My answer is different from the book's answer. Is it wrong?

Not necessarily. Check a maximum and a minimum in both equations. If they give the same values at the same $x$, they draw the same wave. For example $y = 3\sin(2x) + 1$ and $y = 3\cos\left(2\left(x - \frac{\pi}{4}\right)\right) + 1$ are the same graph.

How do I find the period if I only see part of a wave?

Measure the horizontal distance between two neighboring maxima (one full period), or between a maximum and the next minimum (half a period) and double it.

What if the problem uses time in hours or days instead of radians?

The recipe does not change: $B = \frac{2\pi}{\text{period}}$ with the period in the same unit as $x$. A tide with period $12$ hours has $B = \frac{\pi}{6}$, a yearly pattern has $B = \frac{2\pi}{365}$.

Teacher Resources

Writing Trig Equations from Graphs

Q: Should I use sine or cosine?

Both are correct. Use the one that makes $C$ easiest: cosine if you can read a maximum clearly, sine if you can read where the wave crosses the midline going up.

Read the maximum, minimum, period and a starting point from a wave and turn them into y = A sin(B(x - C)) + D or a cosine equation.

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Definition

In the previous lessons you went from an equation to a graph. Now we go the other way: you look at a wave (or read a description of one) and write its equation.

Every sine or cosine wave can be written in one of these forms:

y = A\sin(B(x - C)) + D \qquad y = A\cos(B(x - C)) + D

You only need four readings from the graph: the maximum, the minimum, the period and one starting point.

Letter	What it means	How to find it
$D$	midline (vertical shift)	$D = \frac{\text{max} + \text{min}}{2}$
$A$	amplitude	$A = \frac{\text{max} - \text{min}}{2}$
$B$	how fast the wave repeats	$B = \frac{2\pi}{\text{period}}$
$C$	phase shift (horizontal shift)	where one cycle starts

**Choosing

C

:**

Cosine starts at a maximum: $C$ = the $x$ of a maximum.
Sine starts on the midline going up: $C$ = the $x$ where the graph crosses the midline while rising.

The same graph has many correct equations. You can start at any maximum (add or subtract a whole period to

C

), switch between sine and cosine (move

C

by a quarter period), or use a negative

A

(then cosine starts at a minimum). All of them draw the same wave.

Try it now

A sinusoid has a maximum of

8

and a minimum of

2

. What is its midline?

Worked Examples

A graph has a maximum of

5

x = 0

, a minimum of

1

, and a period of

2\pi

. Write its equation.

Find the midline D

D = \frac{5 + 1}{2} = \frac{6}{2}

→

D = 3

Find the amplitude A

A = \frac{5 - 1}{2} = \frac{4}{2}

→

A = 2

Find B from the period

B = \frac{2\pi}{2\pi}

→

B = 1

Choose the form and C

The wave starts at a maximum at

x = 0

, so use cosine with

C = 0

Answer: $y = 2\cos(x) + 3$

Common Mistakes

Using the maximum as the amplitude (max $7$ , min $3$ , so $A = 7$ )

Why it's wrong: The amplitude is measured from the midline, not from the x-axis. The wave is only $2$ units above its midline $y = 5$ .

Correct: $A = \frac{\text{max} - \text{min}}{2} = \frac{7 - 3}{2} = 2$ , and the $5$ goes into $D$ .

Writing the period as B (period $12$ , so $B = 12$ )

Why it's wrong: B is not the period. A bigger B makes the wave repeat faster, so the period gets shorter.

Correct: $B = \frac{2\pi}{\text{period}} = \frac{2\pi}{12} = \frac{\pi}{6}$ .

Getting the sign of C wrong: a maximum at $x = 2$ written as $\cos(B(x + 2))$

Why it's wrong: Inside the bracket the shift has the opposite sign. $(x + 2)$ moves the wave to the left.

Correct: A maximum at $x = 2$ means $\cos(B(x - 2))$ . Check: at $x = 2$ the bracket is $0$ and $\cos(0) = 1$ .

Using sine and putting C at the maximum

Why it's wrong: Sine does not start at a maximum. At $x = C$ the sine wave is on the midline, going up.

Correct: For sine take the midline crossing going up, which is a quarter period before the maximum. Or use cosine and take the maximum.

Interactive Visual

Function graph

A = 1B = 1C = 0D = 0

y = sin(x)

Challenge 1 of 5

Move the sliders until your curve lies exactly on the dashed target curve.

How close

0 of 5 solved

Tap or hover over the graph to read off values.

y = A·sin(B(x − C)) + DTarget curve

Function graph

A = 1B = 1C = 0D = 0

y = cos(x)

Challenge 1 of 3

Move the sliders until your curve lies exactly on the dashed target curve.

How close

0 of 3 solved

A: smaller
B: right
C: right
D: bigger

Tap or hover over the graph to read off values.

y = A·cos(B(x − C)) + DTarget curve

Function graph

Tap or hover over the graph to read off values.

water levelmidline y = 5

Interactive Sandbox

Interactive Grapher

Tap or hover over the graph to read off values.

