Tri Go No Me Tri A

22 Chapter 1: Functions

81. 3(x - I)' + 2(y + 2)' ~ 6

82. + + ~)' + + -t)' ~ 54

83. Write an equation for the ellipse (x'/16) + (y'/9) ~ I shifted 4 units to the left and 3 units up. S1retch the ellipse and ideotifY its center and major axis.

84. Write an equation for the ellipse (x'/4) + (y'/25) ~ I shifted 3 units to the right and 2 units down. S1retch the ellipse and ideotifY its center and major axis.

Comlrining Functions 85. Assume that f is an even function, g is an odd function, and both

f andg are dermed on the entire real line R. Which of the following (where dermed) are flNeo? odd?

1.3 Trigonometric Functions

a. fg

d. f' ~ If b. f/g •. g'~gg

e. g/f r. fog

g. gof h. fof i. gog

86. Can a function be both flNeo aod odd? Give reasons for your answer.

D 87. (Continuation of Example 1.) Graph the functions f(x) ~ Vx and g(x) ~ ~ together with their (a) sum, (b) product, (c) two differences, (d) two quotients.

D 88. Let f(x) ~ x - 7 aod g(x) ~ x'. Graph f and g together with f 0 gandg 0 f.

TIris section reviews radian measure and the basic trigonometric functions.

< C""10 of ,ifo""

FIGURE 1.38 TIre radian measure of the central aogleA'CB' is the number 9 ~ s/r. For a unit circle of radius r = 1,8 is the length of arc AB that central angle ACB cuts from the unit circle.

Angles

Angies are measured in degrees or radians. The number of radians in the central angle A' CB' within a circie of radius r is defined as the number of ''radius units" contained in the arc s subtended by that central angie. If we denote this central angie by () when measured in radians, this means that fJ = sir (Figure 1.38), or

s = r£J (fJ in radians). (1)

If the circie is a unit circle having radius r = i, then from Figure 1.38 and Equation (i), we see that the central angle fJ measured in radians is just the iength of the arc that the angle cuts from the unit circie. Since one compiete revoiution of the unit circie is 3600 or 21T

radians, we have

1T radians = i80° (2)

and

1 radian = i!O ('" 57.3) degrees or 1 degree = 1~0 ('" 0.017) radians.

Tabie 1.2 shows the equivalence between degree and radian measures for some basic angles.

TABLE 1.2 Angles measured in degrees and radians

Degrees -180 -135 -90 -45 0 30

fJ (radians) -1T -31T

4 o 1T

6

45

1T

4

60 90 120 135 150 180 270 360

1T 1T

3 2

hypotenuse

adjacent

sin {1 = opp byp

d· COS{1=:~

opp tan 9 ~ adj

csc {1 = hyp opp

byp sec {1 = adj

cot {1 = adj opp

opposite

FIGURE 1.41 Trigonometric ratios of an acute angle.

y

FIGURE 1.42 The trigonometric functions of. general angle 6 are defmed in terms ofx,y, and r.

1.3 Trigonometric Functions 23

An angle in the xy-plane is said to be in standard position if its vertex lies at the origin and its initial ray lies along the positive x-axis (Figure 1.39). Angles measured counterclockwise from the positive x-axis are assigned positive measures; angles measured clockwise are assigned negative measures.

y y

Initial ray

---------.~,---~--~x

FIGURE 1.39 Angles in staodard position in the xy-plaoe.

Angles describing counterclockwise rotations can go arbitrarily far beyond 2" radians or 360°. Similarly, angles describing clockwise rotations can have negative measures of all sizes (Figure lAO).

y

9" 4

y

3"

- -"-H --'....+----x

y y

-----*--,----> X ----hh-+---~x

FIGURE 1.40 Nonzero radiao measures cao be positive or negative and cao go beyond 2"..

Angle Convention: Use Radians From now on, in this book it is assumed that all angles are measured in radians unless degrees or some other unit is stated explicitly. When we talk about the angle ,,/3, we mean ,,/3 radians (which is 60°), not ,,/3 degrees. We use radians because it simplifies many of the operations in calculus, and some results we will obtain involving the trigonometric functions are not true when angles are measured in degrees.

The Six Basic Trigonometric Functions

You are probably familiar with derming the trigonometric functions of an acute angle in terms of the sides of a right triangle (Figure 1041). We extend this dermition to obtuse and negative angles by first placing the angle in standard position in a circle of radius r. We then derme the trigonometric functions in terms of the coordinates of the point P(x, y) where the angle's terminal ray intersects the circle (Figure 1.42).

sine: sin l/ = )I cosecant: r

r cscl/=y

cosine: x

secant: sec l/ = l' cosO = r x

tangent: tan 0 =)1 x cotangent: cotO ="" y

These extended dermitions agree with the right-triangle definitions when the angle is acute. Notice also that whenever the quotients are defined,

lanO = sinO cosO

secO =~ cos v

I cotO = lanO

I cseO = sinO


zj2 ~1 '" '" - -

4 2 1

FIGURE 1.43 Radian angles and side leng1hs of two common 1riangles.

y

S A sinpos all pos

x

T C tanpos cospos

FIGURE 1.44 The CAST rule, remembered by the statement "Calculus Activates Student Thinking," tells which trigonometric functions are positive in each quadrant.

As you can see, tan IJ and sec IJ are not defined if x = cos IJ = O. This means they are not defined iflJ is ±1T/2, ±31T/2, ... . Similarly, cot 0 and csc 0 are not defined for values of 0 forwhichy = O,namelyO = 0, ±1T, ±21T, ....

