Presentation on theme: "Higher Unit 3 Exam Type Questions What is a Wave Function"— Presentation transcript:
1 Higher Unit 3 Exam Type Questions What is a Wave Function Connection with Trig Identities EarlierMaximum and Minimum ValuesSolving Equations involving the Wave FunctionExam Type Questions
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 AnalysisElectrical
3 The Wave Function General shape for y = sinx + cosx Like y = sin(x) shifted leftLike y = cosx shifted rightVertical height differentThe Wave Functiony = sin(x)+cos(x)y = sin(x)y = cos(x)
4 The Wave FunctionWhenever 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 sideFOR ALL VALUES OF x
5 The left and right hand sides must be equal for all values of x. The Wave FunctionWorked Example:Re-arrangeThe 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 Find tan ratio note: sin(+) and cos(+) The Wave FunctionFind tan ratio note: sin(+) and cos(+)Square and add
7 Note: sin(+) and cos(+) The Wave FunctionNote: sin(+) and cos(+)CAST0o180o270o90o
8 The Wave Function Example 90o S A 180o 0o T C 270o Expand and equate coefficientsThe Wave FunctionExampleFind tan ratio note: sin(+) and cos(+)CAST0o180o270o90oSquare and add
16 Maximum and Minimum Values ExampleA 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 Maximum and Minimum Values Expand and equate coefficientsMaximum and Minimum ValuesFind tan ratio note: sin(-) and cos(+)CAST0o180o270o90oSquare and add
18 Maximum and Minimum Values remember K = 25k =25 x 5 = 125The minimum value of sin is -1 and it occurs where the angle is 270oTherefore, the minimum value of Vresult is -125Adding or subtracting 360o leaves the sin unchanged
24 Solving Trig Equations Expand and equate coefficientsSolving Trig EquationsExampleFind tan ratio note: sin(-) and cos(-)CAST0o180o270o90oSquare and add
25 Solving Trig Equations 2x – = 16.1o , ( o),( o),( o)2x – = 16.1o , o, o, o, ….2x = o , o, o, o, ….x = o , o, o, o, ….
26 Solving Trig Equations (From a past paper)ExampleA 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 θo is the angle shown in the diagram and Ais the area m2 of the gable end, show that4Find algebraically the value of θo for which the area of the gable end is 30m2.
27 Solving Trig Equations 4sSolving 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 isTotal area = Triangle + 2 x square:
28 Solving Trig Equations (From a past paper)Part (b)Find tan ratio note: sin(+) and cos(+)CAST0o180o270o90oSquare and addFinally:
29 Solving Trig Equations (From a past paper)Part (c)From diagram θo < 90o ignore 2nd quadFind algebraically the value of θo for which the area is the 30m2CAST0o180o270o90o
30 Higher Maths The Wave Function Strategies Click to start Higher MathsStrategiesThe Wave FunctionClick to start
31 The Wave Function The following questions are on Maths4Scotland HigherThe following questions are onThe Wave FunctionNon-calculator questions will be indicatedYou will need a pencil, paper, ruler and rubber.Click to continue
32 Part of the graph of y = 2 sin x + 5 cos x is shown in the diagram. Maths4Scotland HigherPart of the graph of y = 2 sin x + 5 cos x is shownin the diagram.Express y = 2 sin x + 5 cos x in the form k sin (x + a)where k > 0 and 0 a 360b) Find the coordinates of the minimum turning point P.Expand ksin(x + a):Equate coefficients:Square and adda is in 1st quadrant(sin and cos are +)Dividing:Put together:HintMinimum when:P has coords.PreviousQuitQuitNext
33 Maths4Scotland HigherWrite sin x - cos x in the form k sin (x - a) stating the values of k and a wherek > 0 and 0 a 2b) Sketch the graph of sin x - cos x for 0 a 2 showing clearly the graph’smaximum and minimum values and where it cuts the x-axis and the y-axis.Expand k sin(x - a):Equate coefficients:Square and adda is in 1st quadrant(sin and cos are +)Dividing:Put together:Sketch GraphHintPreviousQuitQuitNextTable of exact values
34 Maths4Scotland Higher Express in the form Expand kcos(x + a): Equate coefficients:Square and adda is in 1st quadrant(sin and cos are +)Dividing:Put together:HintPreviousQuitQuitNext
35 Maths4Scotland HigherFind the maximum value of and the value of x for which it occurs in the interval 0 x 2.Express as Rcos(x - a):Equate coefficients:Square and adda is in 4th quadrant(sin is - and cos is +)Dividing:Put together:HintMax value:whenPreviousQuitQuitNextTable of exact values
36 Express in the form Maths4Scotland Higher Expand ksin(x - a): Equate coefficients:Square and adda is in 1st quadrant(sin and cos are both +)Dividing:Put together:HintPreviousQuitQuitNext
37 The diagram shows an incomplete graph of Maths4Scotland HigherThe diagram shows an incomplete graph ofFind the coordinates of the maximum stationary point.Max for sine occursMax value of sine function:Sine takes values between 1 and -1Max value of function:3Coordinates of max s.p.HintPreviousQuitQuitNext
38 Cosine +, so 1st & 4th quadrants Maths4Scotland Highera) Express f (x) in the formb) Hence solve algebraicallyExpand kcos(x - a):Equate coefficients:Square and adda is in 1st quadrant(sin and cos are both + )Dividing:Put together:Solve equation.HintCosine +, so 1st & 4th quadrantsPreviousQuitQuitNext
39 Solve the simultaneous equations where k > 0 and 0 x 360 Maths4Scotland HigherSolve the simultaneous equationswhere k > 0 and 0 x 360Use tan A = sin A / cos ADivideFind acute angleDetermine quadrant(s)Sine and cosine are both + in original equationsSolution must be in 1st quadrantHintState solutionPreviousQuitQuitNext
40 Cosine +, so 1st & 4th quadrants Maths4Scotland HigherSolve the equation in the interval 0 x 360.Use R cos(x - a):Equate coefficients:Square and adda is in 2nd quadrant(sin + and cos - )Dividing:Put together:Solve equation.HintCosine +, so 1st & 4th quadrantsPreviousQuitQuitNext
41 You have completed all 9 questions in this presentation Maths4Scotland HigherYou have completed all 9 questions in this presentationPreviousQuitQuitBack to start
42 Are you on Target ! Update you log book Make sure you complete and correctALL of the Wave Function questions in the past paper booklet.
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