445.102 Mathematics 2

Module 4

Cyclic Functions

Lecture 3

Making Waves

Transforming Trigonometric Graphs

Graphs of cyclic functions can be transformed the same way as polynomial, exponential and rational functions. We will present this as modelling the motion of waves under different conditions, but it is also helpful when identifying solutions of trigonometric equations.

Post Lecture Exercises1.

2. sin x and tan x are ODD, cos x is EVEN3. cos π/4 = 1/√2 cosec π/3 = 2/√34. sin2x + cos2x = 1

Divide by sin2x: 1 + cos2x/sin2x = 1/sin2x=> 1 + cot2x = cosec2x

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π 2π-π-2π 3π 4π

f(ø) = tan ø

Post Lecture Exercises continued…a)cos x = -0.3 => x = 1.875 ± 2πn

and x = (2π - 1.875) ± 2πn = 4.408 ± 2πn

b)tan x = 5 => x = 1.373 ± 2πnand x = (2π + 1.373) ± 2πn = 7.657 ± 2πn

c)sec x = 3 => cos x = 1/3 x = 1.231 ± 2πn

and x= (2π - 1.231) ± 2πn = 5.052 ± 2πn

d)csc x = -2 => sin x = -12 x = -0.524 ± 2πn

and x= (2π + 0.524) ± 2πn = 6.807 ± 2πn

445.102 Lecture 4/4

Administration Last LectureUp & Down Left & Right Squishing & Stretching Changing the Outline Summary

Preliminary Exercise

y = x^2 y = x^2 – 5 y = (x – 4)^2 y = (x – 4)^2 +3

Transforming Vertically

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445.102 Lecture 4/4

Administration Last Lecture Up & DownLeft & Right Squishing & Stretching Changing the Outline Summary

Transforming Horizontally

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445.102 Lecture 4/4

Administration Last Lecture Up & Down Left & RightSquishing & Stretching Changing the Outline Summary

Horizontal Scaling

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2ππ

445.102 Lecture 4/4

Administration Last Lecture Up & Down Left & Right Squishing & StretchingChanging the Outline Summary

Transforming Amplitude

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2π 3ππ-π-2π

Transforming Outlines

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445.102 Lecture 4/4

Administration Last Lecture Up & Down Left & Right Squishing & Stretching Changing the OutlineSummary

An Example ....The height of the tide below a wharf is given by the function:

H(t) = -3 + 2sin 0.56twhere t is the time after midnight in hoursand H is the distance in metresWhen will the waterlevel be exactly 5 metres below the wharf?

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Solving Trigonometric Equations

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H(x) = 1 + 3sin(x/2) = 2.5

Solving Trigonometric Equations

sin(x/2) = (2.5 – 1)/3 = 0.5

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2π

Lecture 4/3 – Summary

The graphs of cyclic functions are transformed in the same manner as graphs of other functions.

Such transformations can be seen as ways of modelling waves or cyclic phenomena which occur in our world.

They can also be used to “see” the solutions of trigonometric equations.