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445.102 Mathematics 2

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.

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445.102 Mathematics 2

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  1. 445.102 Mathematics 2 Module 4 Cyclic Functions Lecture 4 Compounding the Problem

  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.

  3. 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)

  4. Post-Lecture Exercise f(x) = sin (–x) f(x) = cos (–x)

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

  6. Post-Lecture Exercise 3. T(t) = 38.6 + 3sin(πt/8) a) 38.6 is the normal temperature b) 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 = 1.236 after about 1 and a quarter days. 4. Maximum is where sine is minimum i.e. when D = 8 + 2 = 10metres

  7. 445.102 Lecture 4/4 • Administration • Last Lecture • Distributive Functions • Compound Angle Formulae • Double Angle Formulae • Sum and Product Formulae • Summary

  8. 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 = ????????????

  9. The Unit Circle Again sin b sin a b a sin (a + b) < sin a + sin b

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

  11. 445.102 Lecture 4/4 • Administration • Last Lecture • Distributive Functions • Compound Angle Formulae • Double Angle Formulae • Sum & Product Formulae • Summary

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

  13. 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

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

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

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

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

  18. 445.102 Lecture 4/4 • Administration • Last Lecture • Distributive Functions • Compound Angle Formulae • Double Angle Formula • Sum & Product Formulae • Summary

  19. 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

  20. Double Angle Formulae • tan (A + B) = (tanA + tanB) 1 – tanA.tanB • tan 2A = (tanA + tanA) 1 – tanA.tanA • tan 2A = 2tanA 1 – tan2A

  21. 445.102 Lecture 4/4 • Administration • Last Lecture • Distributive Functions • Compound Angle Formulae • Double Angle Formula • Sum & Product Formulae • Summary

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

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

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

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

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

  27. 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)]

  28. 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

  29. 445.102 Lecture 4/4 • Administration • Last Lecture • Distributive Functions • Explanations of sin(A + B) • Developing a Formula • Further Formulae • Summary

  30. 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

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