445 102 Mathematics 2 Module 4 Cyclic Functions

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 f(x) = sin (–x) f(x) = cos (–x)

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

Post-Lecture Exercise 3. T(t) = 38. 6 + 3 sin(πt/8) a) 38. 6 is the normal temperature b) 38. 6 + 3 sin(πt/8) = 40 <=> 3 sin(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. 236 after about 1 and a quarter days. 4. Maximum is where sine is minimum i. e. when D = 8 + 2 = 10 metres

445. 102 Lecture 4/4 Administration Last Lecture Distributive Compound Functions Angle Formulae Double Angle Formulae Sum and Product Formulae Summary

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

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

445. 102 Lecture 4/4 Administration Last Lecture Distributive Functions Compound Double Angle Formulae Sum & Product Formulae Summary

The Formula for 0 ≤ ø ≤ π/2 x b a sin b y z sin a

Lecture 4/5 – Summary Compound Angle Formulae sin (A + B) = sin. A. cos. B + cos. A. sin. B sin (A – B) = sin. A. cos. B – cos. A. sin. B cos (A + B) = cos. A. cos. B – sin. A. sin. B cos (A – B) = cos. A. cos. B + sin. A. sin. B tan (A + B) = (tan. A + tan. B) 1 – tan. A. tan. B tan (A – B) = (tan. A – tan. B) 1 + tan. A. tan. B

Shelter from the Storm 7 m 4 m ø 4 cosø + 7 sinø

Shelter from the Storm 7 m 4 m ø 4 cosø + 7 sinø √ 65 µ 7 4

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

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

445. 102 Lecture 4/4 Administration Last Lecture Distributive Functions Compound Angle Formulae Double Sum Angle Formula & Product Formulae Summary

Double Angle Formulae sin (A + B) = sin. A. cos. B + cos. A. sin. B sin 2 A = sin. A. cos. A + cos. A. sin. A = 2 sin. A cos (A + B) = cos. A. cos. B – sin. A. sin. B cos 2 A = cos. A – sin. A = cos 2 A – sin 2 A

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

445. 102 Lecture 4/4 Administration Last Lecture Distributive Functions Compound Angle Formulae Double Angle Formula Sum & Product Formulae Summary

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

The Octopus Add a small wheel, radius 1. 5 m, 2 s period. B = 1. 5 sin(2πx/2)

The Octopus Combine the two. . . A + B = 6 sin(2πx/8) + 1. 5 sin(2πx/2)

The Surf Decent surf has a height of 1. 5 m, 15 s period. A = 1. 5 sin(2πx/15)

The Surf Add similar wave, say: 1 m, 13 s period. A + B = 1. 5 sin(2πx/15) + 1 sin(2πx/13)

Adding Sine Functions sin(A+B) = sin. Acos. B + sin. Bcos. A sin(A–B) = sin. Acos. B – sin. Bcos. A Adding. . sin(A+B) + sin(A–B) = 2 sin. Acos. B Rearranging. . sin. Acos. B = 1/2[sin(A+B) + sin(A–B)]

Adding Sine Functions sin. Acos. B = 1/2[sin(A+B) + sin(A–B)] Or, making A = (P+Q)/2 and B = (P–Q)/2 That is: A+B = 2 P/2 and A–B = 2 Q/2 1/ [sin P + sin Q] = sin (P+Q)/ cos (P–Q)/ 2 2 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 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