Graphs and Periodic Motion

Almost everything that repeats — a heartbeat, a swinging pendulum, the alternating current in your wall socket, the daily rise and fall of the tide, the pressure wave of a musical note — can be described by the same two curves: sine and cosine. That is not a coincidence. It is one of the deepest and most useful facts in all of mathematics, and this page is about why it is true and how to use it.

When you learned right-triangle trigonometry, sine and cosine were ratios of sides. On the unit circle they became coordinates of a point going around and around. Here we take the final step: we let that circular motion unwind into a graph over time, and out comes a smooth, endlessly repeating wave. Learning to read and build these waves — to see a real oscillation and write down its equation — is one of the most transferable skills you will ever pick up in mathematics.

Learning Objectives

By the end of this page, you should be able to:

• Graph $y = \sin x$ and $y = \cos x$ and explain how each comes from motion around the unit circle.
• Identify and control the four parameters of a general sinusoid: amplitude, period, frequency, and phase shift (plus vertical shift).
• Convert fluently between period and frequency, and between degrees and radians on a graph.
• Build an equation of the form $y = A\sin(B(x - C)) + D$ to model a real oscillation (tide, spring, sound, AC current).
• Explain, at least in outline, Fourier's discovery that every periodic signal is a sum of sines — and why that reshaped modern technology.

Quick Answer

The graph of $y = \sin x$ is a smooth wave that starts at $0$, rises to $1$, returns through $0$ down to $-1$, and comes back to $0$ after one full turn of the unit circle ($2\pi$ radians). Cosine is the identical wave shifted left by a quarter turn, starting at its peak. A general sinusoid is written $y = A\sin\!\big(B(x - C)\big) + D$, where $|A|$ is the amplitude (half the peak-to-trough height), the period is $2\pi/|B|$ (how long one cycle takes), the frequency is $|B|/(2\pi)$ (cycles per unit), $C$ is the phase shift (horizontal slide), and $D$ is the vertical shift (the midline). Because uniform circular motion projects onto a line as exactly this shape, sinusoids describe every simple oscillation — and Fourier showed that combinations of them describe every periodic signal at all.

Where It Came From

The story begins with a very practical need: prediction of things that repeat. Ancient astronomers tracked the periodic positions of the Sun, Moon, and planets, and Greek and later Islamic mathematicians built tables of chords and sines precisely to forecast eclipses, tides, and calendar events. Sine literally entered mathematics as a tool for modeling cyclic motion in the sky.

But the graph of the sine function — the wave — waited for the 17th and 18th centuries, when mathematicians began treating trigonometric ratios as functions of a continuous variable rather than entries in a table. Once you plot $\sin x$ for every $x$, the wave appears, and its connection to physical oscillation becomes obvious.

The pivotal motivation came from physics. In the 1600s and 1700s, scientists studying vibrating strings, swinging pendulums, and springs kept arriving at the same equation of motion, whose solution is a sinusoid. A mass on a spring, pulled and released, traces out $\cos$ in time. This is called simple harmonic motion, and it appears anywhere a system is pushed back toward equilibrium by a force proportional to how far it has strayed. Nature is full of such systems, which is why the same wave shape keeps reappearing.

Then came the revolution. Around 1807, the French mathematician and physicist Joseph Fourier was trying to solve a mundane-sounding problem: how heat spreads through a solid body. To do it, he made an astonishing claim — that any periodic function, no matter how jagged or complicated (even a square wave with sharp corners), can be written as a sum of ordinary sines and cosines of different frequencies. Contemporaries like Lagrange thought this was too good to be true, and the claim needed decades of work to make fully rigorous. But Fourier was essentially right, and the consequence is enormous: sinusoids are the building blocks of every periodic signal in existence. Every time you stream music, take a JPEG photo, get an MRI, or run a phone call through a network, you are using Fourier's insight to break a signal into sine waves, manipulate them, and rebuild it. That is the practical payoff of the curves on this page.

The Basic Waves: Sine and Cosine

Picture a point moving counterclockwise around the unit circle at a steady speed. Its height above the center (its $y$-coordinate) is $\sin\theta$; its horizontal position is $\cos\theta$. Now imagine unrolling the angle $\theta$ along a horizontal axis and plotting that height. As $\theta$ goes from $0$ to $2\pi$, the height traces one complete wave.

