Trigonometry
Trigonometry (from the Greek trigonon = three angles and metron = measure) is a part of elementary mathematics dealing with angles, triangles and trigonometric functions such as sine (abbreviated sin), cosine (abbreviated cos) and tangent (abbreviated tan). It has some connection to geometry, although there is disagreement on exactly what that connection is; for some, trigonometry is just a section of geometry.
Overview and definitions
Trigonometry uses a large number of specific words to describe parts of a triangle. Some of the definitions in trigonometry are:
 Rightangled triangle  A rightangled triangle is a triangle that has an angle that is equal to 90 degrees. (A triangle cannot have more than one right angle) The standard trigonometric ratios can only be used on rightangled triangles.
 Hypotenuse  The hypotenuse of a triangle is the longest side, and the side that is opposite the right angle. For example, for the triangle on the right, the hypotenuse is side c.
 Opposite of an angle  The opposite side of an angle is the side that does not intersect with the vertex of the angle. For example, side a is the opposite of angle A in the triangle to the right.
 Adjacent of an angle  The adjacent side of an angle is the side that intersects the vertex of the angle but is not the hypotenuse. For example, side b is adjacent to angle A in the triangle to the right.
Trigonometric ratios
There are three main trigonometric ratios for right triangles, and three reciprocals of those ratios. There are 6 total ratios. They are:
Sine (sin)  The sine of an angle is equal to the
Cosine (cos)  The cosine of an angle is equal to the
Tangent (tan)  The tangent of an angle is equal to the
The reciprocals of these ratios are:
Cosecant (csc)  The cosecant of an angle is equal to the or
Secant (sec)  The secant of an angle is equal to the or
Cotangent (cot)  The cotangent of an angle is equal to the or
Students often use a mnemonic to remember this relationship. The sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, such as SOHCAHTOA:
 Sine = Opposite ÷ Hypotenuse
 Cosine = Adjacent ÷ Hypotenuse
 Tangent = Opposite ÷ Adjacent
or:
Silly Old Hitler Couldn't Advance His Troops Over America
or:
Sitting On Hard Concrete Always Hurts Try Other Alternatives
Using trigonometry
With the sines and cosines one can answer virtually all questions about triangles. This is called "solving" the triangle. One can work out the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every polygon may be described as a combination of triangles.
Trigonometry is also vital in surveying, in vector analysis, and in the study of periodic functions.
There is also such a thing as spherical trigonometry, which deals with spherical geometry. This is used for calculations in astronomy, geodesy and navigation.
Images

Hipparchus, credited with compiling the first trigonometric table, is known as "the father of trigonometry".

Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can be determined from such measurements.