1. 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

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. 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)

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 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

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

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

8. 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

9. The Wave Function
Finally:

10. 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

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. 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(-)

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. 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

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

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

22. 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

23. Solving Trig Equations
Step 2:

24. 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

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)
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.

27. Solving Trig Equations4 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. 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:

29. 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

30. Higher Maths The Wave Function Strategies Click to start
Higher Maths
Strategies
The Wave Function
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31. The Wave Function The following questions are on
Maths4Scotland Higher
The following questions are on
The Wave Function
Non-calculator questions will be indicated
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32. 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.
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33. 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
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Table of exact values

34. 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
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35. 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
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Table of exact values

36. 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
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37. 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
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38. 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
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39. 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
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40. 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
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Maths4Scotland Higher
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Make sure you complete and correct
ALL of the Wave Function questions in the past paper booklet.