A Short Introduction to Trigonometry

Trigonometry is an advanced field of mathematics typically offered to upperclassmen in high school, or at the college level.  It is the study of triangles, specifically the measurements of the angles and the sides and the relationships to each.  The study of trigonometry began with the Sumerians in ancient times, who devised a system of measuring angles while looking at the stars.  But it was the ancient Greeks who eventually brought order to this science.

There are some basic principles that must be understood before delving into trigonometry.  One of the first and most fundamental of them is that if all of the angles are a triangle are added together, the result will always be 180 degrees.  Therefore, if you have a right triangle (meaning one of the angles measures ninety degrees), and you know the measurement of another angle, then you will be able to figure out the third.  For example, if angle A measures ninety degrees (right triangle), and we know that angle B measures fifty degrees, then angle C must measure forty degrees because A (90) + B (50) + C (?) must equal 180.  Angles B and C add up to 90, and are therefore “complementary.”

Each side of a triangle is defined by its relationship to the right angle.  The hypotenuse, always the longest side, is the side opposite the right angle.  The two remaining sides are the adjacent leg (adjacent to angle A) and the opposite leg (opposite to angle A).  The adjacent leg is sometimes referred to as the base and the opposite leg is sometimes referred to as the perpendicular side. 

The length of the sides can be determined by the measurements of the angles. If we can determine the degrees for each of the angles, and we are given the length of one of the sides, then we can calculate the length of the remaining two sides; the ratios are fixed.  The ratios are formulated using the sine, cosine, and tangent functions.  The Sine function (sin) is defined as the ratio of the side opposite the angle to the hypotenuse; the Cosine function (cos) is defined as the ratio of the adjacent leg to the hypotenuse; and the Tangent function (tan) is defined as the ratio of the opposite leg to the adjacent leg.

The Sine, Cosine, and Tangent functions have both mathematical reciprocals and inverse functions.  Respectively, the reciprocals are called the cosecant (csc or cosec), secant (sec), and cotangent (cot).  The inverse functions simply put the letters “arc” before the name of the functions, and are therefore known as the arcsine, arccosine, and arctangent.

This is a very brief and extremely fundamental overview of some of the basic concepts of trigonometry.  There are practical applications beyond rattling the brains of young students.  The science is most useful in calculating geometric functions.  In the real world, trigonometry can be used in astronomy to measure the distance between the earth and the stars.  It is also used by navigators to measure the distance between landmarks, as well as numerous scientists to make calculations pertaining to light and sound waves.