x2 = 22.48977…
x = 4.74m (2dp)
Square root both
sides
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?
b2 c 2 a2
CosA 
2bc
C
a
B
b
A
c
Remember:
This is just a re-arrangement of the previous formula, so you only need to remember one!
Example
b2 c 2 a2
CosA 
2bc
x
92  112  122
Cosx 
2  9  11
58
Cosx 
198
Cosx  0.292929...  x  72.970 (2dp )
A Nice Little Summary
Cosine Rule
?
8
a2 = b2 + c2 – 2bcCosA
65o
11
?
b2 c 2 a2
CosA 
2bc
10
17
14
Sine Rule
?
a
b
c


SinA SinB SinC
10
43o
16
SinA SinB SinC


a
b
c
Finding Sides
Cosine Rule
Sine Rule
Need 2 sides and
included angle
Need 2 angles
and any side
?
9
62o
55o
Finding Angles
Need all 3 sides
Need 2 sides and an
angle not included
Good luck with
your revision!