Trigonometry (called "trig" for short) (from a)............. trigōnon "triangle" + metron "measure") is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has b)............. degrees (right triangles). Trigonometry deals with relationships between the c).............. and the angles of triangles and with the trigonometricd)..................... which describe those relationships.

Trigonometry has applications in both pure mathematics and in e) ............mathematics, where it is essential in many branches of science and technology. Trigonometry is informally called "trig".

A branch of trigonometry, called spherical trigonometry, studies triangles on f)..................., and is important in astronomy and navigation.

Pre-Hellenic societies such as the ancient Egyptians and Babylonians lacked the concept of an angle measure, but they studied the ratios of the sides of similar triangles and discovered some properties of these ratios. Ancient Greek mathematicians such as Euclid and Archimedes studied the properties of the chord of an angle and proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. The sine function in its modern form was first defined in the Surya Siddhanta and its properties were further documented by the 5th century Indian mathematician and astronomer Aryabhata. These Indian works were translated and expanded by medieval Islamic scholars. By the 10th century Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry. At about the same time, Chinese mathematicians developed trignometry 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. 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 explaining its basic concepts.

If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a right triangle is completely determined, up to similarity, by the angles. This means that once one of the other angles is known, the ratios of the various sides are always the same regardless of the overall size of the triangle. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:

The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively.

The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec) and cotangent (cot), respectively. The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities.

The definitions of functions apply to angles between 0 and 90 degrees (0 and π/2 radians) only. Using the unit circle, one can extend them to all positive and negative arguments. The trigonometric functions are periodic, with a period of 360 degrees or 2π radians.

 

 

A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, as in SOH-CAH-TOA.

Sine = Opposite ÷ Hypotenuse

Cosine = Adjacent ÷ Hypotenuse

Tangent = Opposite ÷ Adjacent

The memorization of this mnemonic can be aided by expanding it into a phrase, such as "Some Officers Have Curly Auburn Hair Till Old Age". Any memorable phrase constructed of words beginning with the letters S-O-H-C-A-H-T-O-A will serve.

a) The alligator has to open its mouth wider for the larger number.

 

b) I Viewed Xerxes Loping Carelessly Down Mountains.
    I Value Xylophones Like Cows Dig Milk

 

c) Tweedle-dee-dum and Tweedle-dee-dee,
    Around the circle is pi times d,
    But if the area is declared,
    Think of the formula π "r" squared.

 

d) Apple pie are square.