Trigonometry

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Trigonometry

Trigonometry (literally "measurement of figures with three angles") is the area of mathematics that deals with relationships between the side lengths and angles of triangles.

Although one can consider triangles on curved surfaces (using spherical and hyperbolic trigonometry), the term trigonometry usually refers to planar trigonometry — that is, the study of triangles in a plane (two-dimensional Euclidean space).

After considering the measurement of angles in degrees and radians, one may define the following six basic trigonometric functions using any right triangle containing a specified acute angle $\theta$ :

$\sin\theta=\frac{\mbox{opp}}{\mbox{hyp}}$
$\cos\theta=\frac{\mbox{adj}}{\mbox{hyp}}$
$\tan\theta=\frac{\mbox{opp}}{\mbox{adj}}$

$\csc \theta ={\frac {\mbox{hyp}}{\mbox{opp}}}$
$\sec \theta ={\frac {\mbox{hyp}}{\mbox{adj}}}$
$\cot \theta ={\frac {\mbox{adj}}{\mbox{opp}}}$

Here "opp" and "adj" represent, respectively, the lengths of the sides opposite and adjacent to the angle $\theta$ , and "hyp" represents the length of the hypotenuse (the side opposite the right angle).

The names of the functions above are abbreviated forms of the words sine, cosine, tangent, cotangent, secant and cosecant.

The values of these six functions for non-acute angles (less than or equal to zero, or equalgreater than a right angle) can be found by reference to the unit circle.

There are many properties of these functions that are true for any given angle. The most fundamental of these are:

$\tan \theta = \frac {\sin \theta}{\cos \theta}$

$\cot \theta ={\frac {\cos \theta }{\sin \theta }}={\frac {1}{\tan \theta }}$

$\sec\theta=\frac{1}{\cos\theta}$

$\csc\theta=\frac{1}{\sin\theta}$

$\sin^2 \theta + \cos^2 \theta = 1$

For other similar properties, see our list of trigonometric identities.

Resource: Online sine, cosine, tangent chart

Tables

Can the function be used?	$\theta = 0^\circ$	$\theta = 90^\circ$	$\theta=180^\circ$	$\theta=270^\circ$	Everything else
$\sin \theta$	Yes	Yes	Yes	Yes	Yes
$\cos \theta$	Yes	Yes	Yes	Yes	Yes
$\tan \theta$	Yes	No	Yes	No	Yes
$\csc \theta$	No	Yes	No	Yes	Yes
$\sec \theta$	Yes	No	Yes	No	Yes
$\cot \theta$	No	Yes	No	Yes	Yes
$\mathrm {c} i\mathrm {s} \;\theta$	Yes	Yes	Yes	Yes	Yes

Multiply...	$\sin \theta$	$\cos \theta$	$\tan \theta$
$\sin \theta$	sin²θ	${\frac {\tan \theta }{hyp^{2}}}$	${\frac {opp^{2}}{\cos \theta }}$
$\cos \theta$	${\frac {\tan \theta }{hyp^{2}}}$	cos²θ
$\tan \theta$			tan²θ

History

Main article: History of trigonometry

Sumerian astronomers studied angle measure, using a division of circles into 360 degrees.^[1] They, and later the Babylonians, studied the ratios of the sides of similar triangles and discovered some properties of these ratios, but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar method.^[2]

The ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. But pre-Hellenic societies lacked the concept of an angle measure and consequently, the sides of triangles were studied instead, a field that would be better called "trilaterometry".^[3]

In the 3rd century BCE, classical Hellenistic mathematicians (such as Euclid and Archimedes) studied the properties of chords and inscribed angles in circles, and proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. The Hellenized Egyptian mathematician Claudius Ptolemy expanded upon Hipparchus' Chords in a Circle in his Almagest.^[4]

The modern sine function was first defined in the Surya Siddhanta, and its properties were further documented by the 5th century (CE) Indian mathematician and astronomer Aryabhata.^[5] The Siddhantas and the Aryabhatiya contain the earliest surviving tables of sine values and versine (1 − cosine) values, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.^[6]

The Indian works were translated and expanded by Islamic mathematicians. By the 10th century, in the work of Abū al-Wafā' al-Būzjānī, Islamic mathematicians were using all six trigonometric functions,^[7] had tabulated their values (Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values^[7]), and were applying them to problems in spherical geometry. At about the same time, Chinese mathematicians also translated Indian works and developed their own field of trigonometry independently, although it was not a major field of study for them. Knowledge of trigonometric functions and methods reached Europe via Latin translations of the works of Persian and Arabic astronomers such as Al Battani and Nasir al-Din al-Tusi.^[8]

One of the earliest works on trigonometry by a European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus. Trigonometry was still so little known in 16th-century Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts.

References

↑ Aaboe, Asger. Episodes from the Early History of Astronomy. New York: Springer, 2001. ISBN 0-387-95136-9
↑ Otto Neugebauer (1975). A history of ancient mathematical astronomy. 1. Springer-Verlag. pp. 744–. ISBN 978-3-540-06995-9. http://books.google.com/books?id=vO5FCVIxz2YC&pg=PA744.
↑ Boyer (1991). "Greek Trigonometry and Mensuration". pp. 158–159. "Trigonometry, like other branches of mathematics, was not the work of any one man, or nation. Theorems on ratios of the sides of similar triangles had been known to, and used by, the ancient Egyptians and Babylonians. In view of the pre-Hellenic lack of the concept of angle measure, such a study might better be called "trilaterometry," or the measure of three sided polygons (trilaterals), than "trigonometry," the measure of parts of a triangle. With the Greeks we first find a systematic study of relationships between angles (or arcs) in a circle and the lengths of chords subtending these. Properties of chords, as measures of central and inscribed angles in circles, were familiar to the Greeks of Hippocrates' day, and it is likely that Eudoxus had used ratios and angle measures in determining the size of the earth and the relative distances of the sun and the moon. In the works of Euclid there is no trigonometry in the strict sense of the word, but there are theorems equivalent to specific trigonometric laws or formulas. Propositions II.12 and 13 of the Elements, for example, are the laws of cosines for obtuse and acute angles respectively, stated in geometric rather than trigonometric language and proved by a method similar to that used by Euclid in connection with the Pythagorean theorem. Theorems on the lengths of chords are essentially applications of the modern law of sines. We have seen that Archimedes' theorem on the broken chord can readily be translated into trigonometric language analogous to formulas for sines of sums and differences of angles."
↑ Marlow Anderson, Victor J. Katz, Robin J. Wilson (2004). Sherlock Holmes in Babylon: and other tales of mathematical history. MAA. p. 36. ISBN 0-88385-546-1
↑ Boyer p. 215
↑ Boyer (1991), p. 215
↑ ^7.0 ^7.1 Boyer (1991) p. 238.
↑ Boyer pp. 237, 274

Bibliography

Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc.. ISBN 0-471-54397-7.
Hazewinkel, Michiel, ed. (2001), "Trigonometric functions", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=p/t094210
Christopher M. Linton (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy . Cambridge University Press.
Weisstein, Eric W. "Trigonometric Addition Formulas". Wolfram MathWorld. Weiner.

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