Sinusoidal model
Model periodic behavior with amplitude, period, shift, and midline.
Sinusoidal model is one of 2 graphs formulas in the trigonometry section of this library, and it is used at high school · ap level.
Why sinusoidal model works
A plain sine wave already rises and falls forever between −1 and 1. Multiplying by A stretches that swing, adding D lifts the whole wave to a new center line, and wrapping the input in B(x − C) rescales and slides the horizontal axis. Four numbers are enough because a wave has exactly four adjustable features: height, center, speed, and start.
What each symbol means
$|A|$ is amplitude, $2\pi/|B|$ is period, $C$ is phase shift, and $D$ is midline.
Sinusoidal model: when it holds
$B\ne0$; use radians unless the variable is explicitly measured in degrees.
When it stops applying
The model assumes every cycle is the same height and the same length. Data that fades out, such as a swinging pendulum losing energy, needs a decaying factor like e^(−kx) multiplying the sine, and this fixed-amplitude form will drift away from it.
Sinusoidal model: 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 fit the height by using the maximum value as A and the minimum as D.
Why it goes wrong: A is the distance from the center line up to the peak, not the peak itself, and D is the middle of the swing, not the bottom.
Do this instead: Compute both from the extremes: A = (max − min)/2 and D = (max + min)/2. Data topping out at 9 and bottoming at 1 gives A = 4 and D = 5, so the wave swings 4 either side of the line y = 5.
Sinusoidal model: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $|A|$ is amplitude, $2\pi/|B|$ is period, $C$ is phase shift, and $D$ is midline.
- Check the conditions before substituting. $B\ne0$; use radians unless the variable is explicitly measured in degrees.
- 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 sinusoidal model 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 sinusoidal model
How do I find B if I know the period?
Divide 2π by the period. A wave that repeats every 12 months has B = 2π/12 = π/6, about 0.524, and a wave that repeats every π units has B = 2.
Can I model the same data with cosine instead?
Yes. Cosine starts at a peak while sine starts at the center line going up, so the two forms differ only by a quarter-period shift. Pick whichever makes C simpler for your starting point.
What if my equation looks like y = A sin(Bx − C) instead?
Then the shift is C/B, not C. Factor the B out first: sin(2x − π/3) is sin(2(x − π/6)), so the graph slides right by π/6, about 0.524, which is half of what the bare C suggests.
How do I choose C for real-world data?
Find an x where the data crosses its center line while rising, and use that as C. For monthly temperatures in the northern hemisphere, that crossing lands near April, so C is about 3 or 4 when x counts months.