Download KS3 Shape 10 Pythagoras and Trigonometry



3D Shapes

Names of 3D solids
Prisms
Volume
Surface Area
Plans and Elevations
Isometric Drawing
Units of measure
Length, area and volume
Capacity / Mass
Metric and imperial measure
Conversion / conversion graphs
Real-life graphs
Speed, distance time

Parts of
Circumference
Area
Circle geometry
Equation of a circle


Right-angled triangles
Finding missing side lengths
Angles and sine, cosine and
tangent ratios
KS3 Shape, Space and
Measure
Constructions
Triangles
Similar & Congruent Shapes
Bisectors
Loci
Angles

Angles on a straight line
Angles at a point
Parallel lines and transversals alternating / corresponding /opposite angles
Supplementary angles
Polygons: interior & exterior angles
Extension work: Bearings
Lines & Angles
Line Segments
Vertical / Horizontal
Perpendicular / Parallel
Types of angle
Estimating measuring and drawing
Direction of turn
Compass directions
Describing angles (90o= ¼ of turn)
Pythagoras and
Trigonometry
Perimeter and Area
Dimensions (including volume)
Units of measure (including volume)
Counting squares
Intrinsic and Extrinsic information
Rectangles, triangles and compound shapes
Using Formulae (Rectangle, Triangle,
Trapezium, Parallelogram)

Circles

Properties of 2D Shapes
Names of polygons up to 10 sides
Special Quadrilaterals and Triangles
Geometric Properties
Tessellation
Transformations

Basic congruent & similar shapes
Coordinate geometry
Reflection (include lines of symmetry)
Rotation (include rotational symmetry)
Translation (including vector notation)
Enlargement
All topics can be covered by the end
of year 8.
Shape 10: Pythagoras and Trigonometry

