Find the values (if they are defined) of the

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```					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 θ for
which the denominator in the corresponding ratio is not zero. Because r > 0, it follows
that sin θ = y/r and cos θ = x/r are defined for all angles θ . However, tan θ = y/x and
sec θ = r/x are not defined when the terminal side of θ lies anlong the y axis (so that x =
0). Likewise, cot θ = x/y and csc θ = r/y are not defined when the terminal side of θ 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 θ = 90° (or θ = π 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 + y2 =                          02 + y 2 =                        y 2 = y = y.
y    y                                                                                        x     0
Therefore, sin 90° =                                   =   =1                                                           cos 90° =                    =    =0
r    y                                                                                        r     y
r   y                                                                                            x    0
csc 90° =   =   =1                                                                               cot 90° =    =   = 0.
y   y                                                                                            y    y
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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
θ degrees                             θ radians                                    sin θ                               cos θ                               tan θ                                cot θ                                sec θ                                csc θ

0°                                      0                                     0                                   1                                    0                                 ––                                     1                                 ––
π
90°                                                                            1                                   0                                  ––                                    0                                  ––                                     1
2

180°                                       π                                      0                                 –1                                     0                                 ––                                   –1                                  ––
3π
270°                                                                           –1                                    0                                  ––                                    0                                  ––                                   –1
2

360°                                     2π                                      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 2π radians).
For instance,
sin 450° = sin( 450° − 360° ) = sin 90° = 1.

sec 7π = sec ( 7π − ( 3 × 2π ) ) = sec π = –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 ---------------------------- ------------------------------------------------------------
θ = 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
1
sin 1110° = sin 30° =                         csc 1110° = csc 30° = 2
2
3                                          2 3
cos 1110° = cos 30° =                         sec 1110° = sec 30° =
2                                             3
3
tan 1110° = tan 30° =                         cot 1110° = cot 30° = 3
3
___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Example 3.3 ---------------------------- ------------------------------------------------------------
5π
θ =–
2
5π
We repeatedly add 2 π to –                                                     until we obtain a nonnegative coterminal angle:
2
5π          π
–        + 2π = –                                                       (still negative)
2          2
π                                        3π
–               + 2π =                           .
2                                        2
Therefore, by Table 3.1 for quadrantal angles,
⎛ 5π ⎞         ⎛ 3π ⎞                ⎛ 5π ⎞         ⎛ 3π ⎞
sin ⎜ −    ⎟ = sin ⎜ ⎟ = –1          cot ⎜ − ⎟ = cot ⎜ ⎟ = 0
⎝    2 ⎠       ⎝  2 ⎠                ⎝    2 ⎠       ⎝ 2 ⎠
⎛ 5π ⎞         ⎛ 3π ⎞                 ⎛ 5π ⎞        ⎛ 3π ⎞
cos ⎜ −    ⎟ = cos ⎜ ⎟ = 0            csc ⎜ −   ⎟ = csc ⎜ ⎟ = –1
⎝ 2 ⎠          ⎝ 2 ⎠                  ⎝ 2 ⎠         ⎝ 2 ⎠
⎛ 5π ⎞           ⎛ 5π ⎞
and both tan ⎜ −     ⎟ and sec ⎜ −   ⎟ are undefined.
⎝ 2 ⎠            ⎝ 2 ⎠
______________________________________________________________________________________

19
Table 3.1
θ degrees    θ radians                  sin θ            cos θ        tan θ        cot θ             sec θ          csc θ
π                         1                 3            3                            2 3
30°                                                                                3                                  2
6                2                2            3                              3
π                      2               2
45°                                                                  1            1                  2                   2
4                 2               2
π                      3              1                          3                                 2 3
60°                                                                   3                             2
3                 2               2                         3                                   3

Figure 3.2
y                                                                         y
θ R = 180° ‐ θ
θ R = π  ‐ θ

θ
θR
θ = θR

O                               x                                         O                            x

(a)                                                                       (b)

y                                                                         y
θ R =  θ  ‐  180°                                                   θ R =  360° ‐  θ
θ R =  θ  ‐  π                                                      θ R =  2π  ‐  θ

θ                                                                           θ

O                       x                                         O                        x
θR
θR

(c)                                                                       (d)

20
Example 3.4 ---------------------------- ------------------------------------------------------------
Find the reference angle θ R for each angle θ .
3π                                  5π
(a) θ = 60°       (b) θ =           (c) θ = 210°     (d) θ =       .
4                                   3
(a) By Figure 3.2(a), θ R = θ = 60° .
3π     π
(b) By Figure 3.2(b), θ R = π – θ = π –  =     .
4      4
(c) By Figure 3.2(c), θ R = θ – 180° = 210° – 180° = 30° .
5π     π
(d) By Figure 3.2(c), θ R = 2 π – θ = 2 π –        =   .
3     3
______________________________________________________________________________________

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

Thus,
sin θ = ± sin θ R ,        cos θ = ± cos θ R ,
and so forth. You can always determine the correct algebraic sign by considering the quadrant in which
θ 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]
5π                7π
2. (a) 5 π        (b) 6 π          (c) –7 π          (d)               (e)      .
2                 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°
19π
9. – 675°                   10.
2
25π
11. 5 π                     12.
6
17π                         31π
13.                         14. –
3                            4

21
15. What happens when you try to evaluate tan 900° on a calculator? [Try it.]
16. Let θ be a quadrant III angle in standard position and let θ R be its reference angle.
Show that the value of any trigonometric function of θ is the same as the value of
θ R , except possibly for a change of algebraic sign. Repeat for θ in quadrant IV.

In problems 17 to 36, find the reference angle θ R for each angle θ , and then find the
exact values of the six trigonometric functions of θ .
17. θ = 150°           18. θ = 120°
19. θ = 240°           20. θ = 225°
21. θ = 315°           22. θ = 675°
5π
23. θ = – 150°         24. θ = –
6
13π
25. θ = – 60°          26. θ   =–
6
π                    53π
27. θ = –              28. θ   =
4                      6
2π                   9π
29. θ   =–             30. θ   =
3                    4
7π                       50π
31. θ   =              32. θ   =–
4                         3
11π                      147π
33. θ   =              34. θ   =–
3                         4
35. θ   = – 420°       36. θ   = – 5370°

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

θ degrees    θ radians        sin θ         cos θ          tan θ
7π
210°
6
5π
225°
4
4π
240°
3
5π
300°
3
7π
315°
4
11π
330°
6

22
θ degrees       θ radians           cot θ        sec θ            csc θ
7π
210°
6
5π
225°
4
4π
240°
3
5π
300°
3
7π
315°
4
11π
330°
6

38. A calculator is set in radian mode. π is entered and the sine (SIN) key is pressed.
The display shows – 4.1 × 10 -10 . But we know that sin π = 0. Explain.

In problems 59 to 62, use a calculator to verify that the equation is true for the indicated
value of the angle θ .
sin θ
59. tan θ =                        for θ = 35° .
cos θ
5π
60. (cos θ )(tan θ ) = sin θ       for θ =
7
5π
61. cos 2 θ + sin 2 θ = 1          for θ =
3
62. 1 + tan 2 θ = sec 2 θ for θ = 17.75°

0   1  2    3        4
63. Verify that for θ = 0° , 30° , 45° , 60° , 90° , we have sin θ =      , ,    ,    , and
2   2  2    2        2
respectively. [Although there is no theoretical significance to this pattern, people often
use it as a memory aid to help recall these values of sin θ .]

23

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