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Higher Unit 3

Higher Unit 3. What is a Wave Function. Connection with Trig Identities Earlier. Maximum and Minimum Values. Solving Equations involving the Wave Function. Exam Type Questions. The Wave Function. Heart beat.

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Higher Unit 3

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  1. Higher Unit 3 What is a Wave Function Connection with Trig Identities Earlier Maximum and Minimum Values Solving Equations involving the Wave Function Exam Type Questions www.mathsrevision.com

  2. The Wave Function Heart beat Many wave shapes, whether occurring as sound, light, water or electrical waves, can be described mathematically as a combination of sine and cosine waves. Spectrum Analysis Electrical

  3. General shape for y = sinx + cosx • Like y = sin(x) shifted left • Like y = cosx shifted right • Vertical height different The Wave Function y = sin(x)+cos(x) y = sin(x) y = cos(x)

  4. The Wave Function Whenever a function is formed by adding cosine and sine functions the result can be expressed as a related cosine or sine function. In general: With these constants the expressions on the right hand sides = those on the left hand side FOR ALL VALUES OF x

  5. The Wave Function Worked Example: Re-arrange The left and right hand sides must be equal for all values of x. So, the coefficients of cos x and sin x must be equal: A pair of simultaneous equations to be solved

  6. The Wave Function Find tan ratio note: sin(+) and cos(+) Square and add

  7. 90o A S 180o 0o C T 270o The Wave Function Note: sin(+) and cos(+)

  8. 90o A S 180o 0o C T 270o Expand and equate coefficients The Wave Function Example Find tan ratio note: sin(+) and cos(+) Square and add

  9. The Wave Function Finally:

  10. 90o A S 180o 0o C T 270o Expand and equate coefficients The Wave Function Example Find tan ratio noting sign of sin(+) and cos(+) Square and add

  11. The Wave Function Finally:

  12. Maximum and Minimum Values Worked Example: b) Hence find: i) Its maximum value and the value of x at which this maximum occurs. ii) Its minimum value and the value of x at which this minimum occurs.

  13. 90o A S 180o 0o C T 270o Expand and equate coefficients Maximum and Minimum Values Square and add Find tan ratio note: sin(+) and cos(-)

  14. Maximum and Minimum Values Maximum, we have:

  15. Maximum and Minimum Values Minimum, we have:

  16. Maximum and Minimum Values Example A synthesiser adds two sound waves together to make a new sound. The first wave is described by V = 75sin to and the second by V = 100cos to, where V is the amplitude in decibels and t is the time in milliseconds. Find the minimum value of the resultant wave and the value of t at which it occurs. For later, remember K = 25k

  17. 90o A S 180o 0o C T 270o Expand and equate coefficients Maximum and Minimum Values Find tan ratio note: sin(-) and cos(+) Square and add

  18. Maximum and Minimum Values remember K = 25k =25 x 5 = 125 The minimum value of sin is -1 and it occurs where the angle is 270o Therefore, the minimum value of Vresult is -125 Adding or subtracting 360o leaves the sin unchanged

  19. Maximum and Minimum Values Minimum, we have:

  20. True for ALL x means coefficients equal. Solving Trig Equations Worked Example: Step 1: Compare Coefficients: Square &Add

  21. 90o A S 180o 0o C T 270o Find tan ratio note: sin(+) and cos(+) Solving Trig Equations

  22. 90o A S 180o 0o C T 270o Solving Trig Equations Step 2: Re-write the trig. equation using your result from step 1, then solve.

  23. Solving Trig Equations Step 2:

  24. 90o A S 180o 0o C T 270o Expand and equate coefficients Solving Trig Equations Example Find tan ratio note: sin(-) and cos(-) Square and add

  25. Solving Trig Equations 2x – 213.7 = 16.1o , (180-16.1o),(360+16.1o),(360+180-16.1o) 2x – 213.7 = 16.1o , 163.9o, 376.1o, 523.9o, …. 2x = 229.8o , 310.2o, 589.9o, 670.2o, …. x = 114.9o , 188.8o, 294.9o, 368.8o, ….

