h select and use appropriate trigonometric ratios and formulae to solve problems involving

1
M3 iii) Non-right-angled triangles and
trigonometry
What is involved in M3 (iii)?
d) draw graphs of the sine and cosine curves for
0 ? A ? 180
?
g) use the area formula to find the area of a
triangle Area ½ab sin C

• h) select and use appropriate trigonometric
  ratios and formulae to solve problems involving
• trigonometry that require the use of more
  than one triangle (two dimensions), where the
• diagram is provided.

(All these things are on your self-assessment
sheet for this topic)
2
Finding the length of sides and size of angles in
non right-angled triangles
So far in your trig work, you have only used
acute angles, because the triangles you have been
working with have been only right-angled.
What do you do in these triangles?
Why cant you use normal Sin, Cos or Tan
Trigonometry?
There is no Hypotenuse because they arent
right-angled triangles!
So how do you find the side marked x or the angle
marked ??
3
Lets investigate these types of triangles!
First of all, we need to be able to name the
sides of the triangle.
This side is b
And, this side is c
We could use two letters, e.g. AB, but it would
be quicker if it was only one letter!
So, this side is a
The convention used is that the side opposite an
angle(which is ALWAYS named with an upper case
letter) is named by a lower case of the same
letter.
4
In this ?ABC, to develop the new rule, first draw
a perpendicular from C to AB. Name the
intersecting point D and the line h. Do it now
on your diagram
Now, ?ABC is broken up into 2 triangles, ?BCD
?ACD, (and they are both right angled).
And in ?BCD
In ?ACD
So, we can make them equal to each other.
h
h
Sin A
Sin B
b Sin A
a Sin B

And, by manipulating,
And, by manipulating,
And, by manipulating,
h b ? Sin A
b
a
h a ? Sin B

5
So, we have derived a rule which uses 2 sides and
their opposite angles.
If we changed the perpendicular so that it came
from A and went to CB, we would come up with this
rule
Both parts of the two rules can be combined to
develop what is known as the Sine rule
6
When the rule is in this form, we shall first use
it to find a side
c
But we will only ever need 4 of the six parts of
the rule in any one use of the formula.
a
First,because the vertices of the triangle have
not been named, name each corner. It doesnt
matter where you put each of the vertex letters,
but it DOES matter where you put the side letters!
b
As we dont know a, we wont be able to use it OR
angle A.
Having placed the letters, substitute into the
relevant parts of the formula correctly.
(to 2 d.p.)
x
9

This is a reasonable answer as it opposite the
40, and the 9cm is opposite the 80
And, by manipulating,
7
It will make our work much easier to find an
angle if we turn the Sine rule upside down, so
that the angles are on the top. After all when
we found the sides they were on top!
b
a
c
This time, we dont know b, so we wont be able
to use it OR angle B.
Once again we will only ever need 4 of the six
parts of the rule in any one use of the formula.
Having placed the letters, substitute into the
relevant parts of the formula correctly.
Sin ?
Sin 10

Using the INV key on your calculator
And, by manipulating,
This is a reasonable answer as it opposite the
19cm, and the 6cm is opposite the 10
Sin ? ? 0.52094.
Your turn, Exercise 9F, page 344 Year 10 Text