The Wave Function

1. 1 SLIDE The Wave Function Unit 3 – Outcome 4

2. Higher Maths 3 4 The Wave Function 2 OUTCOME SLIDE UNIT NOTE Wave Functions It is possible to model the behaviour of waves in real-life situations (e.g. the interaction of sound waves or the tides where two bodies of water meet) using trigonometry. We can add or subtract two trigonometric wave functions and represent them as a single trigonometric wave function.

3. Higher Maths 3 4 The Wave Function 3 OUTCOME SLIDE UNIT NOTE A step-by-step guide to Wave Functions Step 1 – Expand kcos(x – a) using the compound angle formula. Step 2 – Rearrange and compare the coefficients. Step 3 – Mark the quadrants on a CAST diagram. Step 4 – Find k and a using the formulae above (a lies in the quadrant marked twice in Step 3). Step 5 – Complete the question using k and a values.

4. Higher Maths 3 4 The Wave Function 4 OUTCOME SLIDE UNIT NOTE Example Write 5cosx° + 12sinx° in the form kcos(x° – a°) where 0 ≤ a ≤ 360

5. Higher Maths 3 4 The Wave Function 5 OUTCOME SLIDE UNIT NOTE Example Write 5cosx – 3sinx in the form kcos(x – a) where 0 ≤ a ≤ 2π

6. 6 SLIDE p304 Exercise 16C Q1 – 5 p304 Exercise 16D Q2

7. Higher Maths 3 4 The Wave Function 7 OUTCOME SLIDE UNIT NOTE Expressing pcosx + qsinx in other forms The expression pcosx + qsinx may also be written in the form kcos(x + a), ksin(x – a) or ksin(x + a). Example Write 4cosx° + 3sinx° in the form ksin(x° + a°) where 0 ≤ a ≤ 360

8. 8 SLIDE p305 Exercise 16E Q1e, 2e, 3e, 4e, 5e

9. Higher Maths 3 4 The Wave Function 9 OUTCOME SLIDE UNIT NOTE Multiple angles We can also apply the wave function to multiple angles. Example Write 5cos2x° + 12sin2x° in the form ksin(2x° + a°) where 0 ≤ a ≤ 360

10. 10 SLIDE p306 Exercise 16F Q1b, 2b, 3b

11. Higher Maths 3 4 The Wave Function 11 OUTCOME SLIDE UNIT NOTE Maximum and minimum values We can find maximum and minimum values of pcosx + qsinx by expressing it as a single trigonometric function. Example Write 4sinx + cosx in the form kcos(x + a) where 0 ≤ a ≤ 2π and state: (i) the maximum value and the value of x at which it occurs. (ii) the minimum value and the value of x at which this occurs.

12. 12 SLIDE p307 Exercise 16G Q1b, 3, 4, 5, 7.

13. b a CAREFUL! Higher Maths 2 3 Advanced Trigonometry 13 OUTCOME SLIDE UNIT NOTE Compound Angle Formulae sin() The result for can be used to find + b a all four basic compound angle formulae. sin() = sincos + sincos + b b b a a a sin() = sincos – sincos – b b b a a a cos() = coscos–sinsin + b b b a a a cos() = coscos+sinsin – b b b a a a

14. 14 SLIDE p190 Exercise 11C Q1, 3, 6, 7, 10 p191 Exercise 11D Q1, 4, 7, 9

15. REMEMBER Higher Maths 2 3 Advanced Trigonometry 15 OUTCOME SLIDE UNIT NOTE An algebraic fact is called an identity. Proving Trigonometric Identities Example sin() + b a Prove the identity = + tan tan b a cos cos b a sin() sinx + + sin cos sin cos b b b a a a tanx = = cosx cos cos cos cos b b a a sin cos sin cos b b a a = + L.H.S. R.H.S. cos cos cos cos b b a a ‘RightHand Side’ ‘Left Hand Side’ sin sin b a = = + + tan tan b a cos cos b a

