3. EVALUATION OF TRIGONOMETRIC FUNCTIONS

3. EVALUATION OF TRIGONOMETRIC FUNCTIONS

In this section, we obtain values of the trigonometric functions for quadrantal angles, we introduce the idea of reference angles, and we discuss the use of a calculator to evaluate trigonometric functions of general angles. In Definition 2.1, the domain of each trigonometric function consists of all angles for which the denominator in the corresponding ratio is not zero. Because r > 0, it follows that sin = y/r and cos = x/r are defined for all angles . However, tan = y/x and sec = r/x are not defined when the terminal side of lies anlong the y axis (so that x = 0). Likewise, cot = x/y and csc = r/y are not defined when the terminal side of lies along the x axis (so that y = 0). Therefore, when you deal with a trigonometric function of a quadrantal angle, you must check to be sure that the function is actually defined for that angle.

Example 3.1 ---------------------------- ------------------------------------------------------------

Find the values (if they are defined) of the six trigonometric functions for the quadrantal angle = 90? (or = 2 ).

In order to use Definition 1, we begin by choosing any point ( 0 , y ) with y > 0, on the terminal side of the 90? angle (Figure 1). Because x = 0, it follows that tan 90? and sec 90? are undefined. Since y > 0, we

have

r = x2 + y 2 = 02 + y 2 = y 2 = y = y.

Therefore, sin 90? = y = y = 1 ry

cos 90? = x = 0 = 0 ry

csc 90? = r = y = 1 yy

cot 90? = x = 0 = 0. yy

_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

The values of the trigonometric functions for other quadrantal angles are found in a similar manner. The results appear in Table 3.1. Dashes in the table indicate that the function is undefined for that angle.

Table 3.1

degrees radians sin

cos

tan

cot

sec

csc

0 ?

0

90?

2

180 ?

270 ?

3

2

360 ?

2

0

1

0

??

1

??

1

0

??

0

??

1

0

?1

0

??

?1

??

?1

0

??

0

??

?1

0

1

0

??

1

??

18

It follows from Definition 2.1 that the values of each of the six trigonometric functions remain unchanged if the angle is replaced by a coterminal angle. If an angle exceeds one revolution or is negative, you can change it to a nonnegative coterminal angle that is less than one revolution by adding or subtracting an integer multiple of 360? (or 2 radians). For instance,

sin 450? = sin( 450? - 360? ) = sin 90? = 1.

sec 7 = sec ( 7 - ( 3 ? 2 ) ) = sec = ?1. cos (- 660?) = cos (- 660? + (2 ? 360?) ) = cos 60? = 1 .

2

In Examples 3.2 and 3.3, replace each angle by a nonnegative coterminal angle that is less than on revolution and then find the values of the six trigonometric functions (if they are defined).

Example 3.2 ---------------------------- ------------------------------------------------------------

= 1110?

By dividing 1110 by 360, we find that the largest integer multiple of 360? that is less than 1110? is

3 ? 360? = 1080? . Thus,

1110? ? ( 3 ? 360? ) = 1110? ? 1080? = 30? .

(Or we could have started with 1110? and repeatedly subtracted 360? until we obtained 30? .) It follows

that

sin1110? = sin 30? = 1 2

csc1110? = csc 30? = 2

cos1110? = cos 30? = 3 2

sec1110? = sec 30? = 2 3 3

tan1110? = tan 30? = 3 3

cot1110? = cot 30? = 3

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Example 3.3 ---------------------------- ------------------------------------------------------------

= ? 5

2

We repeatedly add 2 to ? 5 until we obtain a nonnegative coterminal angle: 2

? 5

+ 2

= ?

2

2

(still negative)

?

+ 2

=

3

.

2

2

Therefore, by Table 3.1 for quadrantal angles,

sin - 5 = sin 3 = ?1

2

2

cot - 5 = cot 3 = 0

2

2

cos - 5 = cos 3 = 0

2

2

csc - 5 = csc 3 = ?1

2

2

and both tan - 5 and sec - 5 are undefined.

2

2

______________________________________________________________________________________

19

Table 3.1

degrees radians sin

cos

tan

cot

sec

csc

30?

6

1 2

3

3

2

3

3

2 3

2

3

45?

4

2

2

1

1

2

2

2

2

60?

3

3

1

2

2

3

3

2

2 3

3

3

Figure 3.2 y

= R

O

x

y R = 180? - R = -

R

O

x

(a)

y R = - 180? R = -

O

x

R

(b)

y R = 360? - R = 2 -

O

x

R

(c)

(d)

20

Example 3.4 ---------------------------- ------------------------------------------------------------

Find the reference angle R for each angle .

(a) = 60?

(b) = 3 4

(c) = 210? (d) = 5 . 3

(a) By Figure 3.2(a), R = = 60? .

(b) By Figure 3.2(b), R = ?

=

? 3

4

=

. 4

(c) By Figure 3.2(c), R = ? 180? = 210? ? 180? = 30? .

(d) By Figure 3.2(c), R = 2 ?

=

2

?

5 3

=

.

3

______________________________________________________________________________________

The value of any trigonometric function of any angle is the same as the value of the function for the reference angle, R , except possibly for a change of algebraic sign.

Thus,

sin = ? sinR ,

cos = ? cosR ,

and so forth. You can always determine the correct algebraic sign by considering the quadrant in which

lies.

Section 3 Problems---------------------- ------------------------------------------------------------

In problems 1 and 2, find the values (if they are defined) of the six trigonometric functions of the given quadrantal angles. (Do not use a calculator.)

1. (a) 0?

(b) 180?

(c) 270?

(d) 360? .

[When you have finished, compare your answers with the results in Table 3.1]

2. (a) 5 (b) 6

(c) ?7

(d) 5

2

(e) 7 .

2

In Problems 3 to 14, replace each angle by a nonnegative coterminal angle that is less

than one revolution and then find the exact values of the six trigonometric functions (if

they are defined) for the angle.

3. 1440?

4. 810?

5. 900?

6. ? 220?

7. 750?

8. 1845?

9. ? 675?

10. 19

2

11. 5

12. 25

6

13. 17

3

14. ? 31

4

21

15. What happens when you try to evaluate tan 900? on a calculator? [Try it.] 16. Let be a quadrant III angle in standard position and let R be its reference angle.

Show that the value of any trigonometric function of is the same as the value of R , except possibly for a change of algebraic sign. Repeat for in quadrant IV.

In problems 17 to 36, find the reference angle R for each angle , and then find the

exact values of the six trigonometric functions of .

17. = 150?

18. = 120?

19. = 240?

20. = 225?

21. = 315?

22. = 675?

23. = ?150?

24. = ? 5

6

25. = ? 60?

26. = ? 13

6

27. = ?

4

28. = 53

6

29. = ? 2

3

30. = 9

4

31. = 7

4

32. = ? 50

3

33. = 11

3

34. = ? 147

4

35. = ? 420?

36. = ? 5370?

37. Complete the following tables. (Do not use a calculator.)

degrees radians

sin

cos

tan

210?

7

6

225?

5

4

240?

4

3

300 ?

5

3

315 ?

7

4

330 ?

11

6

22

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download