Stretch, squeeze and slide the sine wave yourself

How to use this simulation

1. Pick sine, cosine or tangent under "Which function". The faint grey dashed curve is the parent function and the bold curve is your transformed graph.
2. Drag the amplitude factor A: the wave grows or shrinks, and a negative A flips it upside down. Compare the Amplitude |A| reading as you go.
3. Raise the frequency factor B to squeeze the wave. The bracket under the graph always spans exactly one period.
4. Use the phase C to slide the wave sideways and the vertical shift D to lift the midline. The green point rides along the curve; press play or pause as you like.
5. Switch to tangent and look at the red dashed vertical lines: those are the asymptotes, where tan is undefined. Flip the axis units between degrees and multiples of π.

Waves everywhere: why we graph sin, cos and tan

Drop a pebble in a pond and the surface rises and falls. A guitar string, the tide at the coast, the current in your wall socket, even the height of a seat on a Ferris wheel: all of them repeat themselves after a fixed time. Sine and cosine graphs are the simplest tool mathematics has for describing that repetition.

Plain sin x is not enough, though. Real waves can be taller, faster, late to start or sitting higher than zero. So we change the basic wave in four ways: stretch it, squeeze it, slide it sideways and lift it up. In this lesson you do all four with your own hands.

From the unit circle to the wave

A point travels round the unit circle. At angle θ its height is sin θ and its left-right position is cos θ. Now keep that height and lay it out to the right as the angle grows. A smooth wave appears: that is the graph of y = sin x. For cos x, lay out the left-right position instead.

Both graphs repeat every 360°, so the basic period of sin and cos is 360° (2π in radians). The greatest value is 1 and the least is -1. Sine starts at zero and climbs; cosine starts at its peak, which means cos x is just sin x moved 90° to the left.

Tangent is tan x = sin x ÷ cos x. Wherever cos x = 0 (at 90°, 270° and so on) the division fails, tan is undefined, and the graph shoots off to infinity beside that line. Such a vertical line is called an asymptote. The period of tan is only 180°.

The master formula: y = A·f(Bx + C) + D

Every sine, cosine or tangent graph you will meet fits one mould. f is sin, cos or tan, and four numbers A, B, C, D describe the wave. Here is each job in one line; we take them one at a time below.

The vocabulary in plain words

These words come up again and again in exams. The table gives each meaning and its value for the graph the simulation opens with.

A: amplitude, and what a flip does

Changing A only changes the height of the wave, never how crowded it is. For y = 3 sin x the wave rises to 3, and the distance from midline to peak is 3. That distance is the amplitude.

If A is negative the wave is flipped over its midline. y = −3 sin x and y = 3 sin x have the same amplitude, 3; one has a peak exactly where the other has a trough. So amplitude is always positive: |A|.

B: period, or how squeezed the wave is

B squeezes the wave sideways. With B = 2 the wave runs twice as fast, so one full wave takes 180° (π) instead of 360°. The rule is: period = basic period ÷ |B|.

A B between 0 and 1 does the opposite and stretches the wave out. The period of y = sin(x/2) is 720°.

C: phase shift, the sideways slide

A new cycle starts when Bx + C = 0, so the start is at x = −C/B. In the starting graph y = 2 sin(2x + 60°) + 1 that gives −C/B = -30°: the wave has slid 30° to the left of where the plain sine starts.

Do not forget B. Many students see C = 60° and say the graph moved 60°. Because B = 2, the shift is half of that, -30°. Instead of memorizing, factor the bracket: 2(x + 30°) = 2x + 60°.

D: vertical shift and the midline

D lifts or lowers the whole wave without changing its shape. The midline becomes y = D. In the starting graph the midline is y = 1 and the amplitude is 2, so y runs from -1 up to 3.

The rule: maximum = D + |A| and minimum = D − |A|. Turned around, amplitude = (maximum − minimum) ÷ 2 and D = (maximum + minimum) ÷ 2, which the solved problems use.

Tangent is the odd one out

The tan graph climbs to infinity and drops from minus infinity again and again, so it has no maximum, no minimum and no amplitude. Its range is all real numbers. A only decides how steep the curve is. The phase shift −C/B and the period 180°/|B| work in the same way as before.

The graph of y = 3 tan 2x − 1 has asymptotes at x = -45° and x = 45°, then every 90°. Choose tan in the simulation and you will see exactly these red dashed lines, and when the green point gets close to one the reading says "undefined".

Sin, cos and tan graphs at a glance

Knowing the three parent graphs makes any transformed graph easy to read.

Experiments to run in the simulation

Changing things yourself makes the formulas stick. Try these one by one, and predict before you drag.

• Keep the starting y = 2 sin(2x + 60°) + 1 and read y at x = 0: it is 2.73, which is 2 sin(60°) + 1.
• Raise only B from 1 to 4 and count how many waves fit on the screen; match the count to the Period reading.
• Set A = −2 and watch the wave flip, but notice the Amplitude reading does not go negative.
• Make C positive, then negative, and note which way the dot on the bracket moves each time.
• Set D = 3 and check the Maximum and Minimum readings against D ± |A|.
• Choose cos and set C = 90°. Do you get the graph of a sine?
• On tan, raise B and count the asymptotes: the larger B is, the closer together they sit.
• Switch the axis units to multiples of π and write the period as a multiple of π.

Solved problems

In each one, spot A, B, C, D first and then use the four rules. You can type the same numbers into the simulation to check.

Common mistakes

These slips cost marks. Skim the list before an exam.

• Taking the phase shift to be C. It is −C/B, because the start of a cycle comes from Bx + C = 0.
• Writing a negative amplitude when A is negative. Amplitude is a distance, so it is |A|.
• Multiplying the period by B instead of dividing; a bigger B makes the period smaller.
• Giving tan a period of 360° or 2π. It is 180° or π.
• Hunting for a maximum and minimum of tan; there are none.
• Mixing degrees and radians. If you work in multiples of π, keep everything in radians.

Where you meet these graphs in real life

Sound: a pure musical note is a sine wave of pressure. A larger amplitude is a louder note and a larger frequency factor B is a higher pitch.

Electricity: the alternating current from your wall socket follows a sine wave, as in Problem 7.

Tides and seasons: sea level and the length of the day rise and fall through the year like a cosine.

Wheels and pendulums: the height of a marked point on a turning wheel traces a sine graph against time.

Engineering and signals: pictures, music and wifi are broken into sums of sine waves to be stored and sent.

Exam corner

In CBSE class 11, the chapter on trigonometric functions asks you to draw and read the graphs of sin, cos and tan and to state their domain, range and period. In the later chapters on calculus the same graphs return, so it pays to know them cold.

When you see a question, first write the equation in the form y = A·f(Bx + C) + D and pull out the four numbers. Then apply the four rules for amplitude, period, shift and range. If you must sketch, mark the midline, one peak, one trough and the start of a cycle before drawing the smooth curve.

Quick revision

These few lines are enough for the night before the exam.

• A: amplitude |A|; a negative A flips the graph.
• B: period = basic period ÷ |B|.
• C: the wave moves by −C/B, positive to the right.
• D: midline y = D; range [D − |A|, D + |A|].
• tan: period π or 180°, has asymptotes, range is all real numbers.

Frequently asked questions