16. 16 SLIDE p193 Exercise 11E Q1c, 2a, 3b, 4b, 5a

17. L = = 25 32 + 42 5 8 = = 5 80 82 + 42 4 M 4 8 1 2 4 = = = = cos sin 3 a a 5 5 4 5 4 5 J K b cos() + – = cos cos sin sin b b b a a a a 4 1 2 3 = × × + 5 5 5 5 5 2 10 2 = = = 5 5 5 5 Higher Maths 2 3 Advanced Trigonometry 17 OUTCOME SLIDE UNIT NOTE Applications of Trigonometric Addition Formulae JK = Example Find any unknown sides: KL = From the diagram, show that 5 2 cos() – = b a 5

18. 18 SLIDE p194 Exercise 11F Q1, 2, 4, 8, 5, 6, 7

19. a a REMEMBER Higher Maths 2 3 Advanced Trigonometry 19 OUTCOME SLIDE UNIT NOTE Investigating Double Angles The sum of two identical angles can be written as and is called a double angle. 2 a = sin(+) = + sin sin sin cos cos 2 a a a a a a a = sin 2 cos a a = cos(+) = – cos sin cos cos sin 2 a a a a a a a = – cos2 sin2 a a (1 – cos2) = – cos2 sin2x+ cos2x= 1 a a cos2 sin2 = – – 2 1 1 2 or sin2x= 1 – cos2x a a

20. LEARN THESE... Higher Maths 2 3 Advanced Trigonometry 20 OUTCOME SLIDE UNIT NOTE Double Angle Formulae sin 2 = 2sincos a a a There are several basic identities for double angles which it is useful to know. cos 2 = cos2 – sin2 a a a = 2cos2 – 1 a Example = 1 – 2sin2 4 If tan = , calculateand . a q 3 sin2 cos2 q q cos2– sin2 sin2 2sin cos cos2 opp = = q q q q q q = tan q adj () () 2 2 3 4 4 3 – 2 = = × × 5 5 5 5 5 3 q 24 7 – = = 25 25 4

21. 21 SLIDE p196 Exercise 11G Q7, 8, 9, 3, 6.

22. FACTORISE! Higher Maths 2 3 Advanced Trigonometry 22 OUTCOME SLIDE UNIT NOTE Trigonometric Equations involving Double Angles cos 2x – cosx = 0 Example substitute cos 2x – cosx = 0 Solve 2cos2x– 1 – cosx = 0 2π 0x for 2cos2x – cosx– 1 = 0 (2cosx + 1) (cosx – 1) = 0 remember 2cosx + 1 = 0 π cosx – 1 = 0 or S A P 1 cosx = – 3 2 cosx = 1 π π 2 4 x = 2π T C P x = x = x = 0 or or 3 3

23. Higher Maths 2 3 Advanced Trigonometry 23 OUTCOME SLIDE UNIT NOTE y Intersection of Trigonometric Graphs 4 A Example 360° x The diagram opposite shows the graphs of and . B f(x) g(x) g(x) Find the x-coordinate of A and B. f(x) -4 f(x) = g(x) 4sin2x=2sinx 2sinx = 0 4sin2x–2sinx = 0 4cosx–1 = 0 or 4 ×(2sinxcosx)–2sinx = 0 Solving by trigonometry, x = 0°, 180° or 360° 8sinxcosx–2sinx = 0 common factor x≈ 75.5° or 284.5° 2sinx(4cosx–1) = 0 or

24. 24 SLIDE p198 Exercise 11H Q1a,e,k, 2a,c,e, 3, 6a,c,e.

25. Higher Maths 2 3 Advanced Trigonometry 25 OUTCOME SLIDE UNIT Quadratic means ‘squared’ NOTE Quadratic Angle Formulae The double angle formulae can also be rearranged to give quadratic angle formulae. 1 cos2 (1+cos2) = a a 2 1 sin2 (1–cos2) = a a 2 Example 2cos2x – 3sin2x Expressin terms of cos2x. substitute 2cos2x – 3sin2x 1 1 2×(1+cos2x) – 3×(1–cos2x) = 2 2 3 3 1+ cos2x – + cos2x = 2 2 1 1 5 (5cos2x – 1) + cos2x – = = 2 2 2

26. 26 SLIDE p201 Exercise 11J Q2, 4, 5, 6, 9a,d, 10, 12, 17.