A sinusoidal function is a repeating wave

Explain it like I am five

Picture one seat on a Ferris wheel: up to the top, down to the bottom, up again. Its height over time draws a smooth wave. A function that repeats like this is periodic, and one full repeat is a cycle. A wave built from sine or cosine is sinusoidal (sigh-nuh-SOY-dul).

Numbers: hub (center) 20 meters up, radius 15 meters, one turn every 40 minutes, clock started as the seat passes hub height going up. Height: y = 15 sin( $π over 20$ t) + 20, t in minutes, where $π over 20$ = 2π ÷ 40 spreads one full turn, 2π, over 40 minutes. At t = 10, a quarter turn, the input in the parentheses is $π over 20$ × 10 = $π over 2$ and sin $π over 2$ = 1, so y = 35 meters, the top. At t = 30, sine is −1 and y = 5 meters, the bottom.

The general form y = A sin(ωx − φ) + B has four dials: A the size (15), ω (omega) the speed ( $π over 20$ ), φ (phi) the start (0 here) and B the hub height (20).

In plain words

Picture a mark on the rim of a Ferris wheel. As the wheel turns, the mark rises, reaches the top, falls, reaches the bottom, and rises again. If you draw its height against time, you get a smooth wave.

One complete trip through that motion is a cycle. Repeating the same motion makes a Periodic function. A sinusoidal function uses sine or cosine to make this particular smooth wave.

You can change the wheel's size, how quickly it turns, when you start watching, and how high its center sits. The four numbers in a sinusoidal formula control those four features. We will learn each control separately before combining them. The three study lists appear after you have practiced these features in Lesson 7.

Follow the horizontal input to the wave, then read its height.

Reminder

Function substitution. Replace every x with the chosen input: f(x) = 2x + 1 gives f(3) = 2(3) + 1 = 7.
Radians. A full turn is 2π radians; a quarter turn is $π over 2$ .
Basic values. sin 0 = 0, cos 0 = 1, and sin(− $π over 2$ ) = −1.

Why it works. On the unit circle, Sine reads vertical position and Cosine reads horizontal position. After a full turn of 2π radians, you return to the same point, so both coordinates repeat. A sinusoidal formula changes the scale and location of that repeating motion. Multiplying its output changes its height; changing its inside input changes how quickly or when it runs; adding outside changes its center. These moves preserve the smooth repeating shape.

RuleRule: a sine wave is written y = A sin(ωx − φ) + B, and a cosine wave is written y = A cos(ωx − φ) + B. For a nonconstant wave, A ≠ 0; use ω > 0 after rewriting signs. The inside input is in radians. Read ω as omega and φ as phi. A coefficient is a multiplier; ω is the coefficient of x. A parameter is a fixed setting such as A, ω, φ or B. T labels the period and h labels the signed horizontal shift. The subscript max means highest output and min means lowest output.

The same idea, five ways

Say it

A sine or cosine wave whose height and timing can change.

Write it

A sinusoidal function is a transformed sine or cosine function.

In math

y = A sin(ωx − φ) + B
y = A cos(ωx − φ) + B
A ≠ 0, ω > 0
Graph words: smooth repeated peaks and troughs.

A Ferris wheel gives the same up and down trip every revolution.

See it

Follow the horizontal input to the wave, then read its height.

The same idea, other ways

As a moving object

Imagine the height of a dot on a turning wheel. The same wheel position returns once per revolution.

A full turn brings the dot back to the same position.

As two familiar beginnings

Sine starts in the middle and rises. Positive cosine starts at a peak and falls. Both trace the same kind of motion.

Follow the horizontal input to the wave, then read its height.

As a repeated instruction

Adding 2π to a basic sine or cosine input is like restarting the same music loop. It puts you at the same stage of the loop.

Read	Feature	Formula
A	Amplitude: distance from middle to top	\|A\| = $y sub max − y sub min over 2$ ; $y_{m a x}$ is the highest output; $y_{m i n}$ is the lowest
ω	Period: width of one complete repeat	T = $2 π over ω$
φ and ω	Phase shift: sideways slide	h = $\frac{φ}{ω}$ ; h names this horizontal distance
B	Midline: horizontal center line	y = B; B = $y sub max + y sub min over 2$

.1Sine starts in the middle

A basic sine wave begins with height 0 and rises. Its Domain, the allowed inputs, is all real numbers. Its Range, the possible outputs, is [−1, 1].

Rule: sin 0 = 0.
Rule: −1 ≤ sin x ≤ 1.
Rule: sin(x + 2π) = sin x.

Follow the horizontal input to the wave, then read its height.

The same idea, five ways

Say it

Sine takes every real input and gives a height between negative one and one.

Write it

The domain is all real inputs and the range includes all outputs from −1 to 1.

In math

−∞ < x < ∞
(−∞, ∞)
{x | x is a real number}
−1 ≤ sin x ≤ 1
[−1, 1]

A wheel marker starts at center height and moves upward.

