Trigonometric Graphs

Trigonometric graphs are the visual representations of the functions

sin x

cos x

tan x

and their relatives plotted on the Cartesian plane. Key result in words: sine and cosine produce smooth, repeating waves of period

2 π

bounded between

- 1

and

1

; tangent produces a repeating pattern of period

π

with vertical asymptotes. Where it applies: modelling any oscillation — sound, light, tides, alternating current, springs. Who/when: developed from ancient astronomy (Hipparchus, Ptolemy) through Euler's 18th-century functional treatment.

A trigonometric graph is the locus of points $(x, f (x))$ where $f$ is a circular function such as $sin$ , $cos$ , or $tan$ . These curves matter because periodicity — exact repetition after a fixed interval — is the mathematical signature of every wave and rhythm in nature, so their graphs are the universal language of oscillation. The graphs are completely determined by four transformation parameters: amplitude, period, phase shift, and vertical shift. A surprising fact: the graph of $cos x$ is identical to the graph of $sin x$ slid leftward by $\frac{π}{2}$ — the two "different" functions are the same wave viewed from different starting points.

The Core Insight

A trigonometric function is what you get when you wrap the number line around a unit circle and read off a coordinate. As the wrap-point travels around the circle at constant speed, its height ( $sin$ ) and horizontal position ( $cos$ ) trace out smooth waves. Every sinusoidal graph is just one master wave reshaped by stretching it taller/shorter, wider/narrower, and sliding it sideways/up. Master those four operations and you can read or draw any of them.

Formal Statement

The Standard Sinusoid

A general sinusoidal function is written

f (x) = A sin (B (x - C)) + D

⟳start from the unit-circle definition

sin θ

and apply the four geometric transformations one at a time.

where the symbols mean:

$A$ = amplitude = half the vertical distance between maximum and minimum; the graph reaches up to $D + ∣ A ∣$ and down to $D - ∣ A ∣$ .
$B$ = angular frequency; it compresses or stretches the wave horizontally.
$C$ = phase shift = horizontal displacement (right if $C > 0$ , left if $C < 0$ ).
$D$ = vertical shift = the midline value about which the wave oscillates.

The period $T$ (length of one complete cycle) and frequency $ν$ are:

T = \frac{2 π}{∣ B ∣}, ν = \frac{1}{T} = \frac{∣ B ∣}{2 π}

⟳one full cycle of

sin

requires its argument to advance by

2 π

; set

B \cdot T = 2 π

and solve.

The Three Core Functions

On the unit circle, for an angle $x$ measured in radians from the positive $x$ -axis to point $(cos x, sin x)$ :

sin x = (y -coordinate), cos x = (x -coordinate), tan x = \frac{sin x}{cos x}

⟳project the rotating radius onto each axis; tangent is the slope of the radius.

Their fundamental graph properties:

| Function | Period | Range | Symmetry | Asymptotes | |----------|--------|-------|----------|------------| | $sin x$ | $2 π$ | $[- 1, 1]$ | odd | none | | $cos x$ | $2 π$ | $[- 1, 1]$ | even | none | | $tan x$ | $π$ | $R$ | odd | $x = \frac{π}{2} + nπ$ |

Co-function and Periodicity Identities

The sine and cosine graphs are translates of each other, and both repeat forever:

cos x = sin (x + \frac{π}{2}), sin (x + 2 π n) = sin x (n \in Z)

⟳rotating the unit-circle point by

\frac{π}{2}

swaps which axis you project onto; full turns of

2 π

return to the start.

Why It Holds / Derivation

Deriving the period of the tangent graph

We show $tan x$ repeats every $π$ , not $2 π$ .

By definition $tan x = \frac{sin x}{cos x}$ (equation [3]).
Add $π$ to the angle. Using the half-turn identities $sin (x + π) = - sin x$ and $cos (x + π) = - cos x$ :

tan (x + π) = \frac{sin ( x + π )}{cos ( x + π )} = \frac{- sin x}{- cos x}

⟳a half-turn

π

sends a unit-circle point to its diametric opposite, negating both coordinates.

The two minus signs cancel: $tan (x + π) = \frac{sin x}{cos x} = tan x$ .
Therefore the smallest repeat interval is $π$ . The vertical asymptotes occur exactly where $cos x = 0$ , i.e. at $x = \frac{π}{2} + nπ$ , because division by zero is undefined and $tan x \to \pm \infty$ there. $■$

