Transcript Pythagoras and Trig

Pythagoras
h
b
a
a2 + b2 = h2
Pythagoras - Three simple steps
1. Square the numbers of the sides you are
given
2. To find the longest side, add the two
squared numbers. To find a shorter side,
subtract the smaller squared number
from the larger squared number.
3. After adding or subtracting, take the
square root. Then check that your
answer is sensible.
Bearings
1
2
3
1
2
3
3 key points
A bearing is the direction travelled between two points, given as an angle in
degrees
All bearings are measured clockwise from the northline (Marked N)
All bearings are given to 3 figures, e.g. 009, not 9
3 key words
“FROM”: Find the word from in question and pout your pencil on the diagram at
the point you are going from.
At the point you are going from, draw in the northline.
Now draw in the angle clockwise from the northline to the line joining the two
points. This angle is the bearing.
A
N
The bearing of A
from B
B
Some Definitions
• A bearing is the direction travelled
between two points, given as an angle in
degrees
• All bearings are measured clockwise from
the northline.
• All bearings should be given as 3 figures
Trigonometry:
Using SIN, COS and TAN
7 steps for finding lengths of sides from angles
1.
Labelling sides, example:-
Label the sides, (o)pposite, (a)djacent and
(h)ypotenuse
How do you know how to which side to label
what??
First ask:What is the side opposite the side I have been
given? Identify it and label it ‘opposite’ or ‘opp’, or
just ‘o’. This has already been done in the diagram
opposite.
Then ask ‘Which side is adjacent to the angle that I
have been given, which is not the longest side in
the triangle? Identify it and label it ‘adjacent’, or
‘adj’ or ‘a’. This has already been done in the
diagram opposite.
Label the remaining (longest) side in the triangle
‘hypotenuse’, or ‘hyp’ or ‘h’. This has already been
done in the diagram opposite. I have also labelled
this side ‘x’ as I want you to find the length of this
side in a little test at the end of this presentation.
h (x)
o (15cm)
35
a
In this case, you have been given
length of the opposite side (15cm).
Trigonometry:
Using SIN, COS and TAN
Formula triangles are:2.
3.
Write down SOH CAH TOA
from memory (these letters
form the triangles in the
diagram opposite)
Decide which two sides are
involved (How?) and select
SOH, CAH, or TOA
respectively, i.e. in this case
the two sides involved are
opposite and hypotenuse.
These are the two underlined
letters in SOH so its SOH you
want! Why are these two sides
involved?? – see next slide…
Trigonometry:
Using SIN, COS and TAN
Formula triangles are:4.
5.
6.
7.
Turn the one that you want into a
formula triangle (turn SOH into a
formula triangle – this is the top
one in the diagrams opposite).
Cover up what you want to find we want to find the length of the
hypotenuse. So if you cover up H
with your thumb you get the
equation you need,
i.e. O  S)
Translate into numbers and work
it out (i.e. 15  sin 35; see next
slide for tips)
Finally check you answer is
sensible
How do you decide which sides are
involved?
•
•
•
Look for the side that you have been
given the length of. This is one of the
sides that is involved.
The other side that is involved is the side
that you have to find the length of.
Decide which sides they are and go to
the next slide (in this case I have already
told you the sides involved are opposite
and hypotenuse).
Calculator tips
1. Try typing sin 30 into the calculator – you
should get 0.5 if you are doing it right.
2. From the previous slide we want 15 sin 35.
Type this into calculator and see what your
answer is. (26.15cm is correct!)
3. In questions I will be setting you later on in
tests you may be asked to calculate an angle
in a triangle. In this case you will be given the
lengths of two sides and using one of the
formula triangles you will be able to calculate
the angle - in our example it would be:Inverse x Sin x (15/26.15) (= 35 degrees) !!
Loci and Constructions
7 different types of Loci – this is a revision list – you can
read up them in your book
•
The locus of points which are “A fixed Distance from a
Given Point
•
The locus of points which are “A fixed Distance from a
Given Line”
•
The locus of points which are “Equidistant from Two
Given Lines”
•
The locus of points which are “Equidistant from Two
Given Points”
•
Constructing accurate 60 angles
•
Constructing accurate 90 angles
•
Drawing a Perpendicular from a Point to a Line