Sine Cosine and Tangent

Sine, Cosine, And Tangent

When using trigonometric functions, the three sides of a right triangle (opposite, adjacent and hypotenuse) are identified in relation to a chosen angle. The trigonometric functions (sine, cosine, and tangent) are then defined in relation to the three sides of the right triangle.

The word "sine" comes from the Latin word "sinus," which means a bend or gulf, or the bosom of a garment. (Gelfand) The term was used as a translation for the Arabic word "jayb," the word for a sine that also meant the bosom of a garment, and which in turn comes from the Sanskrit word "jiva," which translates to bowstring.

Originally the word "sine" was applied to the line segment CD on a figure, which meant it was half the chord of twice the angle AOB. A sine resembles a bowstring in this regard. The ratio of the sine CD to the radius of the circle, OA, is the sine of angle AOB.

The word "cosine" was originally written like this: "co.sine," which was the abbreviated version of the sine of the complement, or "complimenti sinus." The cosine of an angle is the sine of the complementary angle.

Relationships between the lengths of the sides and the sizes of the angles of a triangle are given by the cosine and sine rules. If there is enough information already, the cosine and sine rules can be used to find the length of a side, or the size of an angle. Typically, these methods are used only for triangles that do not have a right angle because easier methods for finding the dimensions of right-angled triangles exist.

The word "tangent" is taken from the Latin word "tangens," which is related to the word "tangere," meaning "to touch." (Gelfand) The term was originally applied to the line segment AB in a figure: the segment of the tangent to the circle at A that is cut off by the extension of OB. The ratio of the tangent AB to the radius of the circle, OA, is the tangent of angle AOB.

History of Sine, Cosine and Tangent

As far back as 140 B.C., when Greek mathematician Hipparchus was the first to study triangular geometry, trigonometry has existed. In these early days, trigonometry was used for astronomy. (Kay) Hipparchus is considered the founding father of trigonometry because he is said to have written the first 12 books on tables of chords.

Over the years, trigonometry has advanced. Another Greek mathemetician, Apollonius, was the next to contribute to the field. Apollonius formulated planet's position, leading to Hipparchus's tabulating ratios enabling planet positions.

The next important discovery in trigonometry came from the Islamic people, who also related astronomy to trigonometry. (Kaye) Islamic mathematician Abu Abdallah Muhammad ibn Jabir al-Battani formally introduced cosine. The Islamic people also resurfaced the tangent function. The cotangent, secant and cosecant functions were introduced and it was later made known that they were the reciprocals of tangent, sine, and cosine.

The next group to add to the findings of trigonometry was the Chinese, followed by the Indians. The Chinese concentrated on astronomy and trigonometry and formally introduced the tangent function. However, their other advances were not continued. The Indians used sine tables, using the Greek half-angle formula to guide them. They also constructed cosine tables.

Mathematic astronomy was now called spherical trigonometry because of its usefulness in describing spherical objects in terms of circumference and volume. Because of trigonometry, it was now possible to determine the approximate volume of a star simply by finding its diameter. When it was first discovered, people used simple right-angle trigonometry to find heights of mountains and tall buildings.

It was soon discovered that the entire wave spectrum could be described in terms of frequency and amplitude, and graphed by trigonometric functions, such as sine, cosine and tangent.

The Babylonian measure of 360° formed the study of chords. With this information, sine and cosine were loosely defined as =1. Another Greek mathematician, Menelaus, wrote six books on chords. Ptolemy subsequently created a complete chord table. His new discovery included a variety of different theorems such as a quad inscribed inside a circle has the property that the product of the diagonals = sum of products of opposite sides; the half angle theorem; the sum and difference formulae; the inverse trigonometry functions; and more sine and cosine rules.

How Sine, Cosine and Tangent are Used Today

Today, sine, cosine and tangent are still used for astronomy and for geography, as well as in navigation and mapmaking. The trio is also used in physics with the study of visible light and fluid motion. Engineers today use trigonometric functions for military engineers and conveyors.

Trigonometric functions are the functions of an angle. These functions are important when studying triangles and modeling periodic phenomena. The trigonometric functions may be accurately defined as ratios of two sides of a right triangle containing the angle, or as ratios of coordinates of points on the unit circle.

Of the six trigonometric functions, sine, cosine and tangent are the most important. Sine, cosine, and tangent are used when you know an angle and a length of one of the sides of a right triangle, and you want to know the length of another side. For these functions, the angle is in radians, not degrees