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 f(x) = sin (–x) f(x) = cos (–x)

Post-Lecture Exercise f(x) = 3sin (2x)f(x) = 2cos ( x / 2 ) f(x) = 2 + sin( x / 3 )

Post-Lecture Exercise 3.T(t) = sin(πt/8) a)38.6 is the normal temperature b) sin(πt/8) = 40 3sin(πt/8) = 1.4 sin(πt/8) = 1.4/3 = πt/8 = sin -1 (0.467) = t = 0.486*8/π = after about 1 and a quarter days. 4.Maximum is where sine is minimum i.e. when D = = 10metres

Lecture 4/4  Administration  Last Lecture  Distributive Functions  Compound Angle Formulae  Double Angle Formulae  Sum and Product Formulae  Summary

The Distributive Law 2(a + b) = 2a + 2b (a + b) 2 ≠ a 2 + b 2 = a 2 + 2ab + b 2 (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 a sin a b sin b sin (a + b) < sin a + sin b

A Graphical Explanation ab(a+b) sin a sin b sin (a+b)

Lecture 4/4  Administration  Last Lecture  Distributive Functions  Compound Angle Formulae  Double Angle Formulae  Sum & Product Formulae  Summary

The Formula for 0 ≤ ø ≤ π / 2 a sin a b 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 7 µ 4 √65 4 cosø + 7sinø

Shelter from the Storm 7m 4m ø 4 cosø + 7sinø 7 µ 4 √65 sinµ = 4/√65 cosµ = 7/√65 4 = √65 sinµ 7 = √65 cosµ

Shelter from the Storm 7m 4m ø √65sinµ cosø + √65cosµsinø 7 µ 4 √65 sinµ = 4/√65 cosµ = 7/√65 4 = √65 sinµ 7 = √65 cosµ

Lecture 4/4  Administration  Last Lecture  Distributive Functions  Compound Angle Formulae  Double 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 = cos 2 A – sin 2 A

Double Angle Formulae tan (A + B) = (tanA + tanB) 1 – tanA.tanB tan 2A = (tanA + tanA) 1 – tanA.tanA tan 2A = 2tanA 1 – tan 2 A

Lecture 4/4  Administration  Last Lecture  Distributive Functions  Compound Angle Formulae  Double Angle Formula  Sum & Product Formulae  Summary

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

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

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

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

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

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

Lecture 4/4  Administration  Last Lecture  Distributive Functions  Explanations of sin(A + B)  Developing a Formula  Further Formulae  Summary

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