Trigonometry

Trigonometry (from the Greek trigonon = three angles and metro = measure) is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine, cosine and tangent. It has some relationship to geometry, though there is disagreement on exactly what that relationship is; for some, trigonometry is just a subtopic of geometry.

Early history of trigonometry

Table of Trigonometry, 1728 Cyclopaedia

The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago. It seems that the Babylonians based trigonometry on their base sixty numeral system.

Indian mathematicians were the pioneers of variable computations algebra for use in astronomical calculations along with trigonometry. Lagadha (circa 1350-1200 BC) is the first known mathematician to have used geometry and trigonometry for astronomy in his Vedanga Jyotisha; much of his work was destroyed by foreign invaders of India.

The earliest use of sine appears in the Sulba Sutras written in India, between 800 BC and 500 BC, which correctly computes the sine of π/4 (45°) as 1/√2 in a procedure for circling the square (the opposite of squaring the circle).

Hellenistic mathematician Hipparchus circa 150 BC compiled a trigonometric table for solving triangles.

Ancient Sinhalese, Sri Lankans had used trigonometry when constructing Weva (reservoirs) in the Anuradhapura kingdom, where trigonometry was used to calculate the gradient of the water flow. Minipe ela had a gradient of 6 inches to a mile.

Hellenized Egyptian mathematician Ptolemy circa 100 AD further developed trigonometric calculations in Egypt.

Indian mathematician Aryabhata in 499, gave tables of half chords which are now known as sine tables, along with cosine tables. He used zya for sine, kotizya for cosine, and otkram zya for inverse sine, and also introduced the versine.

Another Indian mathematician Brahmagupta in 628, used an interpolation formula to compute values of sines, up to the second order of the Newton-Stirling interpolation formula.

Persian mathematician Omar Khayyám (1048-1131) combined the use of trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means. Khayyam solved the cubic equation $x^{3}+200x=20x^{2}+2000$ and found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables.

Detailed methods for constructing a table of sines for any angle were given by Indian mathematician Bhaskara in 1150, along with some sine and cosine formulas. Bhaskara also developed spherical trigonometry.

The 13th century Persian mathematician Nasir al-Din Tusi, along with Bhaskara, was probably the first to treat trigonometry as a distinct mathematical discipline. Nasir al-Din Tusi in his Treatise on the Quadrilateral was the first to list the six distinct cases of a right angled triangle in spherical trigonometry.

In the 14th century, Persian mathematician al-Kashi and Timurid mathematician Ulugh Beg (grandson of Timur) produced tables of trigonometric functions as part of their studies of astronomy.

The Silesian mathematician Bartholemaeus Pitiscus published an influential work on trigonometry in 1595 and introduced the word to the English and French languages.

Trigonometry today

There are an enormous number of applications of trigonometry. Of particular value is the technique of triangulation which is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. Other fields which make use of trigonometry include astronomy (and hence navigation, on the oceans, in aircraft, and in space), music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.

An alternative approach to trigonometry has been propounded recently by Dr. Norman Wildberger of the University of New South Wales. He terms this rational trigonometry, and it differs from classical trigonometry in two fundamentally different measurement parameters. Instead of length, he uses the square of length (termed "quadrance"), and instead of angle, he uses a non-linear measure of separation (termed "spread") which ranges from 0 (parallel lines) to 1 (perpendicular lines). Rational trigonometry does not use any transcendental values, and is entirely solved through algebra and quadratic equations.

About trigonometry

Two triangles are said to be similar if one can be obtained by uniformly expanding the other. This is the case if and only if their corresponding angles are equal, and it occurs for example when two triangles share an angle and the sides opposite to that angle are parallel. The crucial fact about similar triangles is that the lengths of their sides are proportionate. That is, if the longest side of a triangle is twice that of the longest side of a similar triangle, say, then the shortest side will also be twice that of the shortest side of the other triangle, and the median side will be twice that of the other triangle. Also, the ratio of the longest side to the shortest in the first triangle will be the same as the ratio of the longest side to the shortest in the other triangle.

Right triangle

Using these facts, one defines trigonometric functions, starting with right triangles, triangles with one right angle (90 degrees or π/2 radians). The longest side in any triangle is the side opposite the largest angle.

Because the sum of the angles in a triangle is 180 degrees or π radians, the largest angle in such a triangle is the right angle.

The longest side in such a triangle is therefore the side opposite the right angle and is called the hypotenuse. Pick two right angled triangles which share a second angle A. These triangles are necessarily similar, and the ratio of the side opposite to A to the hypotenuse will therefore be the same for the two triangles. It will be a number between 0 and 1, because the hypotenuse is always larger than either of the other two sides, which depends only on A; we call it the sine of A and write it as sin(A), or simply sin A. Similarly, one can define the cosine of A as the ratio of the side adjacent to A to the hypotenuse.

$\sin A={{\mbox{opp}} \over {\mbox{hyp}}}\qquad \cos A={{\mbox{adj}} \over {\mbox{hyp}}}$

These are by far the most important trigonometric functions; other functions can be defined by taking ratios of other sides of the right triangles but they can all be expressed in terms of sine and cosine. These are the tangent, secant, cotangent, and cosecant.

