445.102 Mathematics 2

Module 4

Cyclic Functions

Lecture 4

Compounding the Problem

Angle Formulae

In this lecture we treat sine, cosine & tangent as mathematical functions which have relationships with each other. These are expressed as various formulae.

It is important that you UNDERSTAND this work, but not that you can reproduce it. We would like you to be able to USE the formulae when needed. We want you to become familiar with using cyclic functions in algebraic expressions.

f(x) = sin x g(x) = A + sin x Vertical shift of A h(x) = sin(x + A) Horizontal shift of –A j(x) = sin (Ax) Horizontal squish A times k(x) = Asin x Vertical stretch A times m(x) = n(x) sin x Outline shape n(x)

Post-Lecture Exercise

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-1.00

1.00f(x) = sin (–x)

f(x) = cos (–x)

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1.00

Post-Lecture Exercise

f(x) = 3sin (2x)

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5.0

2ππ

f(x) = 2cos (x/2)

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5.0

2ππ

f(x) = 2 + sin(x/3)

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5.0

2ππ

Post-Lecture Exercise3. T(t) = 38.6 + 3sin(πt/8)

a) 38.6 is the normal temperatureb) 38.6 + 3sin(πt/8) = 40<=> 3sin(πt/8) = 1.4<=> sin(πt/8) = 1.4/3 = 0.467<=> πt/8 = sin-1(0.467) = 0.486<=> t = 0.486*8/π = 1.236after about 1 and a quarter days.

4. Maximum is where sine is minimumi.e. when D = 8 + 2 = 10metres

445.102 Lecture 4/4

Administration Last LectureDistributive Functions Compound Angle Formulae Double Angle Formulae Sum and Product Formulae Summary

The Distributive Law

2(a + b) = 2a + 2b (a + b)2 ≠ a2 + b2

= a2 + 2ab + b2

(a + b)/2 = a/2 + b/2 log(a + b) ≠ log a + log b

= log a . log b

sin (a + b) ≠ sin a + sin b

= ????????????

The Unit Circle Again

asin ab

sin b

sin (a + b) < sin a + sin b

A Graphical Explanation

-0.50

-1.00

0.50

1.00

π

a b (a+b)sin a

sin bsin (a+b)

445.102 Lecture 4/4

Administration Last Lecture Distributive FunctionsCompound Angle Formulae Double Angle Formulae Sum & Product Formulae Summary

The Formula for 0 ≤ ø ≤ π/2

asin ab

sin b

y

x

z

Lecture 4/5 – Summary Compound Angle Formulae

sin (A + B) = sinA.cosB + cosA.sinB sin (A – B) = sinA.cosB – cosA.sinB cos (A + B) = cosA.cosB – sinA.sinB cos (A – B) = cosA.cosB + sinA.sinB tan (A + B) = (tanA + tanB)

1 – tanA.tanB tan (A – B) = (tanA – tanB)

1 + tanA.tanB

Shelter from the Storm

7m

4m

ø

4 cosø + 7sinø

Shelter from the Storm

7m

4m

ø

7µ

4√65

4 cosø + 7sinø

Shelter from the Storm

7m

4m

ø

4 cosø + 7sinø

7µ

4√65 sinµ = 4/√65 cosµ = 7/√65

4 = √65 sinµ 7 = √65 cosµ

Shelter from the Storm

7m

4m

ø

√65sinµ cosø + √65cosµsinø

7µ

4√65 sinµ = 4/√65 cosµ = 7/√65

4 = √65 sinµ 7 = √65 cosµ

445.102 Lecture 4/4

Administration Last Lecture Distributive Functions Compound Angle FormulaeDouble Angle Formula Sum & Product Formulae Summary

Double Angle Formulae

sin (A + B) = sinA.cosB + cosA.sinB sin 2A = sinA.cosA + cosA.sinA = 2sinA cosA cos (A + B) = cosA.cosB – sinA.sinB cos 2A = cosA.cosA – sinA.sinA = cos2A – sin2A

Double Angle Formulae

tan (A + B) = (tanA + tanB)

1 – tanA.tanB

tan 2A = (tanA + tanA)

1 – tanA.tanA

tan 2A = 2tanA

1 – tan2A

445.102 Lecture 4/4

Administration Last Lecture Distributive Functions Compound Angle Formulae Double Angle FormulaSum & Product Formulae Summary

The Octopus

Large wheel, radius 6m, 8 second period.A = 6sin(2πx/8)

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10.0

The Octopus

Add a small wheel, radius 1.5m, 2s period.B = 1.5sin(2πx/2)

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The Octopus

Combine the two......A + B = 6sin(2πx/8) + 1.5sin(2πx/2)

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The Surf

Decent surf has a height of 1.5m, 15s period.A = 1.5sin(2πx/15)

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1.00

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The Surf

Add similar wave, say: 1m, 13s period.A + B = 1.5sin(2πx/15) + 1sin(2πx/13)

50 100 150 200 250-50-100-150-200

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Adding Sine Functions

sin(A+B) = sinAcosB + sinBcosA sin(A–B) = sinAcosB – sinBcosA Adding......... sin(A+B) + sin(A–B) = 2sinAcosB Rearranging......... sinAcosB = 1/2[sin(A+B) + sin(A–B)]

Adding Sine Functions

sinAcosB = 1/2[sin(A+B) + sin(A–B)]

Or, making A = (P+Q)/2 and B = (P–Q)/2

That is: A+B = 2P/2 and A–B = 2Q/2

1/2[sin P + sin Q] = sin (P+Q)/2 cos (P–Q)/2

sin P + sin Q = 2 sin (P+Q)/2 cos (P–Q)/2

445.102 Lecture 4/4

Administration Last Lecture Distributive Functions Explanations of sin(A + B) Developing a Formula Further FormulaeSummary

Lecture 4/4 – Summary Compounding the Problem

Please KNOW THAT these formulae exist

Please BE ABLE to follow the logic of their derivation and use

Please PRACTISE the simple applications of the formulae as in the Post-Lecture exercises