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The Sine Wave y = A sin(ωx + φ)

The graph of y = A sin(ωx + φ) is a smooth wave. A is the amplitude (height), the period is T = 2π/ω, and the wave is shifted sideways by φ/ω (to the left if φ is positive). It models anything that repeats: a Ferris wheel, tides, sound and AC current.

🎬 Step-by-step story

  1. A red point goes round a circle at a steady speed. Look only at how HIGH the point is. It goes up, comes down, goes below, and comes back up.
  2. Now draw that height against the angle x. The curve you get is the wave y = sin x. It repeats every 2π. Its highest value is 1 and its lowest is −1.
  3. A is the amplitude. Make A = 2.5: the circle gets bigger, so the wave gets taller. It now goes between −2.5 and 2.5. The length of one wave does not change.
  4. ω is how fast the point goes round. Make ω = 2: the point goes round twice as fast, so the wave repeats twice as often. The period becomes T = 2π / ω = π.
  5. φ is where the point starts. Make φ = π/3: the point starts ahead, so the whole wave slides to the LEFT by φ / ω. The shape stays the same.
  6. Your turn. Change A, ω and φ with the sliders. Read the numbers below: amplitude, period and shift. Try to make the wave match one you imagine.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Why does a circle give a wave?

The height of a point moving at steady speed round a circle goes up, down, below and up again. Plotting that height against the angle gives the sine curve.

Why is the highest value of sin x equal to 1?

The circle has radius 1, so the point is never higher than 1. The wave copies the height of the point.

Does A change the period?

No. A only makes the circle (and the wave) bigger or smaller. One wave still takes the same length. Watch the length stay the same in the 3D.

Why is the period 2π/ω and not 2πω?

A faster point finishes a turn sooner, so the wave is shorter. Bigger ω means a smaller period, so we divide.

Why does positive φ move the wave to the left?

The point starts φ radians ahead, so it reaches every height earlier. The same heights appear at a smaller x, which is to the left.

Can I combine all three changes?

Yes. They act on different things: A on height, ω on repeat length, φ on sideways position. Use the sliders to mix them.

Where the sine wave comes from

Take a point moving round a circle at a steady speed. Its height above the middle of the circle, measured against the angle it has turned, makes the sine wave y = sin x.

The wave repeats after every 2π. This repeat length is called the period. The highest value is 1 and the lowest is −1. Anything whose graph has this smooth up-and-down repeating shape is called sinusoidal.

Amplitude A: how tall

In y = A sin(ωx + φ), the number A multiplies the height. The wave now goes between −|A| and +|A|. The number |A| is the amplitude. Every y-value is multiplied by A, so the graph is stretched up and down. The zeros of the wave do not move.

Example: y = 3 sin x goes from −3 to 3.

ω: how fast, and the period

The number ω (omega, take ω > 0) tells how fast the angle grows. The wave repeats when ωx grows by 2π, so the period is T = 2π / ω. The frequency is how many waves fit in one unit: f = 1/T = ω / 2π.

Bigger ω squeezes the wave sideways, smaller ω stretches it. Example: y = sin 2x has period π; y = sin(x/2) has period 4π.

φ: the sideways shift (phase shift)

Write the inside as ωx + φ = ω(x + φ/ω). So the wave y = sin(ωx + φ) is the wave y = sin(ωx) moved by φ/ω units. If φ > 0 the graph moves to the left; if φ < 0 it moves to the right. This is the phase shift. The wave crosses zero going upward at x = −φ/ω.

You may also see a number added at the end: y = A sin(ωx + φ) + k. The +k lifts the whole wave up by k, so the middle line is y = k.

Drawing y = A sin(ωx + φ) from y = sin x

Do the changes one by one, starting from the basic wave y = sin x:

  1. Draw y = sin x.
  2. Squeeze it sideways so the period is 2π/ω. This gives y = sin ωx.
  3. Slide it left by φ/ω (right if φ is negative). This gives y = sin(ωx + φ).
  4. Stretch it up and down by A. This gives y = A sin(ωx + φ).
  5. Check the sketch: the maximum is A, the minimum is −A, and the first upward zero is at x = −φ/ω.