See it

Follow the horizontal input to the wave, then read its height.

Worked exampleFollow one unshifted sine cycle

The input is an angle in radians. Find the five turning and crossing heights of y = sin x in one full cycle, then see where they repeat.

Follow the horizontal input to the wave, then read its height.

The two endpoint columns both show 0; the intervening columns show one whole rise and fall.

Read the sine table at x = 0: sin 0 = 0.Sine is the vertical coordinate on the unit circle, whose starting point has height 0.
At x = $π over 2$ , sin x = 1; at x = π, sin x = 0.A quarter turn reaches the top of the circle, and a half turn returns to height 0.
At x = $3 π over 2$ , sin x = −1; at x = 2π, sin x = 0.Three quarters of a turn reaches the bottom, and a full turn reaches the start.
At x = 2π + $π over 2$ = $5 π over 2$ , the output is 1 again.The extra 2π is one full turn, which returns to the same circle point before the quarter turn.

Answer

One cycle runs from x = 0 to x = 2π.
The heights at the five key points are shown in the table.
The pattern then repeats.

Check The start and end both have height 0 and are followed by increasing heights. You have returned to the same stage of the motion.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Counterexample: every input where sine is 0 begins a new upward cycle.

Sine is also 0 at π, where it is falling.

✓ Instead: Use x = 0 and x = 2π for consecutive upward starts of the basic sine.

Tips and tricks

Tip: Say what the formula changes before calculating.

.2Cosine starts at an extreme

A basic cosine wave starts with height 1, its maximum, and falls. It accepts every real input and stays between −1 and 1.

Rule: cos 0 = 1.
Rule: −1 ≤ cos x ≤ 1.
Rule: cos(x + 2π) = cos x.

Follow the horizontal input to the wave, then read its height.

The same idea, five ways

Say it

Cosine takes every real input and gives a horizontal coordinate between negative one and one.

Write it

The domain is all real inputs and the range includes all outputs from −1 to 1.

In math

−∞ < x < ∞
(−∞, ∞)
{x | x is a real number}
−1 ≤ cos x ≤ 1
[−1, 1]

Watch the same wheel beginning at its top instead of its side.

See it

Follow the horizontal input to the wave, then read its height.

Worked exampleFollow one unshifted cosine cycle

Find the heights of y = cos x at the same five stages of a full turn. This asks where cosine begins and how it repeats.

Follow the horizontal input to the wave, then read its height.

Cosine begins and ends at 1, while sine begins and ends in the middle.

cos 0 = 1.Cosine reads the horizontal coordinate of the starting circle point (1, 0).
cos( $π over 2$ ) = 0 and cos π = −1.The top of the circle has horizontal coordinate 0; the leftmost point has horizontal coordinate −1.
cos( $3 π over 2$ ) = 0 and cos(2π) = 1.The bottom has horizontal coordinate 0, and a full turn returns to horizontal coordinate 1.

Answer

Cosine starts at its top height 1.
Its cycle also has length 2π.

Check Compare the first and last columns. Both give 1, and the next part of the wave falls from that height.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Counterexample: cosine starts at height 0 because sine does.

Cosine reads the starting horizontal coordinate 1, while sine reads vertical coordinate 0.

✓ Instead: cos 0 = 1 and sin 0 = 0.

Tips and tricks

Tip: Say what the formula changes before calculating.

.3Four parameters and their jobs

Use the table as a map. A and B change heights. ω and φ change the inside input, so they change timing. Do not try to memorize four disconnected facts; connect each letter to where it sits. Each fixed setting is a Parameter. A Coefficient is a multiplier, such as ω multiplying x.

Rule: A multiplies the sine or cosine output.
Rule: ω multiplies x inside.
Rule: φ is the signed number subtracted inside.
Rule: B is added outside.

A: height

ω: timing speed

φ with ω: horizontal slide

B: center height

Each letter has one job, determined by its position in the formula.

The same idea, five ways

Say it

Four fixed numbers control the wave's size, timing, sideways slide and center height.

Write it

Use the table as a map. A and B change heights. ω and φ change the inside input, so they change timing. Do not try to memorize four disconnected facts; connect each letter to where it sits.

In math

y = A sin(ωx − φ) + B
y = A cos(ωx − φ) + B
Inside input: ωx − φ
Outside: multiply by A, then add B

Set four controls on a wheel: size, turning speed, start time and center height.

See it

A: height

ω: timing speed

φ with ω: horizontal slide

B: center height

Each letter has one job, determined by its position in the formula.

Worked exampleIdentify the controls before finding distances

For y = 2 sin(2x − $π over 2$ ) + 1, identify A, ω, φ and B. This asks for the four signed numbers in their formula positions.

Follow the horizontal input to the wave, then read its height.