Key features of $y = \sin x$:

• It passes through the origin: $\sin 0 = 0$.
• It peaks at $x = \pi/2$ with value $1$, crosses zero at $x = \pi$, bottoms out at $x = 3\pi/2$ with value $-1$, and returns to zero at $x = 2\pi$.
• It repeats forever: $\sin(x + 2\pi) = \sin x$. The period is $2\pi$.
• It stays between $-1$ and $1$.

The cosine wave $y = \cos x$ is the same shape but starts at its maximum: $\cos 0 = 1$. In fact $\cos x = \sin\!\big(x + \tfrac{\pi}{2}\big)$ — cosine is sine shifted left by a quarter period. This is why we can model everything with sine alone if we like; cosine is just a phase-shifted sine.

Worked Example: Reading Values Off the Wave

Problem. Without a calculator, find $\sin\!\left(\tfrac{7\pi}{6}\right)$ using the graph's structure.

Solution. The angle $\tfrac{7\pi}{6} = \pi + \tfrac{\pi}{6}$ sits just past $\pi$, in the third quarter of the cycle where sine is negative and heading down from zero. Its reference angle is $\tfrac{\pi}{6}$, and $\sin\tfrac{\pi}{6} = \tfrac{1}{2}$. Because we are below the midline,

$\sin\!\left(\frac{7\pi}{6}\right) = -\frac{1}{2}.$

Reading the sign straight off the wave's shape — negative between $x=\pi$ and $x=2\pi$ — is faster and safer than memorizing a table.

The Four Controls: Amplitude, Period, Frequency, Phase

To model real oscillations we need to stretch, squeeze, and slide the basic wave. The general sinusoid is

$y = A\,\sin\!\big(B(x - C)\big) + D.$

Each letter does exactly one geometric job:

Symbol	Name	Effect on the graph	Formula
$A$	Amplitude	Vertical stretch; height of peak above midline	$\text{amp} = \lvert A\rvert$
$B$	Angular frequency	Horizontal squeeze; more cycles per unit	period $= 2\pi/\lvert B\rvert$
$C$	Phase shift	Slides the whole wave right (if $C>0$)	shift $= C$
$D$	Vertical shift	Raises/lowers the midline	midline $y = D$

Two definitions students constantly mix up:

• Period $T = \dfrac{2\pi}{\lvert B\rvert}$ is the time (or length) for one full cycle.
• Frequency $f = \dfrac{1}{T} = \dfrac{\lvert B\rvert}{2\pi}$ is how many cycles fit in one unit — the reciprocal of the period.

A crucial subtlety: the phase shift is $C$, not the number subtracted inside an un-factored expression. If you see $\sin(2x - \pi)$, factor first: $\sin\!\big(2(x - \tfrac{\pi}{2})\big)$, so the shift is $\tfrac{\pi}{2}$, not $\pi$.

Worked Example: Building an Equation From a Description

Problem. A wave oscillates between a maximum of $9$ and a minimum of $1$. One full cycle takes $6$ units, and at $x = 0$ the wave sits at its midline heading upward. Write its equation.

Solution. Work through the four controls in order.

• Midline: $D = \dfrac{9 + 1}{2} = 5$.
• Amplitude: $A = \dfrac{9 - 1}{2} = 4$.
• $B$ from the period: $T = 6 \Rightarrow B = \dfrac{2\pi}{6} = \dfrac{\pi}{3}$.
• Phase: a plain sine starts at the midline going up, which is exactly our condition, so $C = 0$.

$y = 4\sin\!\left(\frac{\pi}{3}\,x\right) + 5.$

Check. At $x = 0$: $y = 4\sin 0 + 5 = 5$ (midline). At $x = 1.5$ (a quarter period): $y = 4\sin(\pi/2) + 5 = 9$ (maximum). Correct.

Worked Example: Extracting Parameters From an Equation

Problem. For $y = -3\cos\!\big(4x + \pi\big) - 2$, find the amplitude, period, frequency, phase shift, and midline, and describe the graph.

Solution. Factor the inside: $4x + \pi = 4\!\left(x + \tfrac{\pi}{4}\right)$, so the equation is $y = -3\cos\!\big(4(x + \tfrac{\pi}{4})\big) - 2$.

