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Steven Archer
Trig
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
Introduction
Trigonometry is the study of triangles.
The sum of the angles in a triangle is 180◦ .
The sum of any two lengths must be more than the third, we’ll
come back to this.
Right-angled triangles
A triangle is right-angled if one of the angles is 90◦ .
Right-angled triangles relate angles (or bearings) to distances.
Steven Archer
Trig - Pythagoras
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
Theorem of Pythagoras
If a triangle is right-angled, then the sum of the squares of the
shorter sides is equal to the square of the longest side.
If the sum of the squares of two sides equals the square of the
third in a triangle, then that triangle is right-angled.
Examples include the (3, 4, 5) and (5, 12, 13) triangles.
Steven Archer
Trig - sine, cosine and tangent
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
Trigonometric functions
By similarity, in a right-angled triangle if an angle is fixed then
the ratio of the side lengths is also fixed.
h
G
o
a
sin(G ) =
o
h
cos(G ) =
tan(G ) =
o
a
a
h
Steven Archer
Trig - some triangles
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
Standard triangles
Two triangles come from simple shapes, which allow us to get
reasonable√trigometric ratios.
The (1, 1,
√ 2) is half a unit square.
The (1, 3, 2) is half an equilateral triangle.
Isoceles and Equilateral
A triangle is isoceles if either two sides are of equal length, or
two angles are equal.
One condition implies the other.
A triangle is equilateral if either all sides are of equal length, or
all angles are equal.
Steven Archer
Trig - circle diagram
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
Angles of 90◦ and beyond
Using a circle diagram, we can calculate trigonometric ratio’s
for angles greater than 90◦ .
The horizontal distance (x-coordinate) is the cosine of the
angle.
The vertical distance (y -coordinate) is the sine of the angle.
The ratio of vertical to horizontal (y over x) is the tangent of
the angle.
Steven Archer
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
Trig - circle diagram
Steven Archer
Trig - circle diagram
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
( 21 ,
1
60◦
√
3
2 )
Steven Archer
Trig - circle diagram
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
( 21 ,
√
3
2 )
1
sin(60◦ )
60◦
Steven Archer
Trig - circle diagram
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
( 21 ,
√
3
2 )
1
sin(60◦ )
60◦
cos(60◦ )
Steven Archer
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
Trig - circle diagram
Steven Archer
Trig - circle diagram
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
(− √12 , √12 )
1
135◦
Steven Archer
Trig - circle diagram
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
(− √12 , √12 )
1
sin(135◦ )
135◦
Steven Archer
Trig - circle diagram
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
(− √12 , √12 )
1
sin(135◦ )
135◦
cos(135◦ )
Steven Archer
Complex numbers
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
Complex numbers as points
Now we have the machinery to look at complex number a
different way.
As we want to apply arithmetic to complex numbers, we should
allow all things of the form a + bi, where a and b are real
numbers.
This looks like a point, with coordinates (a, b).
In the rush to introduce material, here is an idea whose time
will come. The polar form of a complex number a + bi is an
alternative description of how to find the point on a graph.
We still need two pieces of information to determine where the
point is, but instead of horizontal and vertical movement, we
use distance from the origin and angle made with the positive
x-axis.
Steven Archer
Trig - complex numbers
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
a + bi
Remember from earlier in the course, from the quadratic
formula we get numbers of the form a + bi e.g solving
x 2 − 2x + 2 = 0 we find (x − 1)2 = − 1 and so x = 1 + i or
x = 1 − i.
We can visualise complex numbers on the Argand plane, where
the real values are on the x-axis and the imaginary numbers on
the y -axis.
Seen this way, complex numbers match fit in with our circle
diagram for trigonometry, except that the radius can vary.
We’ll talk about important connections in class.
Steven Archer
Trig
Trigonometry
Introduction
Pythagoras
The ratio’s
Special triangles
Circle diagram
Complex
numbers again
Without a right
angle
Other triangles
There are a number of formulae for non right-angled triangles,
but we’ll focus on two.
The area of a triangle, with side lengths a, b and angle C
between the sides, is
1
ab sin(C )
2
Assume a is the longest side, and C is less than 90◦ .
We’ll fill in the diagram below to convince ourselves that the
formula works.
c
b
a