Trigonometric functions

A TrigonometricFunction is a function of an angle defined by a ratio of two sides of a right triangle that contains that angle.

Although this definition implies that the TrigonometricFunctions are defined only for angles of less than 90 degrees, they are defined on all angles whose measure is a real number.

There are six basic TrigonometricFunctions.

     *    Sine

     *    Cosine

     *    Tangent

     *    Secant

     *    Cosecant

     *    Cotangent

There are then six definitions, one for each function. To illustrate these definitions, see the right triangle below (Figure1).

                                              Angle B

                                                  /l

                                               /   l

                                            /      l          a

                        c

                                         /         l

                                      /            l

                        Angle A     /______________l  

                                                            Angle C

                                        b

                               Figure 1

Using the angle A to define these functions, special names are used for the sides of

this triangle in the definitions.

• The hypotenuse is the side opposite the right angle, in this case c.

• The opposite side is the opposite the angle on which the function is defined, in this case a.

• The adjacent side is the side that is a leg of the angle, but not the hypotenuse, in this case b.

Then,

1). The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse, abbreviated "sin."

In general the sin (theta) = length of the opposite side/length of the hypotenuse.

In our example the sin (A) = a/c.

2). The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, abbreviated "cos."

In general, the cos (theta) = length of the adjacent side/length of the hypotenuse.

In our example, the cos (A) = b/c.

3). The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side, abbreviated "tan."

In general, the tan (theta) = length of the opposite side/ length of the adjacent side.

In our example, the tan (A) = a/b.

The remaining three functions are best defined using the above three functions.

4). The cosecant (A) is the inverse of the ratio of the sin (A), the ratio of the length of the hypotenuse to thelength of the adjacent side, abbreviated "csc."

Then csc (A) = c/a.

5). The secant (A) is the inverse of the ratio of cos (A), the ratio of the length of the hypotenuse to the length of the opposite side, abbreviated "sec."

Then the sec (A) = c/b.

6). The cotangent of (A) is the inverse of the ratio of the tan (A), the ratio of the length of the adjacent side to the length of the opposite side, abbreviated "cot."

Then the cot (A) = b/a.

One familiar mnemonic to remember these definitions is CAHSOHTOA. It reminds

one that "CAH," the cos= adjacent/hypotenuse, "SOA," the sin = opposite/hypotenuse, and "TOA," the tan = opposite/adjacent.

Another mnemonic is commonly used in the UK is OHMS. This is memorable because it might mean "On Her Majesty's Service", which is stamped on the front of mail sent by the government, or "Opposite over Hypotenuse Means Sine".

A simple example will show how easy it is to calculate these functions for a common angle.

Suppose we have a right triangle where the two other angles are equal, and therefore = 45 degrees. Then the length of side b and the length of side c are equal. Now, one can determine the sin, cos and tan of an angle of 45 degrees. Let a = 1, then b = 1.

Using the Pythagorean Theorem, c = sqrt (a^2 + b^2). Then c = sqrt (2). This is illustrated in Figure 2.

                                              Angle B

                                                  /l

                                               /   l

                                            /      l          a = 1

                     c = sqrt(2)

                                         /         l

                                      /            l

                        Angle A     /______________l  

                                                            Angle C

                                        b = 1

                               Figure 2

Then sin (45degrees) = 1/sqrt (2) = sqrt (2)/2,

the cos (45degrees) = 1/sqrt (2) = sqrt (2)/2

and, the tan (45degrees) = sqrt (2)/sqrt (2) = 1.

Using the definitions, the csc (45degrees) = sqrt (2). The sec (45degrees) = sqrt (2), and the cot (45degrees) = 1.

Q. Can you determine the value of the six TrigonometricFunctions for an angle of 60 degrees and for an angle of 30 degrees using only the definitions, the Pythagorean Theorem, and theorems from EuclideanGeometry?----

A. Yes. Take an isosceles triangle and drop a perpendicular from one of the 60 degree angles to the opposite side.The result is two congruent 30-60-90 triangles. For each triangle, the shortest side=1/2, the next largest side =(sqrt(3))/2 and the hypotenuse = 1.----

The next step is to determine the value of these six functions for angles greater than or equal to 90 degrees. Finally, we must determine the value of the functions for angles that are less than or equal to 0, thus defining these functions over the real numbers. Other closely related topics are the graphs of these functions, the

Trigonometric Identities, and, in Calculus, the mathematical limit, the Continuity and Differentiability of each these functions.

---

Footnote:

It is interesting to note that equivalent definitions (when the angle is measure in radians) are given by

• cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...

• sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...

(The equivalence of these definitions is related to theoryderivation of Taylor series). These are often used as the starting point since the theory of such infinite series is well known. The differentiability and continuity is then easily established, as is The Most Remarkable Formula In The World.