Trigonometry is defined as the measurement of triangles. The name, trigonometry, is derived from the Greek word trigon for triangle and metry for measure. Trigonometry consists of learning of the:

1. types of triangles
2. the relationship of their sides
3. the relationship of the interior angles
4. the manner in which the interior angles relate to the sides of the triangle
5. the manner in which trigonometry relates to repetitive functions

TYPES OF TRIANGLES:

Triangles are closed figures having 3 sides and 3 interior angles. The sum of the interior angles in a triangle adds up to 180 degrees. The triangle of primary interest in the study of trigonometry is the right triangle shown below. It is called a right triangle because one of the interior angles is a 90 degree angle. It is usually indicated by the small square shown in the diagram below. There are many types of triangles, all of which can be partitioned into right triangles. Three types of triangles are discussed below. They are 1) obtuse 2) acute and 3) right. The names obtuse, acute and right simply refer to the angle of the triangle. Examine the obtuse and acute triangles and note they may be partitioned to create 2 right triangles.Other triangles may be partitioned in a similar manner to create right triangles. To perform trigonometric analysis on non-right triangles, it is necessary to partition them into right triangles. In the study of trigonometry we will discuss the triangle relative to the relationship of sides as well as the relationship of the sides to the interior angles. That's what trigonometry is all about.

The figure presented below shows a right angle, an obtuse angle, and an acute angle. When the third side is added it creates a triangle with the same name. This is depicted in the triangles shown under each of the related angles. Refer to the right triangle - Note the 3 sides - The 2 straight lines are simply called the sides, while the slanted line is called the hypoteneus. The horizontal line is frequently referred to as the base. In PYTHAGOREAN'S THEORUM presented below you will see the mathematical relationship with sides.This is a simple equation to memorize and it's one of the few things you should commit to memory.

In fact when you learn the following relationships you will have a good basic understanding of Trigonometry. The relationships will be followed by a more extensive discussion of the subject.

PYTHAGOREAN'S THEORUM

c² = a² + b² and c = (a² + b²)^1/2From algebra remember that square root may be shown as the exponent 1/2 That is, the 1/2 power is synonymous with square root.

Trigonometric Functions

Sine z = a/c

Cosine z = b/c

Tangent z = a/b

Cotangent z = b/a

Secant z = c/b

Cosecant z = c/a

Napoleon's engineer used the foregoing knowledge effectively during one of Napoleans battles. It seems that his enemy was located on the other side of the river and he needed to know the distance across the river where the enemy was located in order to adjust his cannon fire. His engineer located a point directly across the river from the enemy. Note the right triangle shown above. Let the vertical line represent the unknown distance across the river and the horizontal line represent a line that Napoleans engineer could measure to a specific length. He then sighted the hypotoneus from the end of the horizontal line to the enemies encampment. He then detetermined the angle between the horizontal line and the hypotoneus. With this knowledge he then determined the length of the vertical line, the distance to the enemy. You can see from the trigonometric relationships shown above that there is a relationship between any 2 sides and the related interior angle.

In Napoleons case the engineer used the tangent relationship. He measured and thus knew the length of the horizontal line and the angle between the horizontal line and the hypoteneus. With this he determined the length of the vertical line. He knew the tangent of the angle = opposite side divided by the adjacent side and solved this equation for the opposite side. That is: Let opposite side = a and adjacent side = b, then:

Tangent of angle = a/b and

a = (b)(tangent of angle a)

When you study and understand the forgoing you have knowledge of the Essence of Trigonometry. I discuss the foregoing in a variety of ways but in the end I will have done nothing more than repeat what is said above.