Graphs of Sine & Cosine

Going round in circles, drawn flat, gives a wave — and four numbers describe any wave.

Ride a Ferris wheel and write down your height every few seconds.

You rise, level off at the top, come down, bottom out, and rise again. Then it repeats, exactly.

your height on a Ferris wheel, minute by minute one turn going round in a circle, drawn flat, is a wave

Plotting height against time gives a wave that never ends, because the wheel never stops going round.

The circle and the wave are the same motion. One is drawn round, the other drawn flat.

That is what y = sin x is. Anything that repeats — tides, daylight, sound, a heartbeat — is this curve, stretched and shifted to fit.

Sine and cosine over one full turn 1 −1 0 π/2 π 3π/2 2π sin cos same wave, same height, shifted a quarter turn apart

The two parent waves

y = sin x starts at 0, climbs to 1 at π2, returns to 0 at π, dips to -1, and closes at 2π.

y = cos x is the same wave shifted a quarter turn, starting at its peak instead of at zero.

Both have amplitude 1, period 2π and midline y = 0 before anything is done to them.

Five points draw it

Start, quarter, half, three-quarter, end. Mark those five heights and the curve draws itself — plotting more points rarely helps.

Amplitude and midline

In y = asin(bx) + d, the a sets the amplitude — the wave reaches |a| above and below the midline.

The d lifts the whole wave, so the midline becomes y = d rather than the x-axis.

A negative a flips the wave upside down without changing how tall it is.

Getting them from a picture

Amplitude is max - min2 and midline is max + min2. Half the difference, and half the sum.

Period: the number inside

The b in sin(bx) controls how fast the wave cycles, and the period becomes 2πb.

Bigger b means a shorter period, because the input races through its values faster.

y = sin(3x) fits three complete waves where y = sin x fits one.

b is not the period

In sin(4x) the period is 2π4 = π2, not 4. The number inside divides into 2π; it is never the answer by itself.

Phase shift

A horizontal slide comes from the bracket: y = sin(x - c) moves the wave c to the right.

When there is also a b, factor it out first: sin(2x - π) = sin(2(x - π2)), a shift of π2.

Without factoring you will read the shift as π, which is twice as far as it really goes.

Factor before reading the shift

The shift is whatever makes the bracket zero. In 2x - π that is x = π2, not π.

Modelling something real

For a Ferris wheel, the midline is the height of the centre and the amplitude is the radius.

The period is the time for one full turn, and the phase shift says where you boarded.

Cosine is often the easier choice when the motion starts at a maximum, since it begins there already.

Choose the wave that fits the start

Starting at the midline going up suits sine; starting at the top suits cosine. Either can model it, but one needs no shift.

Worked examples

Reading a formula

Give the amplitude, period and midline of y = 3sin(2x) - 1.

  1. a = 3, so the amplitude is 3. The number in front, ignoring sign.
  2. b = 2, so the period is 2π2 = π. The inside number divides into 2π.
  3. d = -1, so the midline is y = -1. The wave runs from -4 up to 2.

Amplitude 3, period π, midline y = -1

Building from a picture

A wave peaks at 14, bottoms at 4, and repeats every 8 units. Write a sine model with no phase shift.

  1. Amplitude = 14-42 = 5; midline = 14+42 = 9. Half the difference, half the sum.
  2. Period 8 means 2πb = 8, so b = π4. Rearrange, do not just write 8.
  3. So y = 5sin(π4x) + 9. Check: at x = 2 this gives 5 + 9 = 14, the peak.

y = 5sin(π4x) + 9

A Ferris wheel

A wheel of radius 20 m has its centre 23 m up and turns once every 4 minutes. Write the height of a rider who boards at the bottom.

  1. Amplitude 20, midline 23. Radius and centre height.
  2. Period 4 gives b = 2π4 = π2. Rearranged from period = 2πb.
  3. Boarding at the bottom is a flipped cosine: h = -20cos(π2t) + 23. At t=0 this gives 3 m, the lowest point.

h = -20cos(π2t) + 23

Words to know

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.
Frequency
How many full cycles fit in a given interval. The reciprocal of the period.

Mistakes to watch for

Reading the inside number as the period. In y = sin(4x) the period is 2π4 = π2. The 4 counts how many waves fit in the old period, so a bigger b makes a shorter wave, not a longer one.

Taking the phase shift straight from the bracket. sin(2x - π) shifts by π2, not π — the b must be factored out first, or the shift comes out doubled.

Mixing up amplitude and maximum. A wave with midline 9 and amplitude 5 peaks at 14. The amplitude is measured from the midline, not from the x-axis.

Forgetting that a negative a only flips. y = -3sin x has amplitude 3, not -3. Amplitude is a distance, so it is never negative.

Using degrees on a radian-scaled graph. If the axis is marked in πs the calculator must be in radian mode, and a graph plotted in degree mode will look flat or wildly wrong.

Every sound you have ever heard is a sum of sine waves. Fourier proved in 1807 that any repeating signal can be built from them, and that result runs MP3s, JPEGs and mobile phones.

Tide tables have been computed from sine waves since the 1870s, using brass machines that added the waves mechanically.

Daylight hours through the year trace a sine wave too — which is why the days lengthen fastest around the equinoxes, not midsummer.

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