Mathematics · angles and waves
Trigonometry the circle that draws waves
Take a circle of radius 1 and send a point around it. Its height is the sine, and its sideways distance is the cosine. As the point circles, a wave flows out of the circle: the same wave that describes the current in a wall socket, the sound of a string, ocean tides and the length of the day. And triangles let you measure a tree, a mountain and the whole Earth — without climbing any of them.
Readout
| sin α | |||
| cos α | |||
| tan α | |||
| cot α |
In the first quadrant all of them are positive, in the second only sine, in the third tangent and cotangent, and in the fourth cosine.
Motion
Show
The “downward” view needs a wider screen.
Identities as pictures
Special angles
Sides
Side ratios
sin = opposite / hypotenuse, cos = adjacent / hypotenuse, tan = opposite / adjacent, cot = adjacent / opposite.
| sin α | ac = | |
| cos α | bc = | |
| tan α | ab = | |
| cot α | ba = |
Move the scale slider: the sides grow, but their ratios don’t change — because all triangles with the same angle α are similar. That’s why the sine depends only on the angle.
Enter exactly three values
Examples
Sides in any units, angles in degrees. A decimal point or comma both work.
Observer’s eye height: 1.6 m — that’s why we add it to d·tan α. We assume flat ground and a vertical object.
Length of a stadion (uncertain)
That’s how whole countries were mapped, step by step: from one measured baseline and nothing but angles, surveyors built a network of triangles. Snellius was already doing it in 1615, and the Struve Geodetic Arc (19th century) is now a UNESCO World Heritage Site.
The drawing greatly exaggerates the angle — the real parallax of the nearest star is less than 1/4000 of a degree. More about the scale of distances: Scale of the Universe.
Listen to beats
Two tones: 440 Hz and 440 Hz + Δf. Their sum gets louder and softer Δf times per second — that’s how piano tuners hear that a string is still “wobbling.”
Sound is off by default; volume about 20%. Esc mutes.
Sines in everyday life
The word that lost its meaning
Indian mathematicians computed half the chord of an arc — exactly what we now call the sine. The name traveled a long way through three languages and lost its meaning on the way:
That’s how most historians of mathematics tell the story; the details (who translated it first — Gerard of Cremona and Robert of Chester are often named) vary between sources.
Timeline
- c. 1800 BC · BabyloniaThe tablet Plimpton 322 lists Pythagorean triples. Some scholars see it as a kind of trigonometric table — a disputed hypothesis. What is certain is that we owe the base-60 system and the division of the circle into 360° to Babylonian astronomy.
- c. 150 BC · Hipparchus of NicaeaCompiles the first known table of chords: the length of the chord for a given central angle. He is called the father of trigonometry; the table itself has not survived.
- c. AD 150 · PtolemyIn the Almagest he gives a table of chords every half degree, computed in base 60. The chord of an angle 2α is what we now write as 2·sin α.
- AD 499 · Aryabhata (India)In the Aryabhatiya he uses the half-chord — jyā — instead of the whole chord, and gives a table of its values every 3¾°. This is the birth of the sine.
- 9th–10th c. · al-Battani and Arab scholarsAl-Battani (c. 858–929) works with sines instead of chords and uses tables of “shadows” — today’s cotangent and tangent; Abu al-Wafa (10th c.) compiles accurate tables and new formulas.
- 12th c. · translators in SpainThe Arabic “jayb” is rendered in Latin as sinus.
- c. 1464 · RegiomontanusDe triangulis omnimodis (printed 1533) — Europe’s first textbook treating trigonometry as a field of its own, independent of astronomy.
- 1542–1543 · Copernicus and RheticusIn Wittenberg, Rheticus publishes the chapters on triangles from Copernicus’s work (De lateribus et angulis triangulorum, 1542), and a year later Copernicus’s tables of half-chords appear in De revolutionibus. In 1551 Rheticus publishes tables of all six functions.
- 1595 · Bartholomäus PitiscusWith the title of his book he gives the field its name: trigonometry — from the Greek for “measuring triangles.”
- 1748 · Leonhard EulerIn Introductio in analysin infinitorum he treats sine and cosine as functions of a number on a circle of radius 1 and links them to powers: eix = cos x + i sin x (see complex numbers).
- 1873 · the radianThe word radian appears in print (in James Thomson’s exam questions in Belfast); the idea of measuring an angle by arc length is older.
- 1959 · CORDICJack Volder describes an algorithm that computes sines using only additions and bit shifts. Similar methods made it into calculators, including the HP‑35 (1972).
How does a calculator compute a sine?
One way is the Taylor series — a polynomial that imitates the sine better and better (more in Derivative and integral). It works only in radians!
A calculator first reduces the angle to a small interval (using the periodicity and symmetries of the circle), and then a few terms are enough.
