Mathematics › 🌌 Grade 11 › Periodicity Lab: Amplitude and Period of Sine and Cosine

Periodicity Lab: Amplitude and Period of Sine and Cosine

Two dials control a sine wave's whole silhouette — one stretches it taller, the other squeezes its repeats closer together.

Grade 11Algebra 2 / Pre-Calculus

Periodicity Lab: Amplitude and Period of Sine and Cosine — illustration

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The big idea: Sine and cosine repeat forever: whatever they do over one stretch of 2π, they do again and again. Two transformations reshape that basic wave without breaking the repetition. The coefficient in front sets the amplitude — how tall the wave swings above and below its midline. The coefficient multiplying the angle sets the period — how long one full cycle takes. Learn to read those two numbers and you can graph any pure sine or cosine at a glance.

🎯 By the end, you'll be able to

Explain what makes sine and cosine periodic functions
Identify the amplitude of y = a sin(bx) as |a|
Compute the period of y = a sin(bx) as 2π / |b|
Sketch a transformed sine or cosine using its amplitude, period, and midline

📎 You should already know

The unit circle and radian measure
Graphing sine and cosine

A wave that repeats forever

As a point travels around the unit circle, its height traces the sine curve and its horizontal position traces the cosine curve. Because the point keeps going around, both curves repeat: after one full lap of 2π radians, the pattern starts over exactly.

A function that repeats like this is called periodic, and the length of one full repeat is its period. For the basic sin x and cos x, that period is 2π.

🔑 Amplitude and period, the two dials

In y = a·sin(bx): the number a controls the amplitude — how far the wave rises above and falls below its middle line, equal to |a|. The number b controls how many cycles are packed into 2π, and therefore the period, which is 2π ÷ |b|.

\[ y = a\,\sin(bx): \quad \text{amplitude} = |a|, \qquad \text{period} = \dfrac{2\pi}{|b|} \]

Amplitude is the height of the swing; period is how long one full cycle lasts. b squeezes the wave horizontally.

Amplitude stretches it vertically

Multiplying by a stretches the wave up and down. With a = 3, the peaks reach +3 and the troughs drop to −3; the amplitude is 3. A negative a also flips the wave upside down, but the amplitude is still |a|, since amplitude measures a distance and cannot be negative.

🎮 Periodicity Lab LIVE

Change amplitude and period of a sine wave and watch it stretch and squeeze.

Period squeezes it horizontally

The coefficient b speeds up the input. A bigger b makes the angle race through its cycle faster, so the wave completes a full repeat in a shorter horizontal distance — the graph looks squeezed. A smaller b (between 0 and 1) stretches the wave out, so each cycle takes longer.

Note the inverse relationship: b and the period pull in opposite directions, because period = 2π ÷ |b|.

\[ \sin(\theta + 2\pi) = \sin\theta \]

The defining periodic identity: adding a full period of 2π to the angle leaves the value unchanged.

📝 Worked example: State the amplitude and period of y = 4 sin(2x), and where its first peak occurs.

Amplitude is \( |a| = |4| = 4 \): the wave swings between +4 and −4.
Period is \( \dfrac{2\pi}{|b|} = \dfrac{2\pi}{2} = \pi \): one full cycle every \( \pi \) units.
A basic sine peaks a quarter-period after the start; a quarter of \( \pi \) is \( \pi/4 \).

✓ Amplitude 4, period π, with the first peak (value 4) at x = π/4.

📝 Worked example: A cosine wave has amplitude 5 and period 8π. Write an equation for it.

Amplitude 5 means \( a = 5 \).
Set the period formula equal to the target: \( \dfrac{2\pi}{|b|} = 8\pi \).
Solve for b: \( |b| = \dfrac{2\pi}{8\pi} = \dfrac{1}{4} \).

✓ One such wave is y = 5 cos(x/4).

⚠️ Period is 2π over b, not 2π times b

A frequent slip is to multiply: writing the period as 2π·b. It is a division: period = 2π ÷ |b|. A large b makes the period small (the wave repeats more often), which is the opposite of what multiplying would suggest.

Check your understanding

1. What is the amplitude of y = 7 sin(x)?

Amplitude is |a| = |7| = 7 — the wave swings from +7 to −7.

2. What is the period of y = sin(3x)?

Period = 2π/|b| = 2π/3; a larger b packs the cycles closer together.

3. What is the amplitude of y = −2 cos(x)?

Amplitude is a distance, |a| = |−2| = 2; the negative only flips the wave upside down.

4. A sine wave repeats every 4π units. What is b in y = sin(bx)?

Set 2π/|b| = 4π, so |b| = 2π/4π = 1/2.

5. Increasing b in y = sin(bx) does what to the graph?

A larger b shortens the period 2π/b, squeezing more cycles into the same width.

✅ Key takeaways

Sine and cosine are periodic: they repeat exactly every 2π radians.
In y = a sin(bx), the amplitude is |a| — the height of the swing above and below the midline.
A negative a flips the wave but leaves the amplitude |a| unchanged.
The period is 2π/|b|: a larger b squeezes the cycles closer, a smaller b stretches them out.
Period is a division (2π ÷ |b|), not a multiplication.