Pythagoras’ Theorem and Trigonometry
Must
Should
Identify the ‘hypotenuse’ on a right- Given the length of the hypotenuse
angled triangle (regardless of the
and one of the other sides, use
orientation)
Pythagoras’ Theorem to work out the
length of the other side
Know Pythagoras’ Theorem such
that a2 + b2 = c2 for any right-angled Apply Pythagoras’ Theorems to
triangle where c is the length of the problems involving area of a triangle
hypotenuse
where the height of the triangle is not
Use Pythagoras’ Theorem to work
out the length of the hypotenuse
given the lengths of the other two
sides
Given an angle in a right-angled
triangle, label the opposite and
adjacent sides.
Understand that trigonometry
involves using or working out an
angle in a right-angled triangle.
Could
Understand and give examples of
the Pythagorean triplets
Recognise that any triangle can be
divided into two right-angled
triangles, and that for an isosceles
triangle, the two right-angled
triangles will be congruent
known
Use the trigonometric functions to
Understand trigonometric
work out the length of a side given the functions by considering the unit
length of one other side and an angle circle and by using Pythagoras’
Theorem
Use trigonometric functions to work
out the size of an angle given the
lengths of two of the sides
Learn the trigonometric functions;
Sin  = opposite ÷ hypotenuse
Cos  = adjacent ÷ hypotenuse
Tan  = opposite ÷ adjacent
Key Words: right-angled triangle, hypotenuse, Pythagoras’ Theorem, length, area, Pythagorean Triplets,
opposite, adjacent, trigonometry, trigonometric functions, sin, cos, tan,  , unit circle, ratio
Starters:
Show a series of Pythagorean Triplets and ask students to investigate the ratio of the values
Label sides of a right-angled triangle with the letters a, b and c or with the terms, opposite, adjacent and
hypotenuse
Activities:
Make a ‘unit circle’ on a large enough scale for students to walk around and measure distances horizontally
and vertically from the centre of the circle to experience the trigonometric ratios.
Work out the areas of the three squares formed by drawing a right-angled triangle and using each length as the
side length for each square / investigate the relationship between their areas.
Plenaries:
Learning framework questions:
- When should we choose to use Trigonometry instead of Pythagoras’ Theorem?
- How can you explain Pythagoras’ Theorem?
- What do Sin  and Cos  tell us?
- Where does Tan  come from?
- What are the Pythagorean Triplets?
Shape 10: Pythagoras and Trigonometry
Resources:
Unit circle / graph paper
10 ticks worksheets
Ratio sticks
Possible Homeworks:
Teaching Methods/Points:
Pythagoras’ theorem
Students must know that Pythagoras’ theorem applies to right-angled triangles;
Given a right-angled triangle with lengths labelled a, b and c as follows;
Where c is the length directly opposite the right angle
a
c
This side is also called the ‘hypotenuse’
b
Pythagoras’ theorem states that;
a2 + b2 = c2 for any right-angled triangle.
c2
a2
The theorem shows that if each length forms the side of a
square, the area of the triangle with the side length of the
hypotenuse is equal to the sum of the areas of the other
two squares.
b2
Students should be encouraged to identify the hypotenuse for right-angled triangles in a variety of different
orientations where the right angle is not immediately obvious. Furthermore, students can investigate the
Pythagorean triplets with side lengths in the ratio 3 : 4 : 5, showing that given lengths of 3n, 4n, and 5n
(for the hypotenuse); (3n)2 + (4n)2 = (5n)2 hence 9n2 + 16n2 = 25n2 which, of course, will always be true.
While it is relatively straight forward to use Pythagoras’ Theorem to work out the length of the hypotenuse
given the lengths of the other two sides, students will need to practise rearranging the formula such that if;
a2 + b2 = c2 then a2 = c2 – b2 and b2 = c2 – a2 (which is simple rearrangement on a maths table)
Moreover, students must understand that, having substituted values for two of the lengths, and therefore
having worked out the value of the square of one of the sides, the actual length of that side can be found by
square rooting (√) (equivalent to asking the question … what value written two times as a product is equal to
…) i.e. if a2 = 25, therefore a = 25 = 5. Encourage students to show every stage in their workings.
Shape 10: Pythagoras and Trigonometry
Examples for using Pythagoras’ Theorem to calculate missing lengths
x
4
Pythagoras’ theorem;
42 + 32 = x2
16 + 9 = x2
25 = x2
5 = x (√ both sides)
Pythagoras’ theorem;
72 + b2 = 8.52
49 + b2 = 72.25
b2 = 23.25
b =√23.25 = 4.82 (to 2d.p.s)
3
7
8.5
b
To work out the height, h of the lighthouse;
30 metres
Using Pythagoras’ theorem; h2 + 182 = 302
h2 + 324 = 900
h2
= 900 – 324
h
= 576
h
= 24 metres
h
18 metres
The Trigonometric Functions
Students should understand at the most basic level that the trigonometric functions; sine, cosine and tangent
apply to right-angled triangles. Whereas Pythagoras’ theorem allows you to find a missing length if you are
given the other two lengths (it is all about lengths!), trigonometry allows you to find an angle, or use an
angle to find a missing length if you are given just one other length.
Ensure plenty of practice labelling the sides of the right-angled triangle with o, a and h, as shown below,
given an angle,  ;
Angle, θ … the opposite
length is opposite the given
angle and the adjacent length
is next to the given angle
Hypotenuse
h
Opposite o
Adjacent a
And the trigonometric functions should be learned;
Sine θ =
o
h
Tan θ =
o
a
Cos θ =
a
h
“ Soh Toa Cah” or “Soh Cah Toa” may assist students in recalling the trig functions.
Shape 10: Pythagoras and Trigonometry
Understanding Trigonometry
An effective way of introducing trigonometry is by creating a large unit circle outdoors, and using measuring
tape to record the changing value of x and y as the angle of turn is increased in 5 or 10 degree increments.
While this will not generate exact values, the relationship between the x coordinate and the y coordinate of
each point of the circumference can be observed.
The trigonometric functions are derived from the unit circle (circle of radius; 1 unit, centre; O) and the ratio
