Pre-Calculus Trigonometric functions

Sinusoidal modeling (tides, sound, daylight)

Q: How do you get b from the period?

b equals 2 pi over the period. For a 24-hour cycle, b equals pi over 12.

20 practice questions 0 video lessons Theory + worked examples

Sinusoidal Modeling

Texas Precalculus (TEKS) • Standard P.2(O) • Trigonometric Functions

Sinusoidal Modeling is a topic in Trigonometric Functions in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.2(O), which requires students to develop and use sinusoidal functions to model situations.

Sinusoidal modeling fits \(y=a\sin(b(t-c))+d\) to periodic data such as tides, daylight, or temperature, reading amplitude and midline from the maximum and minimum.

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Practice questions

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Theory

Many real quantities rise and fall on a regular cycle — daily temperature, ocean tides, hours of daylight, a rider's height on a Ferris wheel. These are modeled by a sinusoid

\[y=a\sin\big(b(t-c)\big)+d,\]

where you read the parameters from the situation:

Amplitude \(a\) \(=\dfrac{\max-\min}{2}\).
Midline \(d\) \(=\dfrac{\max+\min}{2}\).
\(b\) from the period: \(b=\dfrac{2\pi}{\text{period}}\).
Phase shift \(c\) lines the curve up with a known max, min, or midpoint.

Choose sine or cosine to minimize the shift: cosine starts at a max (or, negated, a min); sine starts at the midline going up.

A daily-temperature model oscillating between \(45^\circ\) and \(75^\circ\) about \(60^\circ\).

A Ferris-wheel height model: start at the bottom, up to the top, and back.

Reading the parameters from data:

\[a=\dfrac{\max-\min}{2},\quad d=\dfrac{\max+\min}{2},\quad b=\dfrac{2\pi}{\text{period}}\]

amplitude is half the range; midline is the average of max and min; b is 2 pi over the period

Use a negative cosine \(-a\cos\) when the cycle starts at its minimum, as a Ferris wheel does at the bottom.

How to build a sinusoidal model

Amplitude and midline from the maximum and minimum values.
Find \(b\) from the period.
Pick sine or cosine and the shift \(c\) to match a known starting feature.
Write and check the model against a couple of known points.

Example 1 — Build a temperature model

Daily temperature ranges from \(45^\circ\) at 2 a.m. to \(75^\circ\) at 2 p.m. Find the amplitude and midline.

Solution

Amplitude is half the range; midline is the average.

\(\text{amplitude}\)	\(=\)	\(\dfrac{75-45}{2}=15\)
\(\text{midline}\)	\(=\)	\(\dfrac{75+45}{2}=60\)

amplitude 15, midline 60 degrees

Example 2 — Find b from the period

The temperature cycle repeats every 24 hours. Find \(b\).

Solution

Use \(b=\dfrac{2\pi}{\text{period}}\).

\(b\)

\(=\)

\(\dfrac{2\pi}{24}=\dfrac{\pi}{12}\)

b equals pi over 12

Example 3 — Assemble the equation

Write a sine model for the temperature, peaking at \(t=14\) (2 p.m.).

Solution

Combine amplitude 15, midline 60, \(b=\dfrac{\pi}{12}\), and a phase shift so the peak is at \(t=14\). A sine peaks a quarter period after its shift, so shift to \(t=8\).

\[T(t)=15\sin\!\Big(\dfrac{\pi}{12}(t-8)\Big)+60\]

model is 15 sine of pi over 12 times t minus 8, plus 60

Example 4 — Ferris wheel height

A Ferris wheel of radius 25 ft has its center 30 ft up and completes a turn in 40 s, starting at the bottom. Write a height model.

Solution

Starting at the bottom means a downward cosine: midline 30, amplitude 25, period 40 so \(b=\dfrac{2\pi}{40}=\dfrac{\pi}{20}\).

\[h(t)=-25\cos\!\Big(\dfrac{\pi}{20}t\Big)+30\]

height model is negative 25 cosine of pi over 20 times t, plus 30

Common pitfalls

Amplitude is half the range, not the max. Use \(\dfrac{\max-\min}{2}\).

Match the start with the right function. Cosine for a max/min start, sine for a midline start; use \(-\) to flip.

Convert the period to \(b\). A 24-hour cycle gives \(b=\dfrac{\pi}{12}\), not \(24\).

Frequently asked questions

What kinds of situations are modeled with sinusoids?

Anything periodic: daily temperature, tides, hours of daylight, sound waves, and heights on a rotating wheel.

How do you find the amplitude and midline from data?

Amplitude is half the difference of the maximum and minimum; the midline is their average.

How do you get b from the period?

\(b=\dfrac{2\pi}{\text{period}}\). For a 24-hour cycle, \(b=\dfrac{\pi}{12}\).

Should I use sine or cosine?

Whichever needs the smaller shift: cosine starts at a maximum (or a minimum if negated), sine starts at the midline heading up.

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Inverse trig functions (arcsin, arccos, arctan)

Sinusoidal modeling (tides, sound, daylight)

Sinusoidal Modeling

📖 Theory

How to build a sinusoidal model

Common pitfalls

Frequently asked questions

Theory