Right Triangle definition of trigonometric functions
A right angle triangle is composed of a right angle and two acute angles, which are angles less than
a right angle(90 degrees). It is conventional to label the acute angles with Greek
letters. We will label one of the angle with the letter θ ("THAY-ta").
As
for the sides, the side opposite to the right angle is called the hypotenuse.
Each acute angle is formed by the hypotenuse and the side adjacent to
the angle. Thus, angle θ is formed by the hypotenuse and
side adjacent to θ.
With
respect to angle θ, we have an adjacent side to angleθ and on side opposit to angle θ.
The ratios of sides
Any two sides of the right angle triangles are related and formed ratios.
There are 6 ratios: the ratio of the opposite side to the hypotenuse; the adjacent
side to the hypotenuse; and so on. Those six ratios have theirs unique names and abbreviations,
They are as follows:
They are as follows:
sine of θ |
= |
sin θ |
= |
opposite hypotenuse |
cosecant of θ |
= |
cosec θ |
= |
hypotenuse opposite |
|
cosine of θ |
= |
cos θ |
= |
adjacent hypotenuse |
secant of θ |
= |
sec θ |
= |
hypotenuse adjacent |
|
tangent of θ |
= |
tan θ |
= |
oppositeadjacent |
cotangent of θ |
= |
cot θ |
= |
adjacenopposite |
|
Notice that each ratio in the
right-hand column is the inverse, or the reciprocal, of the ratio at the left-hand column.
1. The
reciprocal of sin θ is cosec θ ; and vice-versa.
2. The
reciprocal of cos θ is sec θ;
3. And
the reciprocal of tan θ is cot θ;
Each ratio is a function of the acute angle. i.e. one quantity is a "function" of another and its value depends on the value of the other.
The
value of each ratio depends only on the value of the acute angle. That is why
we say that those ratios are function of the acute angles. We call
them the trigonometric functions of the acute angle.
All of trigonometry
is based on the definitions of
those functions.