• Amplitude $= \lvert -3\rvert = 3$.
• Period $= \dfrac{2\pi}{4} = \dfrac{\pi}{2}$.
• Frequency $= \dfrac{1}{\pi/2} = \dfrac{2}{\pi}$ cycles per unit.
• Phase shift $= -\tfrac{\pi}{4}$ (shifted left a quarter unit, because of $x + \tfrac{\pi}{4}$).
• Midline $y = -2$.

The negative sign on the $3$ flips the cosine: instead of starting at a peak, the shifted-and-flipped wave starts at a trough. So near its reference point the graph opens from a minimum of $-5$ up to a maximum of $1$.

Modeling Real Oscillations

The reason this topic matters is that the same four controls describe wildly different physical systems.

Sound. A pure musical tone is a pressure wave $y = A\sin(2\pi f t)$. The amplitude $A$ is loudness; the frequency $f$ is pitch. Middle A is $440$ Hz, meaning $440$ cycles per second, so its period is $1/440 \approx 0.00227$ seconds.

AC electricity. Household mains voltage is a sinusoid, e.g. $V(t) = 170\sin(2\pi \cdot 60\, t)$ in the US, where $60$ Hz is the frequency and $170$ V is the peak (giving the familiar $120$ V "RMS" average). The alternating sign is literally the wave dipping below its midline.

Tides. Sea level rises and falls roughly sinusoidally with the ~12.4-hour lunar cycle. A tide model has a nonzero vertical shift $D$ (mean sea level) and an amplitude set by the tidal range.

Springs and pendulums. A mass bobbing on a spring obeys $x(t) = A\cos(\omega t)$ where $\omega = \sqrt{k/m}$ — the very definition of simple harmonic motion.

Worked Example: Modeling a Tide

Problem. At a harbor, high tide is $5.2$ m and low tide is $0.8$ m. High tide occurs at $t = 3$ hours (measuring $t$ in hours after midnight) and the cycle length is $12$ hours. Model the water depth $y$, then find the depth at $t = 0$.

Solution.

• Midline: $D = \dfrac{5.2 + 0.8}{2} = 3.0$ m.
• Amplitude: $A = \dfrac{5.2 - 0.8}{2} = 2.2$ m.
• $B$: period $12 \Rightarrow B = \dfrac{2\pi}{12} = \dfrac{\pi}{6}$.
• Because a maximum occurs at $t = 3$, use cosine (which peaks at its start) shifted so its peak lands at $t = 3$: $C = 3$.

$y = 2.2\cos\!\left(\frac{\pi}{6}(t - 3)\right) + 3.0.$

Now evaluate at $t = 0$:

$y = 2.2\cos\!\left(\frac{\pi}{6}(-3)\right) + 3.0 = 2.2\cos\!\left(-\frac{\pi}{2}\right) + 3.0 = 2.2(0) + 3.0 = 3.0 \text{ m}.$

So at midnight the water is exactly at mean sea level, which makes sense: it is three hours (a quarter period) before high tide, so it is rising through the midline. Correct.

Fourier: Why Every Periodic Signal Is a Sum of Sines

Here is the idea that turns this topic from "graphing exercises" into the foundation of modern technology. A single sinusoid is smooth and rounded — so how could it ever build a square wave with sharp vertical edges, or the complex buzz of a violin?

Fourier's answer: add many sinusoids of different frequencies together. For a square wave, adding the fundamental sine plus a bit of the sine at triple frequency, plus a bit at five times, seven times, and so on — the odd harmonics — the sum grows increasingly square-shaped:

$\text{square}(x) \approx \sin x + \frac{1}{3}\sin 3x + \frac{1}{5}\sin 5x + \frac{1}{7}\sin 7x + \cdots$

Include enough terms and the wobbly sum snaps into crisp corners. The remarkable theorem is that this works for any reasonable periodic function: it equals a sum (its Fourier series) of sines and cosines whose frequencies are whole-number multiples of the fundamental. The amplitude of each sine tells you "how much of that frequency" the signal contains — its spectrum.

This is why the topic is universal. A violin and a flute playing the same note have the same fundamental frequency (same pitch) but different recipes of harmonics — that difference in the sine ingredients is what your ear hears as timbre. MP3 compression throws away the sine components your ear can't detect. Noise-canceling headphones measure incoming sine components and add their opposites. JPEG images run the same idea in two dimensions. Every one of these technologies rests on the graphs you are learning to draw here.