Using sine waves as models

Many repeating things are modelled by a sine wave. Ferris wheel: the height of a seat is h = centre height + radius · sin(...). Tides: water depth rises and falls about every 12 hours. AC electricity: the voltage V = V₀ sin(2πft); in India f = 50 Hz. Sound: a pure note is a sine wave; its amplitude is loudness and its frequency is pitch.

To build a model, read off A from the highest and lowest values, ω from the time of one repeat (ω = 2π / T), and φ from where the wave starts.

Try it at home

Tie a small stone to a string and swing it slowly in a circle. Ask a friend to stand in front and watch only the height of the stone. Count how many seconds one full turn takes: that is the period T. Then swing it in a bigger circle (bigger A) and faster (bigger ω) and see what changes. Predict first, then check on the 3D sliders.

Key formulas and definitions

Worked examples

1. For y = 3 sin 2x, find the amplitude, period and range.

A = 3, so the amplitude is 3. ω = 2, so T = 2π/2 = π. The range is −3 ≤ y ≤ 3.

2. For y = 2 sin(x + π/3), find the phase shift and where it crosses zero going upward.

ω = 1 and φ = π/3, so the shift is φ/ω = π/3 to the left. The period is 2π. The upward zero is at x = −φ/ω = −π/3.

3. Describe y = sin(2x − π/2).

Write 2x − π/2 = 2(x − π/4). So the wave is y = sin 2x shifted right by π/4. The period is π. The amplitude is 1.

4. A wave has maximum 5, minimum −5, period 4π and starts at 0 going up at x = 0. Write its equation.

A = 5. T = 4π gives ω = 2π/4π = 1/2. Starts at 0 going up, so φ = 0. The equation is y = 5 sin(x/2).

5. A Ferris wheel has radius 8 m. Its centre is 10 m above the ground. It takes 12 minutes for one turn, and a seat is at the lowest point at t = 0. Write the height h(t) and find h at t = 3 and t = 6.

A = 8, middle line k = 10. T = 12 so ω = 2π/12 = π/6. At the lowest point sin must be −1 at t = 0, so φ = −π/2. h(t) = 10 + 8 sin(πt/6 − π/2). At t = 3: sin(π/2 − π/2) = 0, so h = 10 m. At t = 6: sin(π − π/2) = 1, so h = 18 m.

6. The AC mains voltage is V = 325 sin(100πt) (t in seconds). Find the period, the frequency, and V at t = 0.005 s.

ω = 100π, so T = 2π/100π = 0.02 s. f = 1/T = 50 Hz. At t = 0.005: 100π × 0.005 = π/2, sin(π/2) = 1, so V = 325 volts (the peak).

7. In a simple model, the depth of water in a harbour is d = 5 + 2 sin(πt/6) metres (t in hours). Find the period, the highest and lowest depth, and the first time of high tide.

ω = π/6, so T = 2π ÷ (π/6) = 12 hours. Highest depth = 5 + 2 = 7 m and lowest = 5 − 2 = 3 m. High tide when sin = 1: πt/6 = π/2, so t = 3 hours.

Common mistakes

Practice quiz

1. The amplitude of y = 4 sin(3x) is:
2. The period of y = sin(2x) is:
3. The graph of y = sin(x + π/2) is the graph of sin x moved:
4. The range of y = 5 sin x is:
5. In y = A sin(ωx + φ), the period depends on:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What does y = A sin(ωx + φ) mean?

It is the general sine wave. A sets how tall it is, ω sets how fast it repeats (period 2π/ω), and φ slides it sideways.

How do I find the phase shift?

Divide φ by ω. The shift is φ/ω to the left when φ is positive and to the right when it is negative.

Where do we use sinusoidal functions?

In tides, Ferris wheels, sound, light, and AC electricity. Any smooth repeating up-and-down quantity can be modelled this way.

Where this is taught

China高一Ch.5 Trigonometric functions

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