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 right-angled 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 4/1Going 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 4/1Going Round Again

AdministrationAngles 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)

ø

Ch(ø)

1

1

445.102 Lecture 4/1Going Round Again

Administration Angles as VariablesMeasures of Angle The 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?

RadiansA mathematical measure of angle is defined using the radius of a circle.

RadiansA mathematical measure of angle is defined using the radius of a circle.

1 radian

RadiansCircumference = πd = 2πrHalf circumference = πr

π

2π

RadiansCircumference = πd = 2πrHalf circumference = πr

π

2ππ 2π

445.102 Lecture 4/1

Administration Angles as Variables Measures of AngleThe Unit Circle Sine as a Function Summary

A Special Circle

A Special Circle

1 unit

Ch(ø)

øCh(ø)

1

1

Si(ø)

øSi(ø)

1

445.102 Lecture 4/1Going Round Again

Administration Angles as Variables Measures of Angle The Unit CircleSine as a Function Summary

Si(ø)

øSi(ø)

1

sin(ø) = Si(ø)/1 = Si(ø)

sin(ø)

øsin(ø)

1

The Sine Function

-0.50

-1.00

0.50

1.00 f(ø) = sin ø

π 2πø

ø

The Sine Function

øø

-0.50

-1.00

0.50

1.00 f(ø) = sin ø

π-π

The Sine Function(Many Rotations)

-0.50

-1.00

0.50

1.00

3π 4π

f(ø) = sin ø

π 2π-π-2π

445.102 Lecture 4/1Going Round Again

Administration Angles as Variables Measures of Angle The Unit Circle Sine as a FunctionSummary

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 4/1Going 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 ........