445.102 Mathematics 2
Module 4
Cyclic Functions
Lecture 1
Going Round Again
This Module concerns the CIRCULAR
FUNCTIONS, so named because they were
originally derived from the circle.
One way to think of these functions is of
functions where the variable is an ANGLE. It
turns out that these functions are extremely
common - and are good approximations for
many phenomena: for example waves, orbits,
swings, pendulums and springs.
Their study is rewarding, and provides many
more applications than the familiar rightangled triangle problems of school days.
Oh yes, .....
this is what you might have called
TRIGONOMETRY in the past.
The Module finishes with a few lectures
introducing you to two fascinating and very
important types of mathematical objects.
The first are MATRICES. These are simply
rows and columns of numbers, but they can be
used to describe whole sets of equations, or
geometrical transformations like reflections.
The second type of object are COMPLEX
NUMBERS. These are the numbers you get if
you pretend that it is possible to take the
square root of a negative number. They have
many, many important uses in mathematics
and its applications.
445.102
Lecture 4/1
Going Round Again
Administration
Angles
as Variables
Measures of Angle
The Unit Circle
Sine as a Function
Summary
Administration
Terms
Tests
Assignment 4
Due on Monday NEXT week
This Week’s Tutorial
Assignment 4 & Working Together
Cecil
Assignments/Lectures/Answers/Marks
Post-Lecture Exercise
a)
f(x) = (3x2 - 4)1/2
f '(x) = 3x(3x2 - 4)-1/2
b)
f(x) = 3 (ln x)2
f '(x) = 6 ln x/x
c)
f(x) = 4e3x^2
f '(x) = 24xe3x^2
d)
f(x) = (x2 + 4)/ln 4x
f '(x) = (2x ln 4x – (x2 + 4)1/x)/(ln 4x)2
e)
f(x) = 3(x2 - 2x)2
f '(x) = 6(2x - 2)(x2 - 2x) = 12x(x – 1)(x – 2)
445.102
Lecture 4/1
Going Round Again
Administration
Angles
as Variables
Measures
of Angle
The Unit Circle
Sine as a Function
Summary
Angles as Variables
All kinds of “objects” can be variables.
Usually we think of variables as
numbers:
f(x) = 3x2 – 2x + 1
f(-2) = 12 + 4 + 1 = 17
But last lecture, for example, we had
another function as a variable:
f(x) = g(h(x)),
e.g. f(x) = 3e2t^3
Angles as Variables
We can make up other kinds of
functions:
E.g. a function which determines the
distance of a point from the origin:
D(3,4) = √(32 + 42)
So the variable is a point (3,4)
Angles as Variables
And we can make up functions where
the variable is an angle:
E.g. Full(ø) = the number of angle ø’s
which are needed to make a full turn.
E.g. Ch(ø) = the length of the chord of a
circle of radius 1, which is generated by
an angle ø at the centre.
Ch(ø)
1
Ch(ø)
ø
1
445.102
Lecture 4/1
Going Round Again
Administration
Angles
as Variables
Measures
The
of Angle
Unit Circle
Sine as a Function
Summary
Degrees, Mils, Radians
Degrees are a well-known unit of
angle. There are 90° in a quarter turn
Grads are a surveyors measure, based
on 100grads in a quarter turn.
Mils are an old military measure,
used for artillery calculations.
Other Measures
We can make up other angle
measures:
e.g. let us define a “hand” as the angle
subtended by the width of our hand at
arm’s length.
How many degrees in a hand?
Radians
A mathematical measure of angle is
defined using the radius of a circle.
Radians
A mathematical measure of angle is
defined using the radius of a circle.
1 radian
Radians
Circumference = πd = 2πr
Half circumference = πr
π
2π
Radians
Circumference = πd = 2πr
Half circumference = πr
π
π
2π
2π
445.102
Lecture 4/1
Administration
Angles
as Variables
Measures of Angle
The
Sine
Unit Circle
as a Function
Summary
A Special Circle
A Special Circle
1 unit
Ch(ø)
1
ø
Ch(ø)
1
Si(ø)
1
ø
Si(ø)
445.102
Lecture 4/1
Going Round Again
Administration
Angles
as Variables
Measures of Angle
The Unit Circle
Sine
as a Function
Summary
Si(ø)
Si(ø)
sin(ø) =
/1 = Si(ø)
1
ø
Si(ø)
sin(ø)
1
ø
sin(ø)
The Sine Function
f(ø) = sin ø
1.00
0.50
š
ø
ø
-0.50
-1.00
2š
The Sine Function
1.00
f(ø) = sin ø
0.50
-š
ø
ø
-0.50
-1.00
š
The Sine Function
(Many Rotations)
1.00
f(ø) = sin ø
0.50
-2š
-š
š
-0.50
-1.00
2š
3š
4š
445.102
Lecture 4/1
Going Round Again
Administration
Angles
as Variables
Measures of Angle
The Unit Circle
Sine as a Function
Summary
Lecture 4/1 – Summary
There
are many functions where the
variable can be regarded as an ANGLE.
 One way of measuring an angle is that
derived from the radius of the circle. This is
called RADIAN measure.
From the UNIT CIRCLE, we can see that
the SINE of an angle is the height of a
triangle drawn inside the circle. Sine(ø)
then becomes a function depending on the
size of the angle ø.
445.102
Lecture 4/1
Going Round Again
Before
the next lecture........
Go over Lecture 4/1 in your notes
Do the Post-Lecture exercise
Do the Preliminary Exercise
See you tomorrow ........