### TRIGONOMETRY

**Objective**

- To learn the various ways to solve a trigonometric problems .
- To have the know-how to use trigonometry for solving practical problems.
- To help other people with trigonometry.

*The Canadarm robotic manipulator on the International Space Station is operated by controlling the angles of its joints. Calculating the final position of the astronaut at the end of the arm requires repeated use of the trigonometric functions of those angles.*

**Introduction
**

**, informally called**

Trigonometry

Trigonometry

**trig**, is a branch of mathematics that deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). It specifically deals with the relationships between the sides and the angles of triangles; the trigonometric functions, and calculations based upon them. The insights of trigonometry permeate other branches of geometry, such as the study of spheres using spherical trigonometry.

Trigonometry has important applications in many branches of pure mathematics as well as of applied mathematics and, consequently remains applicable in natural sciences. Trigonometry is usually taught in secondary schools, often in a precalculus course.

Trigonometry was probably invented for use in astronomy.[2] The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago.The common practice of measuring angles in degrees, minutes and seconds comes from the Babylonian's base sixty system of numeration. The Sulba Sutras written in India, between 800 BC and 500 BC, correctly computes the sine of π/4 (45°) as 1/√2 in a procedure for circling the square (the opposite of squaring the circle).

The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus circa 150 BC, who compiled a trigonometric table using the sine for solving triangles. Ptolemy further developed trigonometric calculations circa 100 AD.

The ancient Sinhalese in Sri Lanka, when constructing reservoirs in the Anuradhapura kingdom, used trigonometry to calculate the gradient of the water flow. Archeological research also provides evidence of trigonometry used in other unique hydrological structures dating back to 4 BC.

The Indian mathematician Aryabhata in 499, gave tables of half chords which are now known as sine tables, along with cosine tables. He used *zya* for sine, *kotizya* for cosine, and *otkram zya* for inverse sine, and also introduced the versine. Another Indian mathematician, Brahmagupta in 628, used an interpolation formula to compute values of sines, up to the second order of the Newton-Stirling interpolation formula.

In order to learn trigonometry, you should already be familiar with algebra and geometry. From algebra, you should be comfortable with manipulating algebraic expressions and solving equations. From geometry, you should know about similar triangles, the Pythagorean theorem, and a few other things, but not a great deal.

**Applications of trigonometry**

Image source: http:/nasa.gov

Astronomy and Geography

Trigonometric tables were created over two thousand years ago for computations in astronomy. The stars were thought to be fixed on a crystal sphere of great size, and that model was perfect for practical purposes. Only the planets moved on the sphere. (At the time there were seven recognized planets: Mercury, Venus, Mars, Jupiter, Saturn, the moon, and the sun. Those are the planets that we name our days of the week after. The earth wasn't yet considered to be a planet since it was the center of the universe, and the outer planets weren't discovered then.) The kind of trigonometry needed to understand positions on a sphere is called *spherical trigonometry.* Spherical trigonometry is rarely taught now since its job has been taken over by linear algebra. Nonetheless, one application of trigonometry is astronomy.

#### Engineering and physics

Although trigonometry was first applied to spheres, it has had greater application to planes. Surveyors have used trigonometry for centuries. Engineers, both military engineers and otherwise, have used trigonometry nearly as long.Physics lays heavy demands on trigonometry. Optics and statics are two early fields of physics that use trigonometry, but all branches of physics use trigonometry since trigonometry aids in understanding space. Related fields such as physical chemistry naturally use trig.

#### Mathematics and its applications

Of course, trigonometry is used throughout mathematics, and, since mathematics is applied throughout the natural and social sciences, trigonometry has many applications. Calculus, linear algebra, and statistics, in particular, use trigonometry and have many applications in the all the sciences.