1. Higher Maths The Wave Function Strategies Click to start
Higher Maths
Strategies
The Wave Function
Click to start

2. 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.
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3. 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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4. 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 ksin(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

5. 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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6. 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

7. 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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8. 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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9. a) Express f (x) in the form
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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10. 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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11. Solve the equation in the interval 0  x  360.
Maths4Scotland Higher
Solve the equation in the interval 0  x  360.
Use Rcos(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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12. You have completed all 9 questions in this presentation
Maths4Scotland Higher
You have completed all 9 questions in this presentation
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13. 30° 45° 60° sin cos tan 1 Table of exact values Maths4Scotland Higher
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