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**Higher Unit 3 Exam Type Questions What is a Wave Function**

Connection with Trig Identities Earlier Maximum and Minimum Values Solving Equations involving the Wave Function Exam Type Questions

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**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

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**The Wave Function 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)

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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

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**The left and right hand sides must be equal for all values of x.**

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

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**Find tan ratio note: sin(+) and cos(+)**

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

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**Note: sin(+) and cos(+)**

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

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**The Wave Function Example 90o S A 180o 0o T C 270o**

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

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The Wave Function Finally:

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**The Wave Function Example 90o S A 180o 0o T C 270o**

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

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The Wave Function Finally:

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**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.

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**Maximum and Minimum Values**

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

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**Maximum and Minimum Values**

Maximum, we have:

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**Maximum and Minimum Values**

Minimum, we have:

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**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

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**Maximum and Minimum Values**

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

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**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

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**Maximum and Minimum Values**

Minimum, we have:

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**Solving Trig Equations**

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

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**Solving Trig Equations**

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

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**Solving Trig Equations**

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

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**Solving Trig Equations**

Step 2:

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**Solving Trig Equations**

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

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**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, ….

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**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 θo is the angle shown in the diagram and A is the area m2 of the gable end, show that 4 Find algebraically the value of θo for which the area of the gable end is 30m2.

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**Solving Trig Equations**

4 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:

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**Solving Trig Equations**

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

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**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 C A S T 0o 180o 270o 90o

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**Higher Maths The Wave Function Strategies Click to start**

Higher Maths Strategies The Wave Function Click to start

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**The Wave Function The following questions are on**

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

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**Part of the graph of y = 2 sin x + 5 cos x is shown in the diagram. **

Maths4Scotland Higher Part of the graph of y = 2 sin x + 5 cos x is shown in the diagram. Express y = 2 sin x + 5 cos 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: Hint Minimum when: P has coords. Previous Quit Quit Next

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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 Hint Previous Quit Quit Next Table of exact values

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**Maths4Scotland Higher Express in the form Expand kcos(x + a):**

Equate coefficients: Square and add a is in 1st quadrant (sin and cos are +) Dividing: Put together: Hint Previous Quit Quit Next

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Maths4Scotland Higher Find 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 add a is in 4th quadrant (sin is - and cos is +) Dividing: Put together: Hint Max value: when Previous Quit Quit Next Table of exact values

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**Express in the form Maths4Scotland Higher Expand ksin(x - a):**

Equate coefficients: Square and add a is in 1st quadrant (sin and cos are both +) Dividing: Put together: Hint Previous Quit Quit Next

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**The diagram shows an incomplete graph of**

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

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**Cosine +, so 1st & 4th quadrants**

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

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**Solve the simultaneous equations where k > 0 and 0 x 360**

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

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**Cosine +, so 1st & 4th quadrants**

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. Hint Cosine +, so 1st & 4th quadrants Previous Quit Quit Next

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**You have completed all 9 questions in this presentation**

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

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**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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