$\tan A={\sin A \over \cos A}={{\mbox{opp}} \over {\mbox{adj}}}\qquad \sec A={1 \over \cos A}={{\mbox{hyp}} \over {\mbox{adj}}}$

$\cot A={\cos A \over \sin A}={{\mbox{adj}} \over {\mbox{opp}}}\qquad \csc A={1 \over \sin A}={{\mbox{hyp}} \over {\mbox{opp}}}$

The sine, cosine and tangent ratios in right triangles can be remembered by SOH CAH TOA (sine-opposite-hypotenuse cosine-adjacent-hypotenuse tangent-opposite-adjacent). It is commonly referred to as "Sohcahtoa" by math teachers, who liken it to a (nonexistent) Native American girl's name. See trigonometry mnemonics for other mnemonics.

So far, the trigonometric functions have been defined for angles between 0 and 90 degrees (0 and π/2 radians) only. Using the unit circle, one may extend them to all positive and negative arguments (see trigonometric function).

Once the sine and cosine functions have been tabulated (or computed by a calculator), one can answer virtually all questions about arbitrary triangles, by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and an angle or two angles and a side or three sides are known.

Some mathematicians believe that trigonometry was originally invented to calculate sundials, a traditional exercise in the oldest books. Trigonometry is also a great asset in surveying.

Common formulas

Main article: Trigonometric identity

Pythagorean identities

$\sin ^{2}A+\cos ^{2}A=1\,$

$1+\tan ^{2}A=\sec ^{2}A\,$

$1+\cot ^{2}A=\csc ^{2}A\,$

Sum and difference identities

$\sin(A+B)=\sin A\cos B+\cos A\sin B\,$

$\sin(A-B)=\sin A\cos B-\cos A\sin B\,$

$\cos(A+B)=\cos A\cos B-\sin A\sin B\,$

$\cos(A-B)=\cos A\cos B+\sin A\sin B\,$

$\tan(A+B)={\frac {\tan A+\tan B}{1-\tan A\tan B}}\,$

$\tan(A-B)={\frac {\tan A-\tan B}{1+\tan A\tan B}}\,$

Double-angle identities

Identity

$\sin 2A=2\sin A\cos A\,$

Proof

$\sin 2A=sin(A+A)=sinA\cos A+cosA\sin A=2\sin A\cos A\,$

Identity

$\cos 2A=\cos ^{2}A-\sin ^{2}A=2\cos ^{2}A-1=1-2\sin ^{2}A={1-\tan ^{2}A \over 1+\tan ^{2}A}\,$

Proof

$\cos 2A=\cos(A+A)=cosA\ cosA-sinA\ sinA=cos^{2}A-sin^{2}A.$

Identity

$\tan 2A={2\tan A \over 1-\tan ^{2}A}={2\cot A \over cot^{2}A-1}={2 \over cotA-\tan A}\,$

Proof

Failed to parse (syntax error): {\displaystyle \tan 2A = tan (A+A) = sin (A+A) \over cos (A+A) = 2 \sin A \ cos A \over cos^2 A - sin^A = cos^2 A \over sin^2 A * 2 \ sin A \ cos A \over cos^2 A - sin^2 A}

Half-angle identities

Note that $\pm$ in these formulas does not mean both are correct, it means it may be either one, depending on the value of A.

$\sin {\frac {A}{2}}=\pm {\sqrt {\frac {1-\cos A}{2}}}\,$

$\cos {\frac {A}{2}}=\pm {\sqrt {\frac {1+\cos A}{2}}}\,$

$\tan {\frac {A}{2}}=\pm {\sqrt {\frac {1-\cos A}{1+\cos A}}}={\frac {\sin A}{1+\cos A}}={\frac {1-\cos A}{\sin A}}\,$

For more identities see trigonometric identity.

Proofs of Pythagorean identities

In trigonometry, the two perpendicular sides of a right triangle are referred to as either opposite or adjacent to a given angle. These sides can also be referred to as the legs of a right triangle. The longest side is referred to as the hypotenuse.

The Pythagorean theorem states that

$\mathrm {opp} ^{2}+\mathrm {adj} ^{2}=\mathrm {hyp} ^{2}\,$

where opp and adj are the lengths of the opposite and adjacent sides to a given angle A and hyp is the length of the hypotenuse. This will be used to prove the three Pythagorean identities.

Dividing both sides of the Pythagorean theorem by hyp² gives

${\frac {\mathrm {opp} ^{2}}{\mathrm {hyp} ^{2}}}+{\frac {\mathrm {adj} ^{2}}{\mathrm {hyp} ^{2}}}=1\,$

$\sin ^{2}A+\cos ^{2}A=1\,.$

Dividing both sides of the Pythagorean theorem by adj² gives

${\frac {\mathrm {opp} ^{2}}{\mathrm {adj} ^{2}}}+1={\frac {\mathrm {hyp} ^{2}}{\mathrm {adj} ^{2}}}\,$

$\tan ^{2}A+1=\sec ^{2}A\,.$

Dividing both sides of the Pythagorean theorem by opp² gives

$1+{\frac {\mathrm {adj} ^{2}}{\mathrm {opp} ^{2}}}={\frac {\mathrm {hyp} ^{2}}{\mathrm {opp} ^{2}}}\,$

$1+\cot ^{2}A=\csc ^{2}A\,.$

External links

• Trigonometric Delights by Eli Maor from Princeton University Press.