Trigonometry is a study of math whose origins date back to the ancient Egyptians. The principles of trigonometry deal mostly with the sides, angles and functions of triangles. The most common triangle that is used in trigonometry is the right triangle, which is the basis for the famous Pythagorean Theorem, in which the square of both sides of a right triangle are equal to the square of its longest side or hypotenuse.

### History

The etymology of trigonometry comes from the Greek words "trigonon" (triangle) and "metron" (measure). The person usually associated with inventing trigonometry was a Greek mathematician named Hipparchus. Hipparchus was originally an accomplished astronomer, who observed and applied trigonometric principles to study the zodiac. He is credited with inventing the chord, a function which is the basis for the sine concept. Most of the knowledge regarding Hipparchus' life comes from the writings of Ptolemy, a fellow mathematician and astronomer.

### Pythagorean Theorem

The Pythagorean Theorem is, perhaps, the most well-known math theorem. The theorem is named after its creator, Pythagoras, a Greek mathematician and philosopher. One legend suggests that after discovering the theorem, the philosopher was so ecstatic, he sacrificed his oxen as an offering to the gods. The original theorem was formulated by arranging three square shapes to form a right triangle. Pythagorean triples are side lengths which, when applied to the equation, (a2 + b2 = c2), result in all whole numbers.

### Functions

There are six trigonometric functions: sine, cosine, tangent and their reciprocal functions, secant, cosecant and cotangent. These functions are found by the ratios of a triangle's sides. For example, in right triangles, the sine is equal to the side opposite to the angle divided by the side adjacent to the angle. The secant of a function is 1 divided by the sine, or the hypotenuse divided by the opposite side.

### The Law of Sines

The law of sines is a principle in trigonometry used to calculate the sides or angles of any triangle, given information about the remaining angles and/or sides. The law of sines states that: a/(sin a) = b/(sin b) = c/(sin c), where a, b and c are all side lengths. For example, you can use the law of sines to calculate the measurement of side c, based on the given information for triangle abc: side a = 10, angle a = 20 degrees and angle c = 50 degrees. Plug the numbers into the formula: Sin 20/10 = Sin 50/c. Cross-multiply: c(sin 20) = 10 (sin 50). Divide both sides by sin 20 to solve for c: c = (10 x sin 50)/(sin 20). Input in a calculator to find: c ~ 22.4.