TRIGONOMETRIC FUNCTIONS

Recall that a real number tex2html_wrap_inline453 can be interpreted as the measure of the angle constructed as follows: wrap a piece of string of length tex2html_wrap_inline453 units around the unit circle tex2html_wrap_inline457 (counterclockwise if tex2html_wrap_inline459 , clockwise if tex2html_wrap_inline461 ) 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 tex2html_wrap_inline453 are defined in terms of the coordinates of the point Q(x,y), as follows:

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Since Q(x,y) is a point on the unit circle, we know that tex2html_wrap_inline457 . This fact and the definitions of the trigonometric functions give rise to the following fundamental identities:

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This modern notation for trigonometric functions is due to L. Euler (1748).

More generally, if Q(x,y) is the point where the circle tex2html_wrap_inline483 of radius R is intersected by the angle tex2html_wrap_inline453 , then it follows (from similar triangles) that

displaymath441


Periodic Functions

If an angle tex2html_wrap_inline453 corresponds to a point Q(x,y) on the unit circle, it is not hard to see that the angle tex2html_wrap_inline493 corresponds to the same point Q(x,y), and hence that

  equation74

Moreover, tex2html_wrap_inline497 is the smallest positive angle for which Equations 1 are true for any angle tex2html_wrap_inline453 . In general, we have for all angles tex2html_wrap_inline453 :

  equation78

We call the number tex2html_wrap_inline497 the period of the trigonometric functions tex2html_wrap_inline505 and tex2html_wrap_inline507 , and refer to these functions as being periodic. Both tex2html_wrap_inline509 and tex2html_wrap_inline511 are periodic functions as well, with period tex2html_wrap_inline497 , while tex2html_wrap_inline515 and tex2html_wrap_inline517 are periodic with period tex2html_wrap_inline519 .

EXAMPLE 1 Find the period of the function tex2html_wrap_inline521 .

Solution: The function tex2html_wrap_inline521 runs through a full cycle when the angle 3x runs from 0 to tex2html_wrap_inline497 , or equivalently when x goes from 0 to tex2html_wrap_inline535 . The period of f(x) is then tex2html_wrap_inline535 .

EXERCISE 1 Find the period of the function tex2html_wrap_inline541 .

Solution


Evaluation of Trigonometric functions

Consider the triangle with sides of length tex2html_wrap_inline543 and hypotenuse c>0 as in Figure 1 below:

Figure 1

For the angle tex2html_wrap_inline453 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 tex2html_wrap_inline549 .

Solution: From Figure 2, we see that the angle of tex2html_wrap_inline551 corresponds to the point tex2html_wrap_inline553 on the unit circle, and so

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EXAMPLE 2: Find the values of all trigonometric functions of the angle tex2html_wrap_inline555 .

Solution: Observe that an angle of tex2html_wrap_inline557 is equivalent to 8 whole revolutions (a total of tex2html_wrap_inline559 ) plus tex2html_wrap_inline561 , Hence the angles tex2html_wrap_inline557 and tex2html_wrap_inline561 intersect the unit circle at the same point Q(x,y), and so their trigonometric functions are the same. Furthermore, the angle of tex2html_wrap_inline561 makes an angle of tex2html_wrap_inline551 with respect to the x-axis (in the second quadrant). From this we can see that tex2html_wrap_inline573 and hence that

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We call the auxiliary angle of tex2html_wrap_inline551 the reference angle of tex2html_wrap_inline557 .

EXAMPLE 3 Find all trigonometric functions of an angle tex2html_wrap_inline453 in the third quadrant for which tex2html_wrap_inline581 .

Solution: We first construct a point R(x,y) on the terminal side of the angle tex2html_wrap_inline453 , in the third quadrant. If R(x,y) is such a point, then tex2html_wrap_inline589 and we see that we may take x=-5 and R=6. Since tex2html_wrap_inline595 we find that tex2html_wrap_inline597 (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

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Here are some Exercises on the evaluation of trigonometric functions.

EXERCISE 2

(a)
Evaluate tex2html_wrap_inline605 (give the exact answer).

(b)
If tex2html_wrap_inline607 and tex2html_wrap_inline609 , find tex2html_wrap_inline611 (give the exact answer).

Solution

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 tex2html_wrap_inline613 . How far is the whale from the shoreline?

Solution

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Luis Valdez Sanchez
Wed Dec 4 18:30:59 MST 1996

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