One period of y = sin x and y = cos x on a shared axis

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<rect width="640" height="320" fill="#0a0e17"/>
<!-- axes -->
<line x1="40" y1="160" x2="620" y2="160" stroke="#94a3b8" stroke-width="1.5"/>
<line x1="60" y1="20" x2="60" y2="300" stroke="#94a3b8" stroke-width="1.5"/>
<!-- midline ticks -->
<line x1="60" y1="60" x2="620" y2="60" stroke="#334155" stroke-width="0.7" stroke-dasharray="4 4"/>
<line x1="60" y1="260" x2="620" y2="260" stroke="#334155" stroke-width="0.7" stroke-dasharray="4 4"/>
<text x="30" y="64" fill="#e2e8f0" font-size="12">1</text>
<text x="22" y="264" fill="#e2e8f0" font-size="12">-1</text>
<!-- x-axis labels: span 0 to 2pi over x=60..580 -->
<text x="190" y="178" fill="#e2e8f0" font-size="12">π/2</text>
<text x="315" y="178" fill="#e2e8f0" font-size="12">π</text>
<text x="435" y="178" fill="#e2e8f0" font-size="12">3π/2</text>
<text x="560" y="178" fill="#e2e8f0" font-size="12">2π</text>
<!-- cos x curve (reference, dimmer) -->
<path d="M60,60 C75,62 95,90 130,160 C165,230 185,258 200,260 C215,258 235,230 270,160 C305,90 325,62 340,60 C355,62 375,90 410,160 C445,230 465,258 480,260 C495,258 515,230 550,160 C575,108 590,72 580,62" fill="none" stroke="#64748b" stroke-width="2"/>
<text x="120" y="48" fill="#64748b" font-size="13">y = cos x</text>
<!-- sin x curve (key curve) -->
<path d="M60,160 C75,108 95,72 130,60 C165,48 185,62 200,108 C215,154 235,224 270,260 C305,224 325,154 340,108 C355,62 375,48 410,60 C445,72 465,108 480,160 C515,212 535,250 580,260" fill="none" stroke="#d4a853" stroke-width="2.5"/>
<text x="400" y="48" fill="#d4a853" font-size="13">y = sin x</text>
</svg>

Leonhard Euler | 1707–1783

Although sine and cosine values were tabulated for over a millennium before him, it was Euler who in his Introductio in analysin infinitorum (1748) recast them as functions of a real-number variable rather than as ratios of chord lengths in a circle of fixed radius. By adopting the unit circle (radius 1) he stripped away the arbitrary scaling factor and made $sin$ and $cos$ pure functions ready to be graphed. His formula $e^{i x} = cos x + i sin x$ later revealed why these curves are the natural shape of oscillation: they are the real and imaginary shadows of uniform circular motion in the complex plane.

Building Intuition

Picture a bicycle wheel turning at constant speed and a single reflector stuck to the rim. Watch only the reflector's height off the ground and plot it against time — that is the sine wave. Watch only its horizontal position — that is the cosine wave. Because the wheel turns smoothly, neither graph has corners; because it returns to the same place every full turn, both repeat. Now imagine the parameters: pump the wheel up to a larger radius and the wave gets taller ( $A$ ); spin faster and the wave bunches up ( $B$ ); start timing a quarter-turn early and the whole wave slides ( $C$ ); mount the axle higher off the ground and the whole wave lifts ( $D$ ). The tangent graph is the slope of the line from the centre to the reflector — it blows up to infinity when that line goes vertical, which is why $tan$ has its dramatic asymptotes.

Worked Examples

Sketching y = 3 sin(2x − π) + 1

Given:

f (x) = 3 sin (2 x - π) + 1

. Step 1 — Rewrite in standard form. Factor the inside:

2 x - π = 2 (x - \frac{π}{2})

, so

A = 3, B = 2, C = \frac{π}{2}, D = 1

. Step 2 — Amplitude.

∣ A ∣ = 3

, so the wave swings 3 units above and below the midline. Step 3 — Midline & range.

D = 1 \Rightarrow

midline

y = 1

; maximum

= 1 + 3 = 4

; minimum

= 1 - 3 = - 2

. Range

[- 2, 4]

. Step 4 — Period. Using [2]:

T = \frac{2 π}{∣ B ∣} = \frac{2 π}{2} = π

. Step 5 — Phase shift.

C = \frac{π}{2}

to the right; one cycle starts at

x = \frac{π}{2}

and ends at

x = \frac{π}{2} + π = \frac{3 π}{2}

Amplitude 3, Period π, Shift \frac{π}{2} right, Midline y = 1, Range [- 2, 4]

Finding the equation of a tide

Given: A harbour's water depth varies between a low of

2 m

and a high of

8 m

, completing one full cycle every

12

hours, with high tide at

t = 3

h. Model depth

d (t)

in metres with

t

in hours. Step 1 — Amplitude.

A = \frac{max - min}{2} = \frac{8 - 2}{2} = 3 m

. Step 2 — Midline.