Real-World Applications

• Audio and music: synthesizers build tones by summing sinusoids; equalizers boost or cut specific frequency bands; MP3/AAC compression discards inaudible sine components.
• Electrical engineering: AC power, radio and Wi-Fi carrier waves, and signal filters are all analyzed as sinusoids; "frequency" on your radio dial is literally the sine's $f$.
• Medicine: ECG and EEG traces are periodic signals decomposed into frequency bands; MRI reconstructs images from measured sine-wave frequencies.
• Oceanography and climate: tide tables, seasonal temperature cycles, and El Niño oscillations are fit with sinusoidal models.
• Mechanical engineering: vibration analysis of engines, bridges, and buildings uses Fourier spectra to find dangerous resonant frequencies.
• Everyday life: the swing of a pendulum clock, the bounce of a car's suspension, the daily rhythm of body temperature, and the flicker-free hum of AC lighting all follow sinusoids.

Common Mistakes

1. Confusing period and frequency. Many students report the value of $B$ as the period. Misconception: "$B$ is how long a cycle takes." Why it's wrong: $B$ measures angular speed; a bigger $B$ means a shorter, more crowded cycle. Correction: always compute period $= 2\pi/\lvert B\rvert$ and frequency $= \lvert B\rvert/(2\pi)$. They are reciprocals, not the same thing.

2. Reading the phase shift straight from an un-factored equation. Given $\sin(3x - \pi)$, students say the shift is $\pi$. Why it's wrong: the phase shift is the value of $C$ after factoring out $B$. Correction: rewrite as $\sin\!\big(3(x - \tfrac{\pi}{3})\big)$; the shift is $\tfrac{\pi}{3}$. Divide the constant by $B$.

3. Getting the shift direction backward. With $\sin(x - C)$ and $C>0$, some shift the graph left. Why it's wrong: to keep the argument the same, $x$ must be larger, so features move to the right. Correction: $\sin(x - C)$ shifts right by $C$; $\sin(x + C)$ shifts left by $C$. It feels backward, so double-check with one point.

4. Ignoring a reflection when $A$ is negative. Treating $y = -2\sin x$ as if it just has amplitude $2$. Why it's wrong: the negative flips the wave vertically, so it starts by going down, not up. Correction: amplitude is $\lvert A\rvert = 2$, but note the reflection when sketching the starting direction.

Comparison and Connections

Idea	What it captures	Key formula	Watch out for
Amplitude	Height of the swing	$\lvert A\rvert$	Not the total peak-to-trough (that's $2\lvert A\rvert$)
Period	Length of one cycle	$2\pi/\lvert B\rvert$	Reciprocal of frequency
Frequency	Cycles per unit	$\lvert B\rvert/(2\pi)$	Measured in Hz when $x$ is seconds
Phase shift	Horizontal slide	$C$ (after factoring)	Sign/direction reversed
Vertical shift	Midline level	$D$	Moves min and max together

Sine vs. cosine: identical waves offset by a quarter period; $\cos x = \sin(x + \tfrac{\pi}{2})$. Use whichever starts more conveniently — cosine when a maximum sits at your start point, sine when the midline does.

Sinusoid vs. tangent: tangent is periodic too, but it has period $\pi$ and shoots off to infinity at asymptotes — it is not a smooth bounded wave and does not model simple harmonic motion.

Connection to the unit circle: the wave is literally the vertical (sine) or horizontal (cosine) coordinate of circular motion unrolled over time. Circular motion and simple harmonic motion are two views of the same thing.

Practice Questions

Recall

State the amplitude, period, frequency, phase shift, and midline of $y = 5\sin\!\big(2(x - \tfrac{\pi}{4})\big) + 3$.

Answer: amplitude $5$; period $2\pi/2 = \pi$; frequency $1/\pi$; phase shift $\tfrac{\pi}{4}$ right; midline $y = 3$.

Understanding

Explain why $\cos x$ and $\sin x$ have the same graph shape, and state precisely how far apart they are.

Answer: Both are projections of steady circular motion; cosine tracks the horizontal coordinate and sine the vertical, and these differ by a quarter turn. Formally $\cos x = \sin(x + \tfrac{\pi}{2})$, so cosine is sine shifted left by $\tfrac{\pi}{2}$ (a quarter period).

Application

A Ferris wheel of radius $10$ m has its center $12$ m above the ground and completes one turn every $40$ seconds. A rider boards at the lowest point at $t = 0$. Write the height $y(t)$ and find the height at $t = 10$ s.

