Graphs of Sine & Cosine — Practice
Going round in circles, drawn flat, gives a wave — and four numbers describe any wave.
What these questions cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Coefficient — The number multiplying a variable — the 3 in 3n.
- π — The circumference of any circle divided by its diameter — about 3.14159.
- Radius — Centre to edge. Half the diameter.
- Sine — sin θ = oppositehypotenuse.
- Amplitude — Half the distance from the top to the bottom — how tall the wave is.
- Period — How long before the pattern repeats.
- Midline — The horizontal line halfway between the peaks and troughs.
- Phase shift — A horizontal slide — how far the wave has been moved sideways.
Problem 1
Give the amplitude of y=5sin x.
Worked answer
5
Amplitude =|5|=5
Problem 2
Give the amplitude of y=-3cos x.
Worked answer
3
|-3|=3
Problem 3
Give the period of y=sin x. Use the coefficient of π.
Worked answer
2
2π
Problem 4
Give the midline of y=sin x+4.
Worked answer
y=4
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Problem 5
Where does y=cos x start at x=0?
- At its maximum
- At zero going up
- At its minimum
- At zero going down
Worked answer
At its maximum
cos 0 = 1, the highest value
Problem 6
Give the period of y=sin(2x). Give the coefficient of π.
Worked answer
1
2π2=π
Problem 7
Give the period of y=cos(4x). Give the answer as the fraction multiplying π.
Worked answer
1/2
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Problem 8
A wave peaks at 17 and bottoms at 3. Find its amplitude.
Worked answer
7
17-32=7
Problem 9
A wave peaks at 17 and bottoms at 3. Find its midline.
Worked answer
y=10
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Problem 10
Find the maximum of y=4sin x+6.
Worked answer
10
6+4=10
Problem 11
Find the minimum of y=4sin x+6.
Worked answer
2
6-4=2
Problem 12
A sine wave has period 6. Find b in y=sin(bx). Give the fraction multiplying π.
Worked answer
1/3
b=2π6=π3
Problem 13
A wave repeats every 8 units. Find b. Give the fraction multiplying π.
Worked answer
1/4
b=π4
Problem 14
Give the phase shift of y=sin(2x-π). Give the fraction multiplying π.
Worked answer
1/2
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Problem 15
A Ferris wheel of radius 15 m has its centre 18 m up. How high is the lowest point?
Worked answer
3
18-15=3
Problem 16
The same wheel turns once every 5 minutes. How many full turns in 30 minutes?
Worked answer
6
30 ÷ 5 = 6