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

California Pre-Calculus • Standard F-TF.5 • Trigonometric Functions

Sinusoidal Modeling is a topic in Trigonometric Functions in the California Common Core State Standards. It is aligned to Standard F-TF.5, which requires students to model periodic phenomena with trigonometric functions.

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