Guidance: Lowest point at $t=0$ means an inverted cosine: $D = 12$, $A = 10$, $B = 2\pi/40 = \pi/20$, so $y = -10\cos\!\big(\tfrac{\pi}{20}t\big) + 12$. At $t = 10$: $y = -10\cos(\pi/2) + 12 = -10(0) + 12 = 12$ m (level with the center — a quarter turn up).

Analysis

The first few terms of a square wave are $\sin x + \tfrac{1}{3}\sin 3x + \tfrac{1}{5}\sin 5x$. Explain why the coefficients shrink and why only odd multiples of the frequency appear.

Guidance: The coefficients shrink ($1, \tfrac13, \tfrac15,\dots$) because high-frequency corrections only fine-tune the sharp corners, so they contribute less and less. Only odd harmonics appear because the square wave has a specific symmetry (odd, half-wave symmetric) that cancels every even-frequency component in the integral that computes the Fourier coefficients. The result is a signal built entirely from odd-frequency sines whose amplitudes fall off as $1/n$.

FAQ

Do I have to measure the horizontal axis in radians? For pure trig graphs, radians are cleanest because the period comes out as $2\pi/B$. In applications the axis is usually time (seconds, hours), and $B$ absorbs the unit conversion — you rarely think in radians directly, you just use period $= 2\pi/B$.

When should I model with sine and when with cosine? They are interchangeable via a phase shift, so use whichever removes the need for a shift. If your data starts at a maximum, cosine is natural; if it starts at the midline rising, plain sine is natural. Either will fit the same data.

Why is amplitude half the distance between max and min, not the whole distance? Amplitude measures the swing from the midline to a peak — the "radius" of the oscillation. The full max-to-min distance is $2\lvert A\rvert$, the "diameter." Splitting it gives you both the midline ($D$) and the amplitude ($A$) at once.

Is frequency the same as the $B$ value? No. $B$ is the angular frequency (radians per unit); ordinary frequency $f = B/(2\pi)$ is in cycles per unit (Hz when the unit is seconds). They are proportional but differ by the $2\pi$ factor — a common source of errors in physics.

How can smooth sine waves ever add up to a signal with sharp corners? Individually they can't, but an infinite sum of them can approach one arbitrarily closely. Each higher-frequency sine adds a finer wrinkle near the corner. This is Fourier's theorem, and near a true jump you even get a small persistent overshoot called the Gibbs phenomenon — a fingerprint of building sharp edges from smooth waves.

What does "phase" mean physically? Phase is where in its cycle an oscillation is at a chosen moment. Two waves "in phase" peak together; "out of phase" (by half a period) one peaks as the other bottoms out — which is exactly how noise-canceling headphones erase sound by adding an opposite-phase copy.

Quick Revision

• General sinusoid: $y = A\sin\!\big(B(x - C)\big) + D$.
• Amplitude $= \lvert A\rvert$; peak-to-trough $= 2\lvert A\rvert$.
• Period $T = \dfrac{2\pi}{\lvert B\rvert}$; frequency $f = \dfrac{1}{T} = \dfrac{\lvert B\rvert}{2\pi}$.
• Phase shift $= C$ (factor out $B$ first!); positive $C$ shifts right.
• Vertical shift / midline $= D$; max $= D + \lvert A\rvert$, min $= D - \lvert A\rvert$.
• $\cos x = \sin\!\big(x + \tfrac{\pi}{2}\big)$: cosine is sine shifted a quarter period left.
• Negative $A$ reflects the wave vertically (starts by going down).
• Fourier: every periodic signal is a sum of sines/cosines at whole-number multiples of the fundamental frequency — the basis of audio, imaging, and communications.

Related Topics

Prerequisites

• The Unit Circle — where sine and cosine come from as coordinates of circular motion.
• Right-Triangle Trigonometry — the original ratio definitions of sine and cosine.
• Functions — transformations (shifts, stretches, reflections) that reshape any graph.

Related Topics

• Trigonometric Identities — algebraic relationships used to combine and simplify sinusoids.
• Trigonometry Overview — how these ideas fit together.

Next Topics

• Applications of Calculus — derivatives of sine and cosine describe velocity and acceleration in oscillating systems.
• Distributions — periodic and wave ideas reappear when analyzing signals and noise.