The exact values of these trigonometric retios for some angles can be read from the triangles in Figure 1.43. For instance,

. 1T 1 sm-=-

4v'Z 1T 1 cos-= -4v'Z

tan 1T =1 4

. 1T 1 sm 6 =2

1T V3 cos 6 =2

tan 1T = _1_ 6V3

. 1T V3 smT=2

1T 1 cosT = 2

The CAST rule (Figure 1.44) is useful for remembering when the basic trigonometric functions are positive or negative. For instance, from the triangle in Figure 1.45, we see that

21T 1 cosT = -2' tan 2; = -V3.

(cos 2;, sin 2;) = t-~, ~) y

FIGURE 1.45 The 1riangle for calculating the sine and cosine of21f/3 mdians. The side leog1hs come from the geometty of right triangles.

Using a similar method we determined the values of sin 0, cos 0, and tan 0 shown in Table 1.3.

TABLE 1.3 Values of sin 0, cos 0, and tan 0 for selected values of 0

Degrees -180 -135 -90 -45 0 30 45 60 90 120 135 150 180 270 360

o (radians) -31T -1T -1T

0 1T 1T 1T 1T 21T 31T 51T 31T

21T -1T 4 2 4 6 4 3 2 3 4 6 1T 2

sinO 0 -v'Z

-I -v'Z

0 1 v'Z V3 1 V3 v'Z 1

0 -I 0 2 2 2 2 2 2 2 2

cosO -I -v'Z

0 v'Z 1 V3 v'Z 1

0 1 -v'Z -V3 -I 0 1

2 2 2 2 2 2 2 2

tan 0 0 1 -I 0 V3 1 V3 -V3 -I -V3 0 0 3 3

I Period. of Trigonometric Functions Period 'IT : tan (x + ".) ~ tan x

cot(x + ".) ~ cotx Period 2'IT: sin(x + 2".) ~ sinx

cos(x + 2".) ~ cosx sec(x + 2".) ~ .. cx csc(x + 2".) ~ cscx

Even

cos ( -x) ~ cosx sec ( -x) ~ secx

Odd

sin( -x) ~ -sinx tan(-x) ~ -tanx esc ( -x) ~ -cscx cot ( -x) ~ -COlx

y

FIGURE 1.47 The reforenoe triangle for a general angle 8.


Periodicity and Graphs of the Trigonometric Functions

When an angle of measure 8 and an angle of measure 8 + 21T are in standard position, their terminal rays coincide. The two angles therefore have the same trigonometric fimction values: sin(8 + 21T) = sin 8, tan(8 + 21T) = tan 8, and so on. Similarly, cos(8 - 21T) ~ cos 8, sin(8 - 21T) = sin 8, and so on. We describe this repeating behavior by saying that the six basic trigonometric functions are periodic.

DEFINmON A function f(x) is periodic ifthere is a positive number p such that f(x + p) ~ f(x) for every value ofx. The smallest such value ofp is the period of f.

When we graph trigonometric functions in the coordinate plane, we usually denote the independent variable by x instead of 8. Figure 1.46 shows that the tangent and cotangent functions have period p ~ 1T, and the other four functions have period 21T. Also, the symmetries in these graphs reveal that the cosine and secant functions are even and the other four functions are odd (although this does not prove those results).

Domajn: _00 < x < co

Range: -1:s y :s 1 Period: 2'1T

(a)

y y = socx

\v -,t-=-tnt-t-~~~, _3 (\ 0 n Domain' "" + 1T + 31T • X -2'-2 •... Range: y:s -1 ory ~ 1 Period: 2'17"

(d)

Domain: -co < x < 00

Range: -1:s y:s 1 Period: 2'1J"

(b)

y

Domain: x "'" 0, ±'1f, ±2'1f, ••• Range: y::5 -lory 2:: 1 Period: 271'

(0)

y y - tunx

Domajn' *+'1r + 3'17" .X -2'- 2.···

Range: -co < y < 00

Period: '1T (c)

y y - cotx

Domain: x "'" 0, ±'1f, ±2'1f, ... Range: _00 < y < 00

Period: '1T

(I)

FIGURE 1.46 Graphs of the six hasic trigonometric functions using radian measure. The shading for each trigonometric function indicates its periodicity.

Trigonometric Identities

The coordinates of any point P(x, y) in the plane can be expressed in terms of the point's distance r from the origin and the angle 8 that ray OP makes with the positive x-axis (Figure 1.42). Since xlr ~ cos 8 andylr = sin 8, we have

x = rcos6, y ~ rsin8.

When r = 1 we can apply the Pythagorean theorem to the reference right triangle in Figure 1.47 and obtain the equation

(3)


TIris equation, true for all values of 6, is the most frequently used identity in trigonometry. Dividing this identity in turn by cos2 6 and sin2 6 gives

1 + tan2 6 = sec2 6

I + cot' 6 = csc2 6

The following formulas hold for all angles A and B (Exercise 58).

Additioo Formulas

cos(A + B) = cosA cosB - sinA sinB

sin(A + B) = sinA cosB + cosA sinB (4)

There are similar formulas for cos (A - B) and sin (A - B) (Exercises 35 and 36). All the trigonometric identities needed in this book derive from Equations (3) and (4). For example, substituting 6 for both A and B in the addition formulas gives

Double-Angle Formulas

cos 26 = cos2 6 - sin2 6

sin 26 = 2 sin 6 cos 6

Additional formulas come from combining the equations

cos2 6 + sin2 6 = I, cos2 6 - sin2 6 = cos 26.

(5)

We add the two equations to get 2 cos2 6 = I + cos 26 and subtract the second from the first to get 2 sin2 6 = I - cos 26. TIris results in the following identities, which are useful in integral calculus.

Half-Angle Formulas

The Law of Cosines

cos2 6 = 1 + cos 