Graphs of Sine and Cosine

16 min read

Tape a marker to the rim of a bike wheel and spin it. Watch only how high the marker rides off the ground. It climbs, drops, and climbs again, tracing a smooth wave that never stops. That wave is the graph of sine. Cosine is the same wave that got a quarter-turn head start.

By the end of this page you can:

• Read the amplitude $|a|$, the period $\dfrac{2\pi}{k}$, and the phase shift $b$ off $y = a\sin k(x - b)$ and $y = a\cos k(x - b)$.
• Sketch one full period by marking the five quarter points (midline, peak, midline, trough, midline for sine).
• Recover an equation $y = a\sin k(x - b)$ or $y = a\cos k(x - b)$ from a given graph.
• Tell a change of period (a change in $k$) apart from a horizontal slide (a change in $b$).

The floor. For $y = a\sin k(x - b)$, the amplitude is $|a|$, the period is $\dfrac{2\pi}{k}$, and the shift is $b$, but only once the inside is written as $k(x - b)$. Factor the inside first, then read the three numbers straight off.

Quick Reference

Before You Start

Check each box you can do from memory. A box you cannot check yet points to a quick refresher, not a grade.

• I can give the sine and cosine of $0$, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, and $2\pi$ from the unit circle. (Unit Circle: Sine and Cosine)
• I can say which quadrants make sine positive and which make it negative. (Signs and Domains of Trig Functions)
• I can read $\sin$ and $\cos$ as functions that take an input angle and return one output number.

A 60-second self-check. Pick an answer, then reveal the reasoning.

Where the Wave Comes From

Before any formula, walk one trip around the unit circle and record the height. A point moves counterclockwise from the far right. Its height above the horizontal axis is the sine value. Read the height at five stops, a quarter turn apart.

Plot those five heights left to right, with the angle on the horizontal axis and the height on the vertical axis, and connect them smoothly. Where does the curve peak, and where does it bottom out? What happens after one full turn?

A periodic function returns to the same value after a fixed amount of input, then does the same thing again. A wheel at steady speed, a tide, a vibrating string: each comes back to its earlier state after a fixed interval. Sine and cosine are the cleanest such functions, and their graphs are pictures of that motion. Three numbers describe any one of these curves once the midline sits on the axis:

• how tall the swing is above and below the center (amplitude),
• how long one full repeat takes (period),
• how far the pattern is slid sideways (phase shift, also called the horizontal shift).

Stewart writes the general curve as $$y = a\sin k(x - b) \qquad\text{or}\qquad y = a\cos k(x - b), \qquad k > 0.$$ The inside is already factored as $k(x - b)$, the form that makes the three numbers read off directly: amplitude $|a|$, period $\dfrac{2\pi}{k}$, phase shift $b$.

Prerequisite Hub

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Builds on (master these first):

Unlocks (what this feeds into next):

Cross-course prerequisites: none required.

Official Definitions

Each statement below traces to Stewart 7e, Section 5.3. The verbatim OpenStax quotes that follow are an alternate phrasing of the same facts.

Amplitude

For $y = a\sin k(x - b)$ and $y = a\cos k(x - b)$ with $k > 0$, the amplitude is $|a|$, the largest distance the graph rises above or falls below its center line.

Stewart 7e, Section 5.3 Trigonometric Graphs (page to confirm).

The absolute value matters. A negative $a$ still produces a swing of size $|a|$; the sign reflects the graph across the center line so the curve starts by going down rather than up. The full distance from a peak down to a trough is $2|a|$, twice the amplitude, not the amplitude itself.

Period

For $y = a\sin k(x - b)$ and $y = a\cos k(x - b)$ with $k > 0$, the period is $\dfrac{2\pi}{k}$, the input length of one complete repeat.

Stewart 7e, Section 5.3 Trigonometric Graphs (page to confirm).

A larger $k$ packs more repeats into the same stretch of input, so the period shrinks. With $k = 2$ the curve repeats twice as fast as the basic sine, giving a period of $\dfrac{2\pi}{2} = \pi$.

Phase shift (horizontal shift)

For $y = a\sin k(x - b)$ and $y = a\cos k(x - b)$, the phase shift is $b$, the horizontal displacement of the basic sine or cosine curve. One complete period is graphed on the interval from $x = b$ to $x = b + \dfrac{2\pi}{k}$.

Stewart 7e, Section 5.3 Trigonometric Graphs (page to confirm).

The shift is the number $b$ that appears in the factored inside $k(x - b)$, not the loose constant that appears when the inside is multiplied out. A positive $b$ slides the pattern right; a negative $b$ slides it left.

Periodic function (alternate phrasing)

“A periodic function is a function for which a specific horizontal shift, $P$, results in a function equal to the original function: $f(x + P) = f(x)$ for all values of $x$ in the domain of $f$.”

OpenStax Precalculus 2e, Section 6.1 Graphs of the Sine and Cosine Functions.

The smallest positive such $P$ is the period. For the basic sine and cosine that smallest $P$ is $2\pi$.