A = 2 and B = 1.The multiplier is 2 and the outside addition is 1.
ω = 2 and φ = $π over 2$ .Inside, 2 multiplies x and $π over 2$ is subtracted.

Answer

A = 2
ω = 2
φ = $π over 2$
B = 1

Check At x = 0 the inside is −, so y = 2(−1) + 1 = −1. Reading the 1 as an inside shift would give a different expression.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Counterexample: read +1 inside and +1 outside as the same parameter.

An inside addition changes what sine receives; an outside addition changes the resulting height.

✓ Instead: In 2 sin(2x − ) + 1, B = 1 is outside and φ = is subtracted inside.

Tips and tricks

Tip: Mark the parentheses before identifying the four signed numbers.

Strategy: step by step

Name the wave as sine or cosine.
Read the number multiplying the whole wave, A.
Read the coefficient of x inside, ω, and match the inside to ωx − φ.
Read the number added after the wave, B.
Use the next lessons to translate these parameters into distances.

Strategy

Strategy: recognize a wave formula

Is the function built from sine or cosine?

YesContinue by reading its four parameters.

NoDo not apply this section's four-feature rules, for example to tangent.

↓

Are A and ω both nonzero?

YesThere is a nonconstant wave.

NoThe formula is constant, so it has no least positive wave period.

Find the sin or cos.
Separate the outside multiplier and outside addition from the entire inside input.
Keep the letters attached to their positions; their jobs depend on those positions.

Worked exampleFollow one unshifted sine cycle

The input is an angle in radians. Find the five turning and crossing heights of y = sin x in one full cycle, then see where they repeat.

Follow the horizontal input to the wave, then read its height.

The two endpoint columns both show 0; the intervening columns show one whole rise and fall.

Read the sine table at x = 0: sin 0 = 0.Sine is the vertical coordinate on the unit circle, whose starting point has height 0.
At x = $π over 2$ , sin x = 1; at x = π, sin x = 0.A quarter turn reaches the top of the circle, and a half turn returns to height 0.
At x = $3 π over 2$ , sin x = −1; at x = 2π, sin x = 0.Three quarters of a turn reaches the bottom, and a full turn reaches the start.
At x = 2π + $π over 2$ = $5 π over 2$ , the output is 1 again.The extra 2π is one full turn, which returns to the same circle point before the quarter turn.

Answer

One cycle runs from x = 0 to x = 2π.
The heights at the five key points are shown in the table.
The pattern then repeats.

Check The start and end both have height 0 and are followed by increasing heights. You have returned to the same stage of the motion.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Counterexample: y = tan x is sinusoidal because it is a trig function.

Tangent has breaks and no finite highest and lowest heights. Its shape is not a smooth sine or cosine wave.

✓ Instead: Apply sinusoidal amplitude rules to sine and cosine waves.

✗ Not this: Counterexample: the four numbers can be read without noticing parentheses.

Adding 1 inside changes the input. Adding 1 outside changes every height.

✓ Instead: Separate the whole inside input from the outside addition.

Tips and tricks

Tip: Circle the entire inside input before identifying the four parameters.
Tip: Memorize the two starting heights; rebuild the intermediate heights from a turn.

Trap. Do not call every periodic function sinusoidal. Repeating stair steps repeat too, but they are not sine or cosine waves.

Keep in mind

Periodic is not the same as sinusoidal: a staircase pattern that repeats every 2 units is periodic, but it has sharp corners instead of a smooth wave.
The input of sine or cosine is in radians: sin $π over 2$ = 1, while sin 90 read as radians is about 0.894.
An unwritten number is 1 or 0: in y = sin x, A = 1, ω = 1, φ = 0 and B = 0.
A positive sine wave starts at its center going up, and a positive cosine wave starts at its top: sin 0 = 0 but cos 0 = 1.

Memory hookA, ω, φ, B: size, speed, start, center. On the wheel: its radius, how fast it spins, when you start watching, and how high the hub sits.

Flash cards: say the answer out loud, then flip

What is a periodic function?

A function whose outputs repeat after a fixed horizontal distance.

What is a sinusoidal function?

A repeating wave whose equation is a sine or a cosine that has been stretched, shifted and lifted: y = A sin(ωx − φ) + B, or the same equation with cos in place of sin.

How do you say ω and φ?

ω is omega (oh-MAY-guh).
φ is phi (fie, rhymes with pie).

In y = 6 sin(2x − 1) + 9, read A, ω, φ and B.

A = 6
ω = 2
φ = 1
B = 9

A Ferris wheel's hub is 12 meters up and its radius is 10 meters. What are the highest and lowest seat heights?

Highest: 12 + 10 = 22 meters
Lowest: 12 − 10 = 2 meters

A traffic light repeats its colors every 60 seconds. Is it sinusoidal?

No. It is periodic, but it jumps between colors instead of rising and falling smoothly like a sine wave.