CHAPTER 2 TRIGONOMETRIC FUNCTIONS

Recall that a real number  can be interpreted as the measure of the angle constructed as follows: wrap a piece of string of length  units around the unit circle  (counterclockwise if  , clockwise if ) with initial point P(1,0) and terminal point Q(x,y). This gives rise to the central angle with vertex O(0,0) and sides through the points P and Q. All six trigonometric functions of  are defined in terms of the coordinates of the point Q(x,y), as follows:



Since Q(x,y) is a point on the unit circle, we know that  . This fact and the definitions of the trigonometric functions give rise to the following fundamental identities:



This modern notation for trigonometric functions is due to L. Euler (1748).
More generally, if Q(x,y) is the point where the circle  of radius R is intersected by the angle  , then it follows (from similar triangles) that

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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 1 Find 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  .
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  .
EXAMPLE 3 Find all trigonometric functions of an angle  in the third quadrant for which  .
Solution: We first construct a point R(x,y) on the terminal side of the angle  , in the third quadrant. If R(x,y) is such a point, then  and we see that we may take x=-5 and R=6. Since we find that  (the negative signs on x and y are taken so that R(x,y) is a point on the third quadrant, see Figure 3).

Figure 3

It follows that



Here are some Exercises on the evaluation of trigonometric functions.
please do it your self as Exercise
EXERCISE 

(a)

Evaluate  (give the exact answer).

(b)

If  and  , find  (give the exact answer).