of the opposite side compared to the adjacent side. Here is an example of the unit circle;
2 y-axis (x=0)
1
x-axis (y=0)
-2
-1
1
2
-1
-2
This is the unit circle drawn on a coordinate grid.
The radius of the circle is 1, hence the term, ‘unit circle’.
The centre of the circle is (0,0)
One point has been chosen on the circumference of the circle. The coordinate of the point happens to be;
(0.8, 0.6). A right-angled triangle is drawn to this vertex.
Remembering that, for a right-angled triangle, according to Pythagoras;
a2 +b2 = c2
and given that a refers to the distance in the x direction and b refers to the distance in the y direction and c is
equal to the radius of the circle; this gives us the equation for a circle; x2 + y2 = r2 [see Shape 9]
Now, the angle formed by the hypotenuse as it meets the origin is given as θ (or “theta”). This is the angle
measured at the centre of the circle in an anticlockwise direction where 0 degrees is equivalent to facing in
the positive x direction. The trigonometric functions; cosine and sine, describe the changing values of the x
and y coordinates respectively as the angle of turn, θ changes.
1
cosθ
Sin θ
θ
for the unit circle; the distance in the x direction is called, cos θ and
the distance in the y direction is called, sin θ and the hypotenuse will
be equal to the radius of 1.
Shape 10: Pythagoras and Trigonometry
Where θ is zero degrees (0o), the distance in the x direction to the circumference of the unit circle must be 1,
and the distance in the y direction must be 0. Similarly, where θ is 90o, the distance in the x direction to the
circumference of the unit circle is 0, whilst the distance in the y direction is 1. Where θ is 45o, the coordinate
at the circumference will be at a point where the distance in the x direction is equal to the distance in the y
direction (i.e. along the line y = x) approximately at (0.707 , 0.707), although the exact value is expressed as;
2
.
2
Furthermore, due to the reflective properties of a circle, and the lines of symmetry naturally formed by the x
and y axes, reflections of the right-angled triangle in the axes will have corresponding lengths. Therefore,
Sine (sin) 30, for example, will be equal to Sine 150 and -Sine 210 and -Sine 330, as these all form rightangled triangles with corresponding angles of 30 degrees. This, of course, is also true for the Cosine (cos) of
each angle such that Cos 30 = - cos 150 = - cos 210 = cos 330. This same symmetry occurs in the line, y = x.
Hence, the Sine of 30 will equal the Cosine of 60.
Similar circles, increasing the length of the hypotenuse, and the ratio of sides
Thus far, the relationship of the trigonometric functions, sine and cosine, have been discussed in relation to
each other and in relation to the amount of turn at the centre (θ) in the unit circle.
Remembering that the unit circle has a radius of 1, and this is the length of the hypotenuse for a right-angled
triangle formed in the unit circle; any scale factor enlargement of the circle will lead to a scale factor
enlargement for every length of the triangle. This means that the ratio of the opposite or adjacent sides in
comparison to the hypotenuse will remain the same.
Students can understand this principle using ratio sticks … if the hypotenuse ratio stick is called, “one”, then
the adjacent ratio stick is called, “one cos θ” and the opposite ratio stick is called, “one sin θ” … if the
hypotenuse ratio stick is called, “two”, then the adjacent ratio stick is called, “two cos θ” and the opposite
ratio stick is called, “two sin θ” … and so on!
So, if Cos θ is the distance in the x direction, and Sin θ is the distance in the y direction, then for a circle,
centre 0 and radius, h, the distance in the x direction becomes h Cos θ and the distance in the y direction
becomes h Sin θ.
As already explained, students must identify the opposite side, adjacent side and hypotenuse on the rightangled triangle based on given angle, θ;
So a = h Cos θ
and
o = h Sin θ
h
o
the distance in the y direction
θ
a
the distance in the x direction
Which leads to reasoning that;
a
 Cos
h
and
o
 Sin
h
Tan θ
The tangent (tan) function is used to directly compare the distance in the y direction to the distance in the x
direction. This function, therefore, describes the gradient of the hypotenuse and its changing value as θ, the
angle of turn at centre O, changes.
o
sin 
o
 h
So, tan  
cos a
a
h
Shape 10: Pythagoras and Trigonometry
Pythagoras’ Theorem and Trigonometry: Help Sheet
Pythagoras’ theorem applies to right-angled triangles;
a
c
Length c (directly opposite the right angle)
is called the ‘hypotenuse’.
b
Pythagoras’ theorem states that; a2 + b2 = c2 for any right angle.
By using this, and rearranging where necessary, it is possible to find the length of any side if you are
given the lengths of the other two sides. Look at the following examples;
x
4
Pythagoras’ theorem;
42 + 32 = x2
16 + 9 = x2
25 = x2
5 = x (√ both sides)
Pythagoras’ theorem;
7 + b = 8.5
49 + b2 = 72.25
b2 = 23.25
b =√23.25 = 4.82 (to 2d.p.s)
3
7
2
2
2
8.5
b
Trigonometry
The trigonometric functions; sine, cosine and tangent apply to right-angled triangles.
Whereas Pythagoras’ theorem allows you to find a missing length if you are given the other two lengths
(it is all about lengths!), trigonometry allows you to find an angle, or use an angle to find a missing
length if you are given just one other length.
Angle, θ … the opposite
length is opposite the given
Hypotenuse
angle and the adjacent length
Opposite
h
is next to the given angle
o
Adjacent a
Sine θ =
o
h
Tan θ =
o
a
Cos θ =
a
h
Soh Toa Cah
or
Soh Cah Toa
The trigonometric functions come from the unit circle (circle of radius; 1 unit, centre; O) and the ratio of the
opposite side compared to the adjacent side.
θ
cosθ
Sin θ
1
for the unit circle;
the distance in the x direction is called, cos θ
the distance in the y direction is called, sin θ
the hypotenuse will be equal to the radius of 1.
For similar circles, where the hypotenuse is multiplied by a value, h, the
length of the adjacent side will be h cos θ and the opposite side will be h sin θ
a
Sine rule: for any triangle
a
b
c


SinA SinB SinC
Shape 10: Pythagoras and Trigonometry
B
c
Cosine rule: for any triangle
A
C
b
a 2  b 2  c 2  2bcCosA