CORDIC: rotating a vector
We start with the vector (1, 0) and rotate it by smaller and smaller angles arctan(1/2k) — one way or the other, until we hit the target angle. Multiplying by 1/2k is just a bit shift for a computer.
| step | turn | angle | cos ≈ | sin ≈ |
|---|
MathematicsThree faces of the same functions
1. On the unit circle
Draw a circle of radius 1 centered at the origin and measure an angle α from the x-axis (counterclockwise). The angle’s arm meets the circle at a point P = (x, y). Then:
This definition works for every angle — obtuse, negative or larger than 360°. The Pythagorean theorem in the triangle under the radius gives the Pythagorean identity sin²α + cos²α = 1. The tangent shows up on the tangent line x = 1, and the cotangent on the tangent line y = 1 — hence their names (Latin tangens — touching).
2. In a right triangle
a — the leg opposite angle α, b — the leg next to the angle, c — the hypotenuse. In any triangle (sides a, b, c opposite angles α, β, γ; R — radius of the circumcircle; P — area):
3. As waves
When the angle grows steadily with time (α = ωt), the height of the point on the circle describes an oscillation y = A sin(ωt + φ) + c, where ω = 2πf is the angular frequency and the period is T = 1/f. Trigonometric functions are periodic: sin(α + 360°) = sin α, while tan and cot repeat every 180°.
Radians and degrees
The radian measure of an angle is the length of the arc the angle cuts out of a circle of radius 1. The whole circle has length 2π, so 360° = 2π rad, and 1 rad = 180°/π ≈ 57.3°. In English and on calculators you write tan and cot; Polish schools write tg and ctg.
Reduction formulas
Each of them is just a symmetry of the circle: reflecting the point across the y-axis, the line y = x or the x-axis. Check them in the “Unit circle” mode.
Angles in a circle (“Angles and circles” mode)
An inscribed angle is half the central angle that subtends the same arc — the proof uses two isosceles triangles whose sides equal the radius. It follows that all inscribed angles on the same arc are equal, an angle subtending a diameter is a right angle, and in a quadrilateral inscribed in a circle the opposite angles add up to 180°. The angle between a tangent and a chord equals the inscribed angle on that chord. Most of these facts are already in Book III of Euclid’s Elements.
The Polish “Thales’s theorem” is about segments (in English: the intercept theorem): parallel lines crossing the arms of an angle cut off proportional segments on them. The inscribed angle is also behind the law of sines: the angle α is also seen from the end of a diameter of the circumcircle, and there the triangle is a right triangle, so a = 2R sin α.
Simplifications on this page
Eratosthenes’ Earth is a perfect sphere, and Syene lies exactly due south of Alexandria and exactly under the Sun (in reality neither is exact). Tides are a single sine wave with a period of 12 h 25 min, and day length is computed for the center of the Sun’s disk without atmospheric refraction — the real day is a few minutes longer. The pendulum uses the small-swing formula. Parallax angles in the drawings are exaggerated.
Did you know…?
- The word sine comes from a translator’s mistake: the Indian “bowstring” (jyā) passed into Arabic as jība, which was read as jayb — “fold, bosom” — and translated into Latin that way.
- Only in radians is the derivative of sine exactly cosine. In degrees you would get (sin x)′ = π/180 · cos x — that’s why mathematicians and physicists prefer radians.
- “Thales’s theorem” means different things in different countries. In Poland and France it is the theorem about proportional segments on the arms of an angle, while in English-speaking countries and Germany it is the theorem that an angle subtending a semicircle is a right angle. Both are attributed to Thales of Miletus only on the basis of later accounts.
- 1 radian ≈ 57.3°, and for small angles sin x ≈ x (in radians): sin 5° = 0.0872, while 5° = 0.0873 rad. The formula for the period of a pendulum rests on this approximation.
Challenges
- Find an angle where sin = cosIn “Unit circle,” place the point so that the sine and cosine segments are equal.not completed
- Unroll an arc of length πSet the angle whose arc has length π, and click “Unroll arc.”not completed
- Measure the height of a treeIn “Measuring the world,” aim the line of sight exactly at the top.not completed
- Repeat Eratosthenes’ measurementSet his data: an angle of 7.2° and 5,000 stadia.not completed
- Find two triangles at onceIn “Solve a triangle,” enter two sides and the angle opposite one of them so that two different triangles fit.not completed
- A right angle in 5 placesIn “Angles and circles” → “Angle in a semicircle,” put point P in five different places on the circle.not completed
- Measure a pyramid from its shadow (Thales)Set the Sun so that Thales’s shadow equals his height.not completed
- A quadrilateral that can’t be inscribed in a circleUnpin vertex D and build a convex quadrilateral in which ∠A + ∠C clearly differs from 180°.not completed
- Hear the beatsIn “Waves,” turn on the sound and listen for a few seconds to two tones that differ by 1–10 Hz.not completed
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