📈Graphs of Sine & Cosine

Climber Sora keeps a rope wound round a drum at the top of Sine Summit, and paints one red mark on the rim. Turn the drum steadily and watch the mark: it rises, hangs at the top, comes back down, drops to the bottom, and climbs again. Nothing else happens, ever. That is the whole of this skill.

On the unit circle a point at angle t sits at (cos t, sin t). Sine is the height of that point and cosine is how far across it is. Walk once round the circle and the height goes 0, up to 1, back to 0, down to -1, back to 0. Plot that height against the angle instead of against the circle and you have unrolled the circle into a wave.

• ◆y = sin(x) starts at 0, on the middle, and rises.
• ◆y = cos(x) starts at 1, at the top, and comes down.
• ◆Both finish a complete lap at x = 2pi and then do exactly the same thing again.

That "and then again" is what makes these functions worth having. A rope swinging, a tide, a wheel, a bouncing weight, daylight through a year — all of them come back to where they were and go round again, and all of them are described by a sine or a cosine that has been stretched, squashed, lifted and slid.

Everything on the rest of this page is one of those four moves, written into one equation:

y = a sin(b(x - h)) + k

Read it left to right and it tells you the whole story of the wave: a is how big the swing is, b decides how often it happens, h slides it sideways, and k lifts it. The same four letters work with cosine in place of sine. Note also that x here is an angle in radians, not in degrees — on this mountain everything is measured in radians.

Two of the four numbers are easy, because both are read straight off the equation without any work.

The midline is the horizontal line the wave swings about — halfway between its highest point and its lowest point. Plain sine and cosine swing about 0, so the only thing that can move the midline is the number added on the outside. In y = a sin(b(x - h)) + k, the midline is the line y = k.

The amplitude is the distance from the midline up to the highest point — half the total swing. It is |a|, the number multiplying the sine or cosine. Two words in that sentence do a lot of work:

• ◆Distance. An amplitude is never negative. y = -3 sin(x) has amplitude 3; the minus sign flips the wave upside down about its midline, but it does not make the swing smaller.
• ◆From the midline. The amplitude is not the highest point. For y = 3 sin(x) + 2 the highest point is 5, and the amplitude is 3.

Put the two together and the greatest and least values fall out with no effort at all, because sin and cos never leave the range from -1 to 1 whatever is fed into them:

greatest value = k + |a| least value = k - |a|

For y = 4 cos(x) - 1: amplitude 4, midline y = -1, so the wave lives between -5 and 3.

And it runs backwards just as easily. If a graph rises to 11 and drops to 3, then

amplitude = (11 - 3) / 2 = 4 midline: k = (11 + 3) / 2 = 7

Subtract and halve for the amplitude; add and halve for the midline. Those two lines answer most "read the graph" questions on their own.

The period is how far along x you travel before the wave starts repeating. A plain sine or cosine has period 2pi, because that is once round the circle.

Now put a number in front of x. In y = sin(2x), the input runs twice as fast, so the wave gets through a whole lap in half the distance. In y = sin(x/3) the input crawls, and the wave takes three times as long. In general:

period = 2pi / b

b is not the period. b counts how many complete cycles fit into 2pi. That single confusion accounts for more wrong answers here than everything else combined, so say the formula out loud with the division in it: *two pi over b*.

• ◆y = cos(2x): b = 2, period = 2pi/2 = pi.
• ◆y = sin(x/3): b = 1/3, period = 2pi / (1/3) = 6pi.
• ◆y = sin((3/2)x): b = 3/2, period = 2pi / (3/2) = 4pi/3.

The formula runs backwards too, and rearranging it needs no new idea:

b = 2pi / period

So a wave with period 4pi has b = 2pi / (4pi) = 1/2, and a wave that repeats every pi has b = 2pi / pi = 2. Notice how the pi cancels: whenever the period is a multiple or a fraction of pi, b comes out as a plain number or a plain fraction with no pi left in it. If your b still has a pi in it, check the division.

On a graph, measure a period between two features that look identical: peak to the next peak, or trough to the next trough, or one rising crossing of the midline to the next rising crossing. Peak to the next trough is only half a period — a classic way to end up with an answer exactly half the size it should be.

The last move slides the wave sideways. In y = sin(x - h) every point of the wave moves h to the right, and in y = sin(x + h) every point moves h to the left. That distance is the phase shift.

The direction reads backwards from the sign, which feels wrong until you ask *which x now does the old job of x = 0*. In sin(x - pi/2), the bracket is 0 when x = pi/2, so the point that used to sit at 0 has moved right to pi/2. Minus means right.

y = sin(x - pi/4) shifted pi/4 to the RIGHT y = cos(x + pi/3) shifted pi/3 to the LEFT

Now the trap, and it is the whole reason this skill is worth a page of its own. Consider

y = cos(2x - pi)

It is tempting to read a shift of pi. It is not pi. The formula reads a shift from the form b(x - h), and 2x - pi is not yet in that form. Factor b out first:

2x - pi = 2(x - pi/2)

so h = pi/2 and the wave is shifted pi/2 to the right — half of what it first looked like. The rule is short: divide the constant by b. A shift is a distance along the x-axis, and b has already stretched that axis, so the constant has to be stretched back.

One more example, in the other direction. For y = sin((1/2)x + pi/4):

(1/2)x + pi/4 = (1/2)(x + pi/2)

so h = -pi/2 and the shift is pi/2 to the left — this time bigger than it looked, because b was less than 1. Multiplying back out is the check that costs five seconds and settles it.

Sine and cosine are the same wave a quarter turn apart, which is why a shift can turn one into the other: cos(x) = sin(x + pi/2). Two people can describe the same graph with two different equations, and both be right.

Building an equation from a description is the four moves in reverse, in a fixed order. Sora does it the way ropes get set: same order every time, no thinking required.

1. a — the amplitude, half the distance from the lowest point to the highest. 2. k — the midline, the average of the lowest and the highest. 3. b — from the period, using b = 2pi / period. 4. h — the shift, positive for a slide to the right.

For a wave with amplitude 2, period pi, midline y = -1 and no shift: a = 2, k = -1, b = 2pi/pi = 2, h = 0, giving y = 2 sin(2x) - 1.

Modelling something real is the same four steps with the numbers hidden inside a story. A Ferris wheel of radius 5 m whose centre is 8 m up and which turns once every 20 seconds gives a = 5, k = 8, and b = 2pi/20 = pi/10. The last decision is where the ride starts. A rider boarding at the bottom is at the lowest point at t = 0, and cos(0) = 1 puts a plain cosine at the top — so the cosine is turned over:

h(t) = 8 - 5 cos((pi/10)t)

Check it at the two easy times. At t = 0 the height is 8 - 5 = 3 m, the boarding platform. Ten seconds later, half a turn, cos(pi) = -1 and the height is 8 + 5 = 13 m, the very top. A model that survives those two checks is almost certainly right.

How to type your answer here.

• ◆An amplitude, a midline value k, and a greatest or least value are whole numbers: type 3, or -1. An amplitude is always positive.
• ◆A period, a phase shift, and the x-value of a peak are exact multiples of pi. Write pi as the two letters pi, and type the value the way you would write it: pi/4, 2pi/3, 3pi/2, 6pi. The checker works these out, so (2/3)pi passes too — but a rounded decimal such as 2.09 does not, because it is not the same number. When a question says "give an exact value using pi", a decimal will not be accepted.
• ◆Where an equation, a direction ("to the left") or a model is asked for, the options are listed and you pick one.
• ◆Angles here are always radians. There are no degree answers anywhere in this skill.