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445.102 Mathematics 2 Module 4 Cyclic Functions Lecture 4 Compounding the Problem

445.102 Mathematics 2

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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. - PowerPoint PPT Presentation

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Page 1: 445.102 Mathematics 2

445.102 Mathematics 2

Module 4

Cyclic Functions

Lecture 4

Compounding the Problem

Page 2: 445.102 Mathematics 2

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.

Page 3: 445.102 Mathematics 2

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)

Page 4: 445.102 Mathematics 2

Post-Lecture Exercise

1.00 2.00 3.00 4.00 5.00 6.00 7.00

-1.00

1.00f(x) = sin (–x)

f(x) = cos (–x)

1.00 2.00 3.00 4.00 5.00 6.00 7.00

-1.00

1.00

Page 5: 445.102 Mathematics 2

Post-Lecture Exercise

f(x) = 3sin (2x)

-5.0

5.0

2ππ

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

-5.0

5.0

2ππ

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

-5.0

5.0

2ππ

Page 6: 445.102 Mathematics 2

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

Page 7: 445.102 Mathematics 2

445.102 Lecture 4/4

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

Page 8: 445.102 Mathematics 2

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

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

Page 9: 445.102 Mathematics 2

The Unit Circle Again

asin ab

sin b

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

Page 10: 445.102 Mathematics 2

A Graphical Explanation

-0.50

-1.00

0.50

1.00

π

a b (a+b)sin a

sin bsin (a+b)

Page 11: 445.102 Mathematics 2

445.102 Lecture 4/4

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

Page 12: 445.102 Mathematics 2

The Formula for 0 ≤ ø ≤ π/2

asin ab

sin b

y

x

z

Page 13: 445.102 Mathematics 2

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

Page 14: 445.102 Mathematics 2

Shelter from the Storm

7m

4m

ø

4 cosø + 7sinø

Page 15: 445.102 Mathematics 2

Shelter from the Storm

7m

4m

ø

4√65

4 cosø + 7sinø

Page 16: 445.102 Mathematics 2

Shelter from the Storm

7m

4m

ø

4 cosø + 7sinø

4√65 sinµ = 4/√65 cosµ = 7/√65

4 = √65 sinµ 7 = √65 cosµ

Page 17: 445.102 Mathematics 2

Shelter from the Storm

7m

4m

ø

√65sinµ cosø + √65cosµsinø

4√65 sinµ = 4/√65 cosµ = 7/√65

4 = √65 sinµ 7 = √65 cosµ

Page 18: 445.102 Mathematics 2

445.102 Lecture 4/4

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

Page 19: 445.102 Mathematics 2

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

Page 20: 445.102 Mathematics 2

Double Angle Formulae

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

1 – tanA.tanB

tan 2A = (tanA + tanA)

1 – tanA.tanA

tan 2A = 2tanA

1 – tan2A

Page 21: 445.102 Mathematics 2

445.102 Lecture 4/4

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

Page 22: 445.102 Mathematics 2

The Octopus

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

2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00

-5.0

-10.0

5.0

10.0

Page 23: 445.102 Mathematics 2

The Octopus

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

2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00

-5.0

-10.0

5.0

10.0

Page 24: 445.102 Mathematics 2

The Octopus

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

2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00

-5.0

-10.0

5.0

10.0

Page 25: 445.102 Mathematics 2

The Surf

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

5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0

-1.00

-2.00

1.00

2.00

Page 26: 445.102 Mathematics 2

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

-2.00

-4.00

2.00

4.00

Page 27: 445.102 Mathematics 2

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)]

Page 28: 445.102 Mathematics 2

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

Page 29: 445.102 Mathematics 2

445.102 Lecture 4/4

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

Page 30: 445.102 Mathematics 2

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