Download Shape Space and Measure

Shape Space and Measure
Level 8
Trigonometry
Example
Naming Sides of a Triangle
Find the length of the
side marked as t.
It is convenient to give names to the sides of a right angled triangle
Answer
The side opposite the right angle is called the hypotenuse this is also the longest side.
The names of the two sides are opposite side and adjacent sides these two names depend on the
angle we consider.
sin 35° =
hypotenuse
adjacent
opposite
t
6.8
6.8cm
t
t sin 35° = 6.8 (multiplying both sides by t)
t=
hypotenuse
35°
6.8
(dividing both sides by sin 35°)
sin35°
t = 11.9 (1 d.p.)
keying 6.8 ÷ 35 sin =
opposite
adjacent
Greek letters are often used to name angles.
α (alpha)
β (beta)
θ (theta)
An unknown angle can be found if two of the sides are given. The triangle has to be a right angled
triangle and we can then use the trig ratios sine, cosine or tangent.
The Ratios Sine, Cosine, Tangent
The ratio
opposite
hypotenuse
The ratio
The ratio
adjacent
opposite
adjacent
We use tan if we are given the lengths of the opposite and adjacent sides
is called the sine of 40° or sine 40°.
hypotenuse
is called the cosine 40°
Finding the size of an Angle
We use cos if we are given the lengths of the adjacent side & the hypotenuse
hypotenuse
opposite
We use sin if we are given the lengths of the opposite side & and hypotenuse
Example
40°
adjacent
is called tangent 40°
Find the angle θ
12cm
Answer
The abbreviations sin, cos, tan are used for sine, cosine, tangent
Looking at the diagram, use cosine.
The ratios sin A, cos A, tan A are called trigonometrical ratios or trig ratios
cos θ =
Finding the length of a Side
To find the angle use the inverse cosine function on your calculator
In a right angled triangle if we are given the length of one side and the size of an angle (other than
the right angle) we can use one of the trig ratios to find the length of one of the other sides.
Cos-1 = 65.4o (1 d.p.)
Example
Navigation problems involve bearings and in surveying distances which are difficult to measure are
calculated by measuring angles then using trigonometry
Find the length of the
side marked as b.
Answer
cos 69° = b or b = cos 69°
7
7
b = 7 x cos 69° (multiplying both sides by 7)
b = 2.5 cm (1 d.p.)
keying 7 x 69 cos =
69°
7cm
b
A
H
=
5
12
θ
5cm
When we look up at something the angle between the horizontal and the direction in which we are
looking is called the angle of elevation.
In this diagram θ is the angle of elevation.
When we look down at something the angle between the
horizontal and the direction in which we are looking
is called the angle of depression.
In this diagram α is the angle of depression.
θ
α
Similar Shapes
Shapes that are identical in
every way are called congruent shapes.
These shapes are congruent.
Shapes that are the same shape but
different sizes are called similar shapes.
These shapes are similar.
Dimensions
In similar shapes the angles that are equal are called corresponding angles.
In similar shapes the sides that are in corresponding position are called corresponding sides. In similar
triangles the corresponding sides are opposite the corresponding angles.
Finding Unknown Lengths
Ian cannot remember if the formula for the area of a circle is A = 2πr, A = πd, A= π2r or A = πr2.
1.
corresponding angles are equal
2.
corresponding sides are in the same ratio
He checks the dimensions of these formulae as follows
A
Example
The triangles ABE and CBD are similar
Find the length of BD and AB
3
D
B
0.9
3.6
E
C
Corresponding angles are A and C, E and D, ∠ABE and ∠CBD
Corresponding sides are BE and BD, AB and CB, AE and CD
BD
CD
AB
AE
=
=
3.6
=
AE
0.8
3
BD = 3.6 ×
= 0.96
0.8
3
2πr
number x number x length
Dimension is L
πd
number x length
Dimension is L
πr2
number x length x length
Dimension is L2
π2r
number x number x length
Dimension is L
Ian then decides that the area formula must be A = πr2 since the dimension of area is L2.
0.8
BE
BD
The dimension of perimeter is length (L).
The dimension for area is length x length (L2).
The dimension for volume is length x length x length (L3).
Example
If two shapes are similar then
Answer
Dimensions for Length, Area Volume
CB
AB
0.9
=
CD
3
0.8
3
AB = 0.9 ×
0.8
= 3.375