Try these examples:

2cos(x) + 3

Expression Calculator

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Practice Problems

18 problems

Problem 1 of 18

Easy

A sinusoid has a maximum of

8

and a minimum of

2

. What is its midline?

Why It Matters

Anything that repeats in a smooth cycle can be described by a sine or cosine equation, and in real life you almost never get the equation first. You get data: a highest value, a lowest value and how long one cycle takes.

Ferris wheels: your height above the ground rises and falls every turn
Tides: the sea level goes between high and low tide about every 12 hours
Daylight: the number of daylight hours peaks in summer and repeats every 365 days
Springs and pendulums: a weight bounces up and down in a fixed time
Temperature: the average daily temperature follows a yearly wave

Once you can write the equation, you can predict: the height after 2 minutes, the water level at 6:00, the daylight on your birthday.

Real World Applications

Hours of Daylight

Astronomers and solar engineers model the length of the day with a cosine wave that repeats every year.

Example:

In a northern city the longest day (day $172$ , around June 21) has $16$ hours of daylight and the shortest has $8$ . A model is $y = 4\cos\left(\frac{2\pi}{365}(d - 172)\right) + 12$ .

1Try It Yourself

Another city has $15$ hours of daylight on its longest day and $9$ hours on its shortest. The pattern repeats every $365$ days.

What value of B goes into the model $y = A\cos(B(d - C)) + D$ ?

Step 1: Write the mathematical expression

Use $B = \frac{2\pi}{\text{period}}$ :

Ferris Wheels

Ride designers use the height equation to check how fast riders climb and how long they spend near the top.

Example:

A wheel $50$ m across with its center $30$ m high turns once every $12$ minutes. Boarding at the bottom gives $h(t) = -25\cos\left(\frac{\pi}{6}t\right) + 30$ .

2Try It Yourself

You board the wheel above ( $50$ m across, center $30$ m high, one turn every $12$ minutes) at the bottom at $t = 0$ .

How high are you after $2$ minutes (in m)?

Step 1: Write the mathematical expression

Put $t = 2$ into $h(t) = -25\cos\left(\frac{\pi}{6}t\right) + 30$ :

A Weight on a Spring

Engineers model vibrations (car suspensions, bridges, sensors) with the same equations as a weight bouncing on a spring.

Example:

A weight bounces between $4$ cm and $16$ cm above a table and needs $2$ seconds for one full bounce.

3Try It Yourself

The weight above is let go at its highest point ( $16$ cm) at $t = 0$ . One full bounce takes $2$ seconds, the lowest point is $4$ cm.

How high is the weight after $\frac{1}{3}$ of a second (in cm)?

Step 1: Write the mathematical expression

Use $y = 6\cos(\pi t) + 10$ with $t = \frac{1}{3}$ :

Key Takeaways

1Midline: $D = \frac{\text{max} + \text{min}}{2}$ ; amplitude: $A = \frac{\text{max} - \text{min}}{2}$
2From the period: $B = \frac{2\pi}{\text{period}}$ (from a maximum to the next minimum is half a period)
3Cosine: $C$ is the $x$ of a maximum; sine: $C$ is where the wave crosses the midline going up
4A wave that starts at a minimum is a cosine with a negative $A$
5One graph has many correct equations: add whole periods to $C$ , move a quarter period to switch sine and cosine, or flip the sign of $A$ and move half a period
6Always check one point (a maximum or a minimum) in your finished equation

Frequently Asked Questions

Both are correct. Use the one that makes

C

easiest: cosine if you can read a maximum clearly, sine if you can read where the wave crosses the midline going up.

Glossary

Midline: The horizontal line $y = D$ halfway between the maximum and the minimum of the wave
Amplitude: The distance from the midline to a maximum: $\frac{\text{max} - \text{min}}{2}$
Period: The horizontal length of one full cycle; $B = \frac{2\pi}{\text{period}}$
Phase shift: The horizontal shift $C$ : where a cycle of the basic sine or cosine wave starts
Sinusoidal model: An equation $y = A\sin(B(x - C)) + D$ or $y = A\cos(B(x - C)) + D$ that describes a real repeating quantity

Formula Card

General forms

y = A\sin(B(x - C)) + D

y = A\cos(B(x - C)) + D

Every sine or cosine wave fits one of these forms

Midline

D = \frac{\text{max} + \text{min}}{2}

The average of the highest and lowest value

Amplitude

A = \frac{\text{max} - \text{min}}{2}

Half of the total height of the wave

B from the period

B = \frac{2\pi}{\text{period}}

The period is the distance between two maxima

Starting point C

\cos

C = x_{\text{max}}

;

\sin

C = x_{\text{midline, up}}

Cosine starts at a maximum, sine on the midline going up

Today's challengeOne problem, three tries

Writing Trig Equations from Graphs

Definition

Worked Examples

Common Mistakes

Interactive Visual

Function graph

Function graph

Function graph

Interactive Sandbox

Interactive Grapher

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Hours of Daylight

Ferris Wheels

A Weight on a Spring

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

General forms

Midline

Amplitude

B from the period

Starting point C

More in This Topic

Graphing the Sine Function

Graphing Cotangent

Graphing the Cosine Function

Amplitude and Period