50 Fun Facts About Trigonometry

In the trig mnemonic SOH-CAH-TOA, the letter O appears twice; which word does it stand for both times?

Opposite is the O in both SOH and TOA: sine is opposite over hypotenuse and tangent is opposite over adjacent. The mnemonic is often sounded out as one word, a little like the volcano Krakatoa.

Mount Everest is named after George Everest, who directed the trigonometric mapping of India in which post?

Surveyor General of India was his title, yet George Everest never saw the peak. His successor Andrew Waugh proposed the name after years of triangle calculations showed it to be the highest known.

The tangent function takes its name from the Latin word tangens, which describes a line doing what to a circle?

Touching is the meaning: a tangent line just grazes a circle at a single point. The secant, from the Latin secans for 'cutting', is the line that slices straight through it.

The degrees used in trigonometry are usually traced to Babylonian astronomers, who counted in which number system?

Base 60 counting, known as sexagesimal, is also why one degree splits into 60 arcminutes and each arcminute into 60 arcseconds.

Which Greek letter, the eighth in its alphabet, is the usual symbol for an unknown angle in trigonometry?

Theta, written θ, stands for the unknown angle in formulas such as sin θ. Although it is the eighth letter, in the Greek numeral system it carried the value 9.

The law of cosines works for any triangle, but with a 90° angle it reduces to which famous formula?

The Pythagorean theorem is the special case: cos 90° is zero, so the extra term −2ab cos C drops out and leaves c² = a² + b².

A flagpole casts a shadow exactly as long as the pole is tall; how high is the Sun above the horizon?

45° is the answer because equal height and shadow form a right triangle with two matching sides, and tan 45° = 1. A longer shadow would mean a lower Sun.

Which technique fixes a distant point's position by measuring only angles to it from both ends of a known baseline?

Triangulation needs just one measured baseline, since every other distance follows from the angles. Trilateration, by contrast, works from measured distances.

On a sine wave such as a pure musical tone, what name is given to the peak height above the centre line?

Amplitude is the peak deviation from zero, so the curve y = 3 sin x climbs to 3 and dips to −3. A pure tone is a sine wave made of one single frequency.

Which Swiss mathematician's famous formula ties sine and cosine to the number e and the imaginary unit i?

Leonhard Euler's formula states that e^(ix) = cos x + i sin x. First published in 1748, it shows that the trig functions and the exponential function are two faces of the same thing.

In height-and-distance problems, the angle between the horizontal and an upward line of sight is the angle of what?

Elevation is measured upward from the horizontal, while the angle of depression is measured downward to something below. Either one pairs with the tangent ratio to turn a known distance into a height.

A right triangle has shorter sides of 3 and 4 units; how long is its hypotenuse, the side facing the right angle?

5 completes the famous 3-4-5 triangle, since 9 + 16 = 25. It is the only right-triangle shape whose three sides go up in equal steps.

Which word describes sine and cosine, used to model sound and light waves because their values repeat in cycles?

Periodic functions repeat after a fixed interval, and sine and cosine are among the simplest of them, which makes the pair the standard building blocks for studying anything that cycles.

Dividing sin A by cos A gives which other trig ratio of the angle A, which also measures a line's slope?

The ratio is tan A, because the shared hypotenuse cancels when sin A is divided by cos A. Rise over run for a line tilted at angle A is the same quotient.

What is the exact value of sin 30°, found by cutting an equilateral triangle in half?

1/2 is exact: halving an equilateral triangle with sides of 2 leaves a 30° angle facing a side of 1, while the longest side is still 2.

Squaring sin θ and cos θ for any angle θ and adding the two results always gives which number?

1 is the total every time, because sin θ and cos θ are the two shorter sides of a right triangle whose hypotenuse has length 1.

Trig ratios depend only on the angle because all right triangles sharing the same acute angle are what?

Similar triangles have the same shape at any size, so their sides stay in proportion. That is why sin 30° is identical for a triangle on a page and one spanning a valley.

