trigonometry

At its simplest level, trigonometry deals with the relationships between the sides and angles of triangles. Unknown angles or lengths are calculated by using trigonometrical ratios such as sine, cosine, and tangent. The earliest applications of trigonometry were in the fields of navigation, surveying, and astronomy, and usually involved working out an inaccessible distance such as the distance of the Earth from the Moon.

Branch of mathematics that concerns finding lengths and angles in triangles. In a right-angled triangle the sides and angles are related by three trigonometric ratios: sine, cosine, and tangent. Trigonometry is used frequently in navigation, surveying, and simple harmonic motion in physics.

Using trigonometry, it is possible to calculate the lengths of the sides and the sizes of the angles of a right-angled triangle as long as one angle and the length of one side are known, or the lengths of two sides. The longest side, which is always opposite to the right angle, is called the hypotenuse. The other sides are named depending on their position relating to the angle that is to be found or used: the side opposite this angle is always termed opposite and that adjacent is the adjacent. So the following trigonometric ratios are used:

sine = opposite/hypotenuse

cosine = adjacent/hypotenuse

tangent = opposite/adjacent

Using trigonometry it is possible to measure the height of a tall object from a vantage point at a known distance. The following example shows the calculation of the height of a tree, from a point 22 m from its base. The angle at this point to the top of the tree is 33°:

Using the trigonometric ratio: tangent θ = opposite/adjacent

tangent 33° = h/22

h = 22 × tan 33°, so h = 22 × 0.6494 = 14.29 (to two decimal places)

The height of the tree is 14.29 m.

beyond 90° The trigonometric ratios have a meaning for any angle, not just angles between 0° and 90°. These ratios (or functions) can be expressed as trigonometric graphs, using angles of any size. The sine rule and cosine rule are both used to calculate unknown angles or sides in triangles where there is not a right angle.

Three-dimensional and spherical trigonometry The methods of elementary trigonometry can be used to solve problems in three dimensions by considering triangles in different planes that have a side in common. This may also involve the use of a dropping perpendicular, that is the envisaging of an imaginary line from a point above the base that will fall vertically to the base. Spherical triangles can be solved using the trigonometric functions, though the formulae are not the same as those employed for plane triangles.

History of trigonometry Trigonometry arose out of the study of astronomy, and was originated by the Greek astronomer Hipparchus. It was also known to early Hindu and Arab mathematicians. Ptolemy the Alexandrian astrologer greatly extended the subject and German astronomer Regiomontanus made it a science independent of astronomy much later on, when he began compiling trigonometric tables in 1467.