Sinusoid amplitude and period
Read the shape and transformations of a sinusoidal model.
Sinusoid amplitude and period is one of 2 graphs formulas in the trigonometry section of this library, and it is used at high school · ap level.
Why sinusoid amplitude and period works
A sine curve completes one cycle while its input travels 2π. Multiplying the input by B makes the input run B times faster, so the cycle finishes in 2π/|B| instead. The absolute value bars appear because a negative B reflects the picture left to right without changing how long the repeat takes, and the same logic makes amplitude |A|.
What each symbol means
$C$ is horizontal shift and $D$ the midline.
Sinusoid amplitude and period: when it holds
$B\ne0$; for degrees use period $360^\circ/|B|$.
When it stops applying
Amplitude and period only describe a pure sinusoid. A curve like y = x sin x oscillates with a growing swing, so |A| has nothing to point at, and a sum of two sines with unrelated periods may not repeat on any regular schedule at all.
Sinusoid amplitude and period: a worked example
$y=3\sin(2x)+1$ has amplitude $3$, period $\pi$, and midline $y=1$.
The mistake to avoid
What people do: Students multiply when reading the period, writing the period of y = sin 4x as 8π.
Why it goes wrong: Backwards: a larger B squeezes the wave, so cycles get shorter, not longer. Eight π is about 25.1, when the true cycle is over before the input reaches 1.6.
Do this instead: Divide instead of multiplying: the period is 2π/4 = π/2, about 1.571. A quick sketch confirms it, since y = sin 4x fits four complete cycles into the same span where y = sin x fits one.
Sinusoid amplitude and period: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $C$ is horizontal shift and $D$ the midline.
- Check the conditions before substituting. $B\ne0$; for degrees use period $360^\circ/|B|$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most trigonometry slips.
Where this formula fits
- Subject
- Trigonometry formulas — 18 entries in this library
- Topic
- Graphs
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where sinusoid amplitude and period comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Solving Trigonometric Equations — the lesson behind this formula: find every angle in a required interval.
- Trigonometry Calculator — check your substitution and the value it produces.
- Study trigonometry — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 18 trigonometry formulas — the full grouped reference, or the complete formula library.
Questions about sinusoid amplitude and period
Is the amplitude the whole distance from the top to the bottom?
No, it is half of that. A wave running between 1 and 9 travels 8 units in total, so the amplitude is 4, measured from the center line at y = 5 up to the peak.
Does shifting the graph change the period?
No. Sliding a wave left, right, up, or down moves it without stretching it, so the cycle length depends only on B. Only the horizontal stretch factor can change how often the pattern repeats.
What is the period when the angle is measured in degrees?
Use 360/|B| instead of 2π/|B|, since a full turn is 360 degrees there. For y = sin(3x) in degree mode, one cycle takes 120 degrees.
What happens if A is negative?
The graph flips upside down across the center line, but the amplitude stays positive because it is a distance. y = −3 sin x still swings 3 units either side; it just starts by going down instead of up.