Base-graph facts

Stewart 7e, Section 5.3 Trigonometric Graphs.

Reading the Three Numbers off the Form

For $y = a\sin k(x - b)$ or $y = a\cos k(x - b)$ with $k > 0$:

Two views of the same three numbers help to hold at once. As transformations: stretch the height by $|a|$, compress the input by $k$, then slide right by $b$. This answers “what changed from $y = \sin x$?” As a recipe for landmark points: one period splits into four equal quarters, and across them an unshifted sine passes through midline, peak, midline, trough, midline. This answers “where do I put my pencil?” The quarter-point view is faster for a hand sketch; the transformation view is clearer for explaining what a coefficient does.

The single most common error is reading the shift before factoring out $k$. The quiz below checks the three reads in turn, including that trap.

Worked Examples

Worked Example 1: Amplitude and period from the formula

Problem. Graph one period of $y = 3\sin(2x)$. State the amplitude, period, and midline.

Predict first. The $3$ out front should stretch the swing taller than the basic sine, and the $2$ inside should make it repeat faster. Predict an amplitude bigger than $1$ and a period smaller than $2\pi$.

Now compute. Match $y = 3\sin(2x)$ to $y = a\sin k(x - b)$ with $a = 3$, $k = 2$, $b = 0$.

• Amplitude: $|a| = |3| = 3$.
• Period: $\dfrac{2\pi}{k} = \dfrac{2\pi}{2} = \pi$.
• Midline: $y = 0$.

Quarter points. One period runs from $x = 0$ to $x = \pi$, split into four pieces of width $\dfrac{\pi}{4}$:

Check the prediction. Amplitude $3 > 1$ and period $\pi < 2\pi$, as predicted. The curve climbs to $3$, returns through $0$, drops to $-3$, and comes back, all within an input length of $\pi$.

Worked Example 2: A reflection from a negative coefficient

Problem. Graph one period of $y = -2\cos x$. State the amplitude, period, maximum, and minimum.

Predict first. The $2$ should double the swing, and the negative sign should flip the cosine so it starts at its lowest point. Predict a curve that begins at $-2$, not $+2$, and an amplitude of $2$ rather than $-2$.

Now compute. Match $y = -2\cos x$ to $y = a\cos k(x - b)$ with $a = -2$, $k = 1$, $b = 0$.

• Amplitude: $|a| = |-2| = 2$.
• Period: $\dfrac{2\pi}{1} = 2\pi$.
• Maximum: $+2$. Minimum: $-2$.

Quarter points over one period from $x = 0$ to $x = 2\pi$:

Check the prediction. The curve starts at its minimum $-2$ (the negative sign flipped the usual peak into a trough), rises to the maximum $2$ at $x = \pi$, and returns. The amplitude is $2$, a positive height, even though $a = -2$. The full span from $-2$ to $2$ is $4$, which is $2|a|$, twice the amplitude.

Worked Example 3: Phase shift from the factored form

Problem. Find the amplitude, period, and phase shift of $y = 4\sin(2x - \pi)$, then state the interval covering the first full period.

Predict first. The $-\pi$ inside slides the curve sideways. A common trap is to read the shift as $\pi$. Predict instead a shift smaller than $\pi$, because the $2$ multiplying $x$ shares the input.

Now compute. Read amplitude and period directly: $a = 4$, $k = 2$.

• Amplitude: $|a| = 4$.
• Period: $\dfrac{2\pi}{2} = \pi$.

Factor out $k$ before reading the shift. The inside is $2x - \pi$. Factor the $2$: $$2x - \pi = 2\left(x - \frac{\pi}{2}\right).$$ So $y = 4\sin\!\left(2\left(x - \frac{\pi}{2}\right)\right)$, which matches $a\sin k(x - b)$ with $b = \dfrac{\pi}{2}$.

• Phase shift: $b = \dfrac{\pi}{2}$, to the right.
• First full period: from $x = b = \dfrac{\pi}{2}$ to $x = b + \text{period} = \dfrac{\pi}{2} + \pi = \dfrac{3\pi}{2}$.

Check the prediction. The shift is $\dfrac{\pi}{2}$, not $\pi$. Reading the loose $-\pi$ as the shift would have placed the start a full $\pi$ too far right. Factoring out $k = 2$ first fixes that.

A note on the trap: the quantity $\dfrac{C}{B}$ that some texts (and the OpenStax alternate, written $y = A\sin(Bx - C)$) call the phase shift is exactly Stewart’s $b$. Here $\dfrac{C}{B} = \dfrac{\pi}{2}$ agrees with $b = \dfrac{\pi}{2}$. The two forms name the same shift; the factored form $k(x - b)$ shows it without an extra division.

Worked Example 4: Recover the equation from a graph

Problem. A cosine-shaped curve has its center line on the horizontal axis. It reaches a maximum of $3$ and a minimum of $-3$. One full repeat takes a horizontal length of $4\pi$, and the curve has a peak exactly at $x = 0$. Write an equation of the form $y = a\cos k(x - b)$.