  26. 4 4 Solving Trig Equations (From a past paper) Example A builder has obtained a large supply of 4 metre rafters. He wishes to use them to build some holiday chalets. The planning department insists that the gable end of each chalet should be in the form of an isosceles triangle surmounting two squares, as shown in the diagram. • If θois the angle shown in the diagram and A • is the area m2 of the gable end, show that • Find algebraically the value of θofor which the area of the gable end is 30m2.

  27. 4 4 s s Solving Trig Equations (From a past paper) Part (a) Let the side of the square frames be s. Use the cosine rule in the isosceles triangle: This is the area of one of the squares. The formula for the area of a triangle is Total area = Triangle + 2 x square:

  28. 90o A S 180o 0o C T 270o Solving Trig Equations (From a past paper) Part (b) Find tan ratio note: sin(+) and cos(+) Square and add Finally:

  29. 90o A S 180o 0o C T 270o Solving Trig Equations (From a past paper) Part (c) From diagram θo< 90o ignore 2nd quad Find algebraically the value of θo for which the area is the 30m2

  30. www.maths4scotland.co.uk Higher Maths Strategies The Wave Function Click to start

  31. Maths4Scotland Higher The following questions are on The Wave Function Non-calculator questions will be indicated You will need a pencil, paper, ruler and rubber. Click to continue

  32. Hint Maths4Scotland Higher • Part of the graph of y = 2sin x + 5cos x is shown • in the diagram. • Express y = 2sin x + 5cos x in the form k sin (x + a) • where k > 0 and 0  a  360 • b) Find the coordinates of the minimum turning point P. Expand ksin(x + a): Equate coefficients: Square and add a is in 1st quadrant (sin and cos are +) Dividing: Put together: Minimum when: P has coords. Previous Next Quit Quit

  33. Hint Maths4Scotland Higher • Write sin x - cos x in the form k sin (x - a) stating the values of k and a where • k > 0 and 0  a  2 • b) Sketch the graph of sin x - cos x for 0  a  2 showing clearly the graph’s • maximum and minimum values and where it cuts the x-axis and the y-axis. Expand k sin(x - a): Equate coefficients: Square and add a is in 1st quadrant (sin and cos are +) Dividing: Put together: Sketch Graph Previous Next Quit Quit Table of exact values

  34. Express in the form Hint Maths4Scotland Higher Expand kcos(x + a): Equate coefficients: Square and add a is in 1st quadrant (sin and cos are +) Dividing: Put together: Previous Next Quit Quit

  35. Find the maximum value of and the value of x for which it occurs in the interval 0 x  2. Hint Maths4Scotland Higher Express as Rcos(x - a): Equate coefficients: Square and add a is in 4th quadrant (sin is - and cos is +) Dividing: Put together: Max value: when Previous Next Quit Quit Table of exact values

  36. Express in the form Hint Maths4Scotland Higher Expand ksin(x - a): Equate coefficients: Square and add a is in 1st quadrant (sin and cos are both +) Dividing: Put together: Previous Next Quit Quit

  37. The diagram shows an incomplete graph of Find the coordinates of the maximum stationary point. Hint Maths4Scotland Higher Max for sine occurs Sine takes values between 1 and -1 Max value of sine function: Max value of function: 3 Coordinates of max s.p. Previous Next Quit Quit

  38. a) Express f (x) in the form b) Hence solve algebraically Hint Maths4Scotland Higher Expand kcos(x - a): Equate coefficients: Square and add a is in 1st quadrant (sin and cos are both + ) Dividing: Put together: Solve equation. Cosine +, so 1st & 4th quadrants Previous Next Quit Quit

  39. Hint Maths4Scotland Higher Solve the simultaneous equations where k > 0 and 0 x 360 Use tan A = sin A / cos A Divide Find acute angle Sine and cosine are both + in original equations Determine quadrant(s) Solution must be in 1st quadrant State solution Previous Next Quit Quit

  40. Hint Maths4Scotland Higher Solve the equation in the interval 0 x 360. Use R cos(x - a): Equate coefficients: Square and add a is in 2nd quadrant (sin + and cos - ) Dividing: Put together: Solve equation. Cosine +, so 1st & 4th quadrants Previous Next Quit Quit

  41. Maths4Scotland Higher You have completed all 9 questions in this presentation Previous Quit Quit Back to start

  42. Are you on Target ! • Update you log book • Make sure you complete and correct • ALL of the Wave Function questions in the past paper booklet.

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