Download 5_cosine-1 - Nayland Maths

Finding the length of sides and size of
angles in non right-angled triangles when
the Sine rule won’t do the job!
The non right-angled triangles we
have worked with so far, have all
had one thing in common.
110°
35°
x m
12 m
Find x.
Q. What can you say about the
information (sides and angles) given in the
triangles for all Sine rule questions????
25°
28 m

15 m
Find .
A. Each two pieces of information (sides and angles) have
ALWAYS been opposite each other.
So, what happens here?
B
Let’s try to use the Sine rule!
120°
a
b?
c
=
=
Sin A
Sin B
Sin C
18 m
25 m

C
Find x.
x m
A
We know these parts
of the triangle
And we’re asked to
find this.
You can’t use the Sine rule!
Does it work finding an angle?
B
Sin A
Sin B
Sin C ?
=
a =
b
c
35 m
Or here?
It won’t work
here either!
We need
another rule!
50 m
Find .
A
22 m

C

Here’s how to derive the next rule:
C
b
A
In  BCD, by Pythagoras,
a
h
x D
a2 = h2 + (c - x)2
= h2 + c2 - 2cx + x2
= (h2 + x2) + c2 - 2cx
c - x
c
If we say this part of
c is x, then
B
then this part
must be..
1
But in  ACD, by Pythagoras
b2 = h2 + x2
2
And in  ACD, by normal Trig
Cos A = x
b
i.e. x = b  Cos A
So, we have 3 equations to use…….
3
a2 = (h2 + x2) + c2 - 2cx
C
a
b
A
xD
b2 = h2 + x2
c - x
c
B
x = b  Cos A
1
2
3
Use # 2 to replace h2 + x2 with…. b2
So, a2 = (h2 + x2) + c2 - 2cx becomes
a2 = b2 + c2 - 2cx
And use # 3 to replace x with….
b  Cos A
So, the finished rulea2 = b2 + c2 - 2cb  Cos A
This is known as the Cosine rule :a2 = b2 + c2 - 2cb  Cos A
a2 = b2 + c2 - 2bc  Cos A
More simply written as...
a2 = b2 + c2 - 2bcCos A
This rule allows you to find a.
(But only after you have found the square root of a2).
What happens if you are asked to find b or c?
There is a pattern in the way the rule is written………...
What is it?
The three sides are written first, all squared, minus
twice the second two sides from the rule, times the
So,
Cos of the side you’re finding…….
and
b2 = a2 + c2 - 2acCos B
c2 = a2 + b2 - 2abCos A
Now, to use it…………...
C
Step 1. Work out what you are trying to
find within the rule, by labelling
the triangle.
16m
Step 2. Write the correct rule
A
In this instance we need the rule to find c.
c2 = a2 + b2 - 2abCos A
45°
20 m
x
B
Find x, to the nearest metre
Step 3. Substitute then evaluate
Solution:
c2 = 20
2
+ 162 - 2  20  16  Cos 45
c2  400 + 256 - 640  0.7071….
c2  656 - 452.54834….
c2  203.45166….
c  203.45166.....
Don’t forget to do this - many do!
c  14.26m

c  14 m
Now, to find an angle…….
B

55m
A
Step 1. Work out what you are
trying to find within the
27 m
rule, by labelling the triangle.
C
75m
Find , to the nearest metre
Here, we’re after  B, but
our rules only allow us to
find sides!
Write the rule to find side b then change the subject.
b2 = a2 + c2 - 2acCos B
Here the CosB is with the
-2ac, so its negative.
It will become positive if we move everything after the minus sign to the right
2acCos B + b2 = a2 + c2
2acCos B = a2 + c2 - b2
a2 + c2 - b2
Cos B =
2ac
Move the b2 to the other
side by…. subtracting it, so
Now, move the 2ac to the
other side by…. dividing it, so
In future you either do all these steps at
the start of the question or you remember
the Cos rule in its angle form as well
(I suggest the latter)
So what’s the pattern to be able to write any
Cos rule for an angle?
a2 + c2 - b2
Cos B =
2ac
First, write the Cos of the angle you’re finding,then
equal to the other 2 sides each squared, minus the side
related to the angle squared, all divided by twice the
other 2 sides
So….
b2 + c2 - a2
Cos A =
2bc
And….
a2 + b2 - c2
Cos C =
2ab
Now to use it….
B

55m
A
27 m
We were trying
to find  B
C
75m
Find , to the nearest minute
Cos B =
27 2 + 552 - 752
Cos B 
2  27  55
-1871
2970
-0.6299….
B
 B
129.04….°
129° 3’
Cos B =
Now use your calculator
with the INV key to find
the size of angle A.
Your turn……
Set 6I Page 273
of Coroneus