Predict first. A peak at $x = 0$ with no vertical shift is the signature of a plain cosine, so predict $b = 0$. The swing of $3$ predicts $a = 3$, and a repeat longer than $2\pi$ predicts $k$ smaller than $1$.

Now compute.

• Amplitude: the curve rises $3$ above the center line, so $|a| = 3$. The peak sits at $x = 0$, where a positive cosine peaks, so take $a = 3$ (no reflection).
• Period: the repeat length is $4\pi$. Solve $\dfrac{2\pi}{k} = 4\pi$, which gives $k = \dfrac{2\pi}{4\pi} = \dfrac{1}{2}$.
• Phase shift: a plain cosine already peaks at $x = 0$, so $b = 0$.

Assemble. $$y = 3\cos\!\left(\tfrac{1}{2}\,x\right) = 3\cos\!\left(\tfrac{1}{2}(x - 0)\right).$$

Check the prediction. At $x = 0$, $y = 3\cos 0 = 3$, the stated maximum. The period is $\dfrac{2\pi}{1/2} = 4\pi$, the stated repeat length. Both match, and $b = 0$ matches the prediction.

Worked Example 5: Using parity to evaluate

Problem. Without a calculator, use the parity facts to rewrite $\sin\!\left(-\frac{\pi}{6}\right)$ and $\cos\!\left(-\frac{\pi}{3}\right)$ with positive inputs, then give their values.

Predict first. Sine is odd and cosine is even. Predict that the sine value picks up a negative sign while the cosine value does not.

Now compute. Apply $\sin(-x) = -\sin x$ and $\cos(-x) = \cos x$: $$\sin\!\left(-\frac{\pi}{6}\right) = -\sin\!\left(\frac{\pi}{6}\right) = -\frac{1}{2}.$$ $$\cos\!\left(-\frac{\pi}{3}\right) = \cos\!\left(\frac{\pi}{3}\right) = \frac{1}{2}.$$

Check the prediction. The sine result is negative and the cosine result stays positive, matching odd-and-even behavior. On the graph this shows directly: the sine curve has point symmetry through the origin, and the cosine curve is mirror-symmetric across the vertical axis.

Common Misconceptions

Practice Problems

Check Yourself

Close the notes and answer each from memory, then reveal it. Pulling an idea back from memory is one of the strongest ways to make it stick.

Go Deeper (optional)

A pure tone is a single sine wave. When a tuning fork vibrates, the air pressure at your ear rises and falls as a sine curve, where amplitude is the loudness and a shorter period (a larger $k$) is a higher note. A chord is a sum of a few sine waves; a recorded voice is a sum of many. Fourier analysis takes any repeating signal apart into sine and cosine pieces, and it drives audio compression, image formats, and signal processing.

Where these waves show up in work:

• Engineering: alternating current is a sinusoid, and the amplitude and period of a voltage are the first two numbers any circuit analysis needs.
• Data science: seasonal patterns such as daily temperature, yearly sales, and tide height are modeled as sinusoids, and fitting amplitude, period, and phase to data is a standard regression task.
• Computer graphics and games: smooth looping motion, a bobbing object or a pulsing light, is usually driven by a sine of time.

A runner picture ties it together. Plot a runner’s height above the center of a circular track against time. The plot rises to a maximum (runner at the top), falls to a minimum (runner at the bottom), and repeats every lap. The amplitude is the track’s radius, the period is the lap time, and the phase shift is where the runner started the clock.

Mastery Checklist

Novice (Level 1-2):

• I can state the domain, range, and period of the base sine and cosine graphs.
• I can read the amplitude $|a|$ and period $\dfrac{2\pi}{k}$ off $y = a\sin k(x - b)$.
• I can tell the amplitude apart from the full peak-to-trough span.

Competent (Level 3-4):

• I can factor out $k$ to find the phase shift $b$ correctly.
• I can sketch one full period using the five quarter points.
• I can build a sine or cosine formula from a stated amplitude, period, and shift.

Proficient (Level 5):

• I can recover an equation $y = a\sin k(x - b)$ or $y = a\cos k(x - b)$ from a graph.
• I can tell a period change (a change in $k$) apart from a horizontal slide (a change in $b$).
• I can rewrite a given cosine as an equivalent sine using a quarter-period shift.

Connections

Looking back:

• Unit Circle: Sine and Cosine supplies the exact heights the graph plots.
• Signs and Domains of Trig Functions explains the sign pattern that becomes the rise and fall of the wave.

Looking ahead:

• Graphs of the Other Trig Functions: tangent, cotangent, secant, and cosecant are read against these sine and cosine pictures.

Real-world connections:

• Sound: a pure tone is a sine wave; amplitude is loudness and period sets pitch.
• Tides: water height rises and falls on a near-sinusoidal daily cycle.
• Alternating current: household voltage is a sinusoid oscillating around a zero center line.

Resources

Last updated: 2026-06-24