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.

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 whose works were 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.

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 Khayyam (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 ${\displaystyle x^{3}+200x=20x^{2}+2000}$ 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.

There are enormously many 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, called rational trigonometry, in which the sine and distance functions are replaced by their squares, has been propounded recently by Dr. Norman Wildberger of the University of New South Wales.

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.

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.

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

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.

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

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

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 (non existent) 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. It is also very important for surveying.

Main article: Trigonometric identity

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

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

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

${\displaystyle \sin(A+B)=\sin A\cos B+\cos A\sin B\,}$

${\displaystyle \sin(A-B)=\sin A\cos B-\cos A\sin B\,}$

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

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

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

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

${\displaystyle \sin 2A=2\sin A\cos A\,}$

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

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

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

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

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

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

For more identities see trigonometric identity.

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 ${\displaystyle a^{2}+b^{2}=c^{2}}$ in which the sides a and b are legs of a right triangle and c is the hypotenuse. Since the opposite and adjacent sides are also the legs of a triangle, they may be termed as sides a and b. Therefore, the length of the opposite side squared plus the length of the adjacent side squared is equal to the length of the hypotenuse squared.

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

This will be used to prove the three Pythagorean identities.

By definition, sin A is the opposite divided by the hypotenuse and cos A is the adjacent divided by the hypotenuse. By using substitution, the original equation

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

can be rewritten as

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

By multiplying each side by ${\displaystyle \mathrm {hyp} ^{2}}$ the equation will become

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

which follows from the Pythagorean theorem.

Because tan A is equal to the opposite divided by the adjacent and sec A is equal to the hypotenuse divided by the adjacent, the original equation

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

can be rewritten like this:

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

By multiplying each side of the equation by ${\displaystyle adj^{2}}$ the equation will become

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

which follows from the Pythagorean theorem.

Because cot A is equal to the adjacent divided by the opposite, and csc A is equal to the hypotenuse divided by the opposite, the original equation

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

can be rewritten like this:

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

By multiplying each side of the equation by ${\displaystyle \mathrm {opp} ^{2}}$ the equation will become

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

which follows from the Pythagorean Theorem.

• Trigonometry--What is it good for?     First of 7 parts of a quick course at the high school level.
• Trigonometric Delights by Eli Maor from Princeton University Press.