Graphs of Sine & Cosine — Challenge
Going round in circles, drawn flat, gives a wave — and four numbers describe any wave.
What these problems 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
What is the period of y=sin x, in degrees?
- 90
- 180
- 270
- 360
- 720
Worked answer
360
The sine function is defined by going round the unit circle.
One complete circuit is 360 degrees, after which the values repeat exactly.
So the period is 360 degrees.
Problem 2
What is the maximum value of y=sin x?
- 0
- 1
- 2
- -1
- 360
Worked answer
1
Sine gives the height of a point on the unit circle, whose radius is 1.
The highest that height can reach is 1, at the top of the circle.
So the maximum is 1, reached at 90 degrees.
Problem 3
What is the minimum value of y=cos x?
- -1
- 0
- 1
- 2
- -2
Worked answer
-1
Cosine gives the horizontal coordinate of a point on the unit circle.
The furthest left it can reach is -1, at the point (-1,0).
So the minimum is -1, reached at 180 degrees.
Problem 4
What is the amplitude of y=3sin x?
- 1
- 2
- 3
- 6
- 0
Worked answer
3
Amplitude is how far the curve rises above its centre line.
Multiplying sine by 3 stretches it so it runs from -3 to 3.
So the amplitude is 3. The full height from bottom to top is 6, which is twice the amplitude.
Problem 5
What is the period of y=sin 2x, in degrees?
- 90
- 180
- 360
- 720
- 45
Worked answer
180
The input is doubled, so the function completes its cycle in half the usual x range.
3602 = 180 degrees.
In general y = sin(bx) has period 360b degrees.
Problem 6
What is the period of y=3sin(π x2)?
- 2
- 4
- π
- 2π
- 4π
Worked answer
4
The period is 2πb where b is the coefficient of x. Here b=π2, so the period is 2ππ/2. =2π · 2π=4. The 3 out front changes the height, not the period.
Problem 7
The function y=asin(bx)+c has maximum 11 and minimum 3. What is c?
- 3
- 4
- 7
- 8
- 11
Worked answer
7
The constant c is the midline, halfway between the top and the bottom. c=11+32=7. The amplitude is the other half: a=11-32=4.
Problem 8
How many solutions does sin x=12 have for 0 ≤ x ≤ 4π?
- 2
- 3
- 4
- 5
- 6
Worked answer
4
In one full turn, sin x=12 happens twice: once in quadrant I and once in quadrant II. The interval 0 to 4π is two full turns. So there are 2 × 2 = 4 solutions. The endpoints give sin x = 0, so they add nothing.