D = \frac{max + min}{2} = \frac{8 + 2}{2} = 5 m

. Step 3 — Angular frequency. From [2],

B = \frac{2 π}{T} = \frac{2 π}{12} = \frac{π}{6} rad/h

. Step 4 — Phase. Cosine peaks at its start, and high tide is at

t = 3

, so use

cos

shifted right by

3

d (t) = 3 cos (\frac{π}{6} (t - 3)) + 5

Check: at

t = 3

cos (0) = 1 \Rightarrow d = 8

✓ (high tide); at

t = 9

cos (π) = - 1 \Rightarrow d = 2

✓ (low tide).

d (t) = 3 cos (\frac{π}{6} (t - 3)) + 5 metres

Common Misconceptions

The " $B$ is the period" trap. In

sin (B x)

, the constant

B

is not the period — it is inversely related to it via

T = 2 π /∣ B ∣

. A larger

B

gives a shorter period (more cycles squeezed in), which feels backwards to many students. Likewise, in

sin (B (x - C))

the phase shift is

C

, not the whole quantity inside the bracket. Always factor out

B

first:

sin (2 x - π)

shifts by

\frac{π}{2}

, not by

π

. Claim (false): "Amplitude is half the range, so the amplitude of

f (x) = 2 sin x + 5

\frac{1}{2} (range)

measured from zero."

The range of $f$ is $[3, 7]$ , a span of $4$ , so half the span is $2$ — which is the amplitude here, by luck, because the vertical shift happened to keep things consistent. But for $g (x) = sin x + 5$ with range $[4, 6]$ , students who measure "from the $x$ -axis up to the peak" get $6$ , not the correct amplitude $1$ . The amplitude is always measured from the midline, never from the $x$ -axis. The vertical shift $D$ must be subtracted out first.

Connections

Trigonometric graphs sit at a crossroads of mathematics. They are the visual face of the Unit Circle, which supplies the raw $sin$ / $cos$ values. They are governed by the algebra of Trigonometric Identities, which let you rewrite $cos x$ as a shifted $sin x$ or expand $sin (2 x)$ . As functions, they obey all the rules of Function Transformations — the same stretch/shift logic you use on parabolas applies here. Their derivatives, studied in Calculus of Trigonometric Functions, famously cycle $sin \to cos \to - sin \to - cos$ , which is why the slope of a sine curve is itself a cosine curve. Finally, summing many sinusoids underlies Fourier Series, the idea that any periodic signal is a stack of these simple waves.

Real-World Applications

Alternating current (AC) electricity: household voltage is $V (t) = V_{0} sin (2 π \cdot 50 t)$ — a 50 Hz sinusoid in Europe, 60 Hz in North America.
Sound and music: a pure tone (e.g. a tuning fork at 440 Hz, concert A) is a single sine wave; timbre comes from added harmonics.
Tides: coastal water depth is modelled by sinusoids of period $\approx 12.4$ hours, as in the worked example above, vital for shipping and harbour design.
Daylight hours: the length of day across a year follows a sine curve, peaking at the summer solstice and bottoming at the winter solstice.
Pendulums and springs (simple harmonic motion): displacement of a mass on a spring is $x (t) = A cos (ω t)$ .
Biological rhythms: circadian body-temperature variation and seasonal animal populations are approximated by sinusoids.
Signal processing & telecommunications: AM/FM radio encodes information by modulating a carrier sine wave's amplitude or frequency.

Historical Development

~150 BCE — Hipparchus compiles the first known table of chords, the ancestor of the sine table, for astronomical calculations.
~500 CE — Aryabhata (India) tabulates the jya (half-chord), the direct precursor of the modern sine, in his Aryabhatiya.
~830 CE — Al-Khwarizmi and later Islamic astronomers refine and extend sine and tangent tables for prayer-time and qibla calculations.
1464 — Regiomontanus writes De triangulis omnimodis, the first European treatise treating trigonometry as a subject independent of astronomy.
1748 — Leonhard Euler redefines sine and cosine as functions of a real variable on the unit circle and establishes $e^{i x} = cos x + i sin x$ , unifying the graphs with exponential growth.
1822 — Joseph Fourier publishes Théorie analytique de la chaleur, showing arbitrary periodic functions decompose into sums of sine and cosine graphs.
1860s — James Clerk Maxwell describes electromagnetic waves as sinusoidal, making trigonometric graphs the shape of light itself.
20th century — Oscilloscope and CRT technology make sinusoidal waveforms physically visible to engineers, cementing the graph as an everyday tool.

Subtleties & Limits

Trigonometric graphs are perfectly periodic idealizations, but real oscillations are rarely so tidy. A damped pendulum's graph is $A e^{- k t} sin (ω t)$ — a sine wave inside a shrinking envelope, no longer truly periodic. Quasiperiodic signals (sums of sinusoids whose periods are irrationally related) never exactly repeat, raising deep questions in dynamical systems about when a motion is "almost periodic." At the frontier, chaotic systems produce signals that look wavy but are unpredictable: decomposing them into clean trigonometric pieces (the Fourier program) breaks down or requires infinitely many terms, and characterizing such spectra remains an active research area in nonlinear dynamics.