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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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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
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
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
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)
2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00
-5.0
-10.0
5.0
10.0
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
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
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
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
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