Trigonometric function

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Intro

A trigonometric function is a function of an angle important when studying triangles. They may be defined as ratios of two sides of a right triangle containing the angle, or, more generally, as ratios of coordinates of points on the unit circle. Both approaches will be presented below.

There are six basic Trigonometric Functions.

     *    Sine (sin)
     *    Cosine (cos)
     *    Tangent (tan)
     *    Secant (sec)
     *    Cosecant (csc)
     *    Cotangent (cot)

Several relations between these functions are listed on the page about Trigonometric Identities.


Right Triangle Definitions

There are then six definitions, one for each function. To illustrate these definitions, see the right triangle below (Figure1).
http://home.mindspring.com/~fvmnstl/trig-triangle.gif
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.
http://home.mindspring.com/~fvmnstl/trig-triangle-1-1-root2.gif
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 trigonometric functions 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.


Unit Circle Definitions

The six trig functions can also be defined in terms of the unit circle. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles. Using a unit circle as the base definition does, however, permit the trig functions to take negative values as well as 1 and 0 (not possible when dealing with all positive lengths of a real triangle).

The equation for the unit circle is:

x2 + y2 = 1

and it looks like this:

http://www.wikipedia.com/images/UnitCircle.gif

In the picture, some common angles are labled with their values taken from both a positive and negative sense of rotation from the x-axis, in radians. The coordinates of where a line that makes an angle θ with the x-axis intersects the circle are equal to cosθ and sinθ, respectively. The triangle in the graphic reveals that the reason why this is true is because the radius/hypotenuse has a length 1, sinθ = y/1 and cosθ = x/1. The unit circle can be thought of as a way of looking at an infinite number of triangles by varying the lengths of their legs but keeping the length of their hypotenuses equal to 1.

For angles greater than 2π or less than -2π simply continue to rotate around the circle. Mathematically, it works just like the following:

  • sinθ = sin(θ%(2π))
  • cosθ = cos(θ%(2π))

for any angle θ.

Though only sine and cosine were defined directly by the unit circle, the other four trig functions can be calculated by either using basic Trigonometric Identities or by using:

  • hypotenuse = 1
  • opposite = y
  • adjacent = x

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 theory derivation 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.

/Talk