A straight angle of 180° equals how many radians, the standard unit of angle in the SI system?

π radians make a half-turn, because one radian is the angle whose arc equals the radius, and half a circle's edge is π radii long.

Which ancient Greek astronomer compiled the first known table of chords and is often called the father of trigonometry?

Hipparchus drew up his chord table around 140 BC to tackle problems in astronomy. He is even better known for discovering the precession of the equinoxes.

The word sine comes from the Latin sinus, picked by a 12th-century translator of Arabic texts; what did sinus mean?

A bay or fold was the Latin meaning, a long way from the original idea: the Sanskrit word behind the Arabic term meant a bowstring, picturing the chord stretched across an arc.

Which French mathematician showed that any repeating signal can be built by adding simple sine and cosine waves?

Joseph Fourier hit on the idea while working out how heat spreads through a metal plate, and presented his first results in 1807. His series now turn up across acoustics, optics and signal processing.

Which trig function is the reciprocal of sine, even though its name begins with the prefix 'co'?

Cosecant, written csc, is 1 divided by sine, whereas the plain secant is the reciprocal of cosine. Each 'co' name pairs with the basic function that lacks one.

'All Students Take Calculus' tells which trig functions are positive in each of the coordinate plane's what?

Quadrants are the four regions of the coordinate plane. Going anticlockwise from the top right, the initials stand for All, Sine, Tangent and Cosine, the functions that stay positive in each.

Cosine is an even function, with a graph that mirrors across the vertical axis, so cos(−B) always equals what?

The answer is cos B, since flipping the sign of an angle leaves its cosine unchanged. Sine is an odd function instead, so sin(−B) comes out as −sin B.

The graph of y = sin x is an endless wave; after how many degrees does its pattern first start again?

360° is one full turn, which brings the angle back to where it began. The tangent function repeats twice as fast, every 180°.

A square has sides of length 1; how long is its diagonal, the hypotenuse of the two right triangles it creates?

√2, about 1.414, follows from 1² + 1² = 2. The same ratio shapes every sheet of A4 paper, so that cutting one in half gives two sheets of the same proportions.

Besides degrees and radians, many calculators offer which third angle unit, dividing a right angle into 100 parts?

Gradians, also called gons, split a full turn into 400 parts and are used mainly for land measurement in Europe. They are the G on a calculator's DRG key.

Which law of optics uses the sines of two angles to predict how light bends on entering water or glass?

Snell's law says the sines of the incoming and refracted angles keep a fixed ratio for any two materials. In optics it is the rule behind ray tracing, which works out where each beam of light will go.

What is the exact value of tan 60°, the ratio of the longer leg to the shorter leg in a 30-60-90 triangle?

√3, about 1.73, compares the height of an equilateral triangle with sides of 2 to half its base. Flip the fraction over and you have tan 30° instead.

What is the value of cos 180°, the angle that points in exactly the reverse direction to 0°?

−1 is the value, because a half-turn carries the point on the circle to its far left, at (−1, 0), and the cosine is simply that x-coordinate.

Astronomers use trigonometry to find the distance to nearby stars by measuring which tiny apparent shift in position?

Parallax is the shift a nearby star shows against distant ones as Earth moves around its orbit. Friedrich Bessel published one of the first such measurements, for the star 61 Cygni, in 1838.

Aristarchus of Samos estimated the Sun's distance using the right angle formed when which body appears exactly half lit?

The Moon at half phase sits at the right-angled corner of a triangle with Earth and the Sun. Aristarchus put the angle at Earth at 87°, which made the Sun 18 to 20 times farther away than the Moon.

The 'co' in cosine is short for which word, since an angle's cosine is the sine of that partner angle?

Complementary angles add up to a right angle. Edmund Gunter coined 'cosinus' in 1620 as shorthand for the Latin complementi sinus.

Which rule solves a triangle when you know two angles and one side, by matching each side to the angle facing it?

The law of sines says each side divided by the sine of the angle facing it gives the same number, which is the diameter of the circle through all three corners.

The calculator key marked sin⁻¹ gives which inverse function, often mistaken for 1 divided by sine?

Arcsine takes a ratio and returns the angle that has that sine, so the raised minus one means 'inverse', not 'one over'. The 'arc' is the arc of a circle that the angle spans.

The graph of tan x keeps approaching certain vertical lines without ever reaching them; what are such lines called?

Asymptotes take their name from the Greek for 'not falling together'. The graph of tan x has one for every half-turn of the angle.

Which curve, described by the hyperbolic cosine function cosh, is the shape of a chain hanging freely from two points?

A catenary looks much like a parabola, and its equation was only worked out in 1691, by Leibniz, Huygens and Johann Bernoulli. The name comes from catena, the Latin word for chain.

One radian is the angle whose arc is as long as the circle's radius; roughly how many degrees is that?

57° is close, with 57.3° a better estimate, which is why just over six radians fit into one full turn.

The double-angle identity rewrites sin 2θ as which expression, found by making both angles equal in the sum formula?

The result is 2 sin θ cos θ, which drops out of sin(u + v) = sin u cos v + cos u sin v when u and v are the same angle. The cosine version, cos 2θ, is cos²θ − sin²θ.

Adding 1 to tan²x gives the square of which trig function, an identity valid wherever cos x is not zero?

The result is sec²x, so 1 + tan²x = sec²x. Calculus students meet the pair constantly, because sec²x is also the derivative of tan x.

Which combination of known sides and angles is called the ambiguous case because it can fit two different triangles?

SSA, meaning two sides and an angle that is not between them, can describe two triangles because the free side may swing to meet the base at two different points.

Which treatise by Ptolemy contains a table of chords that served as the ancient world's trig table?

The Almagest lists chord lengths for arcs in half-degree steps, doing the job of a sine table. Its name blends the Arabic 'al' with the Greek word for 'greatest'.

Which German scholar gave trigonometry its name by publishing a book titled Trigonometria in 1595?

Bartholomaeus Pitiscus was a theologian as well as an astronomer, and the title of his book became the name of the whole subject.

Which Indian astronomer, born in 476, included an early table of sines in a verse treatise named after himself?

Aryabhata set out the table in his Aryabhatiya, a treatise written in terse Sanskrit verse. India's first satellite was named Aryabhata in his honour.

Which French-born mathematician gives his name to the formula for raising cos x + i sin x to a whole-number power?

Abraham de Moivre never actually stated the formula in his own works. A Huguenot who fled France for London, he counted Isaac Newton among his friends.

Navigators long found the shortest distance between two points on the globe with a formula named after which function?

Haversine is a contraction of 'half-versed sine', coined by James Inman in 1835. The formula built on it gives the great-circle distance between two points from their latitudes and longitudes.

Before logarithms, astronomers used a trig method called prosthaphaeresis to speed up which arithmetic operation?

Multiplication turned into adding and subtracting table values, thanks to identities that rewrite a product of cosines as a sum. Tycho Brahe was the method's best-known user.

Which physicist named the identity cos 20° × cos 40° × cos 80° = 1/8 'Morrie's law' after a boyhood friend?

Richard Feynman took the name from Morrie Jacobs, a boy he knew in childhood, and remembered the curious identity for the rest of his life.

Which coordinate system fixes a point by a distance and an angle, then converts to x and y using cosine and sine?

Polar coordinates give a distance r and an angle φ, so that x = r cos φ and y = r sin φ. Jacob Bernoulli was using a version of the system in print by 1691.

Which telescope-like instrument on a tripod measures both horizontal and vertical angles for map-makers and builders?

A theodolite is a movable telescope that turns about a horizontal and a vertical axis and reads off the angle on each, giving the raw numbers for trig calculations in the field.