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
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 .
EXERCISE 1 Find the period of the function .
Consider the triangle with sides of length and hypotenuse c>0 as in Figure 1 below:
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.
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).
It follows that
Here are some Exercises on the evaluation of trigonometric functions.
EXERCISE 3 From a 200-foot observation tower on the beach, a man sights a whale in difficulty. The angle of depression of the whale is . How far is the whale from the shoreline?
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Luis Valdez Sanchez