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445.102 Mathematics 2 Module 4 Cyclic Functions Lecture 1 Going Round Again

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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.

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Oh yes,..... this is what you might have called TRIGONOMETRY in the past.

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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.

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445.102 Lecture 4/1 Going Round Again Administration Angles as Variables Measures of Angle The Unit Circle Sine as a Function Summary

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Administration Terms Tests Assignment 4 Due on Monday NEXT week This Week’s Tutorial Assignment 4 & Working Together Cecil Assignments/Lectures/Answers/Marks

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Post-Lecture Exercise a)f(x) = (3x 2 - 4) 1/2 f '(x) = 3x(3x 2 - 4) -1/2 b)f(x) = 3 (ln x) 2 f '(x) = 6 ln x / x c)f(x) = 4e 3x^2 f '(x) = 24xe 3x^2 d)f(x) = (x 2 + 4)/ln 4x f '(x) = (2x ln 4x – (x 2 + 4) 1 / x )/(ln 4x) 2 e)f(x) = 3(x 2 - 2x) 2 f '(x) = 6(2x - 2)(x 2 - 2x) = 12x(x – 1)(x – 2)

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445.102 Lecture 4/1 Going Round Again Administration Angles as Variables Measures of Angle The Unit Circle Sine as a Function Summary

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Angles as Variables All kinds of “objects” can be variables. Usually we think of variables as numbers: f(x) = 3x 2 – 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) = 3e 2t^3

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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) = √(3 2 + 4 2 ) So the variable is a point (3,4)

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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.

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Ch(ø) ø 1 1

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445.102 Lecture 4/1 Going Round Again Administration Angles as Variables Measures of Angle The Unit Circle Sine as a Function Summary

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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.

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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?

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

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1 radian

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Radians Circumference = πd = 2πr Half circumference = πr π 2π

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Radians Circumference = πd = 2πr Half circumference = πr π 2π π

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445.102 Lecture 4/1 Administration Angles as Variables Measures of Angle The Unit Circle Sine as a Function Summary

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A Special Circle

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1 unit

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Ch(ø) ø 1 1

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Si(ø) ø 1

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445.102 Lecture 4/1 Going Round Again Administration Angles as Variables Measures of Angle The Unit Circle Sine as a Function Summary

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Si(ø) ø 1 sin(ø) = Si(ø) / 1 = Si(ø)

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sin(ø) ø 1

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The Sine Function ø ø

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ø ø

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The Sine Function (Many Rotations)

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445.102 Lecture 4/1 Going Round Again Administration Angles as Variables Measures of Angle The Unit Circle Sine as a Function Summary

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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.

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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........

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