Evaluation of Trigonometric functions

Periodic Functions

If an angle corresponds to a point Q(x,y) on the unit circle, it is not hard to see that the angle corresponds to the same point Q(x,y), and hence that

Moreover, is the smallest positive angle for which Equations 1 are true for any angle . In general, we have for all angles :

We call the number the period of the trigonometric functions and , and refer to these functions as being periodic. Both and are periodic functions as well, with period , while and are periodic with period .

EXAMPLE 1Find the period of the function .

Solution: The function runs through a full cycle when the angle 3x runs from 0 to , or equivalently when x goes from 0 to . The period of f(x) is then .

EXERCISE 1Find the period of the function .

Solution

Evaluation of Trigonometric functions

Consider the triangle with sides of length and hypotenuse c>0 as in Figure 1 below:

Figure 1

For the angle pictured in the figure, we see that

There are a few angles for which all trigonometric functions may be found using the triangles shown in the following Figure 2.

Figure 2

This list may be extended with the use of reference angles (see Example 2 below).

EXAMPLE 1: Find the values of all trigonometric functions of the angle .

Solution: From Figure 2, we see that the angle of corresponds to the point on the unit circle, and so

EXAMPLE 2: Find the values of all trigonometric functions of the angle .

Solution: Observe that an angle of is equivalent to 8 whole revolutions (a total of ) plus , Hence the angles and intersect the unit circle at the same point Q(x,y), and so their trigonometric functions are the same. Furthermore, the angle of makes an angle of with respect to the x-axis (in the second quadrant). From this we can see that and hence that

We call the auxiliary angle of the reference angle of .

Graphing Trigonometric Functions

Let's start with the basic sine function, f(t) =sin(t). This function has an amplitude of 1because the graph goes one unit up and one unit down from the midline of the graph. This function has a period of 2 because the sine wave repeats every 2 units. The graph looks like this:

 

Here is the regular graph of cosine:

The regular tangent looks like this:

Many identities exist which interrelate the trigonometric functions. Among the most frequently used is the Pythagorean identity, which states that for any angle, the square of the sine plus the square of the cosine is always 1. This is easy to see by studying a right triangle of hypotenuse 1 and applying the Pythagorean theorem. In symbolic form, the Pythagorean identity reads,

,

which is more commonly written with the exponent "two" next to the sine and cosine symbol:

.

In some cases the inner parentheses may be omitted.

Other key relationships are the sum and difference formulas, which give the sine and cosine of the sum and difference of two angles in terms of sines and cosines of the angles themselves. These can be derived geometrically, using arguments which go back to Ptolemy; one can also produce them algebraically using Euler's formula.

When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulas.

These identities can also be used to derive the product-to-sum identities that were used in antiquity to transform the product of two numbers in a sum of numbers and greatly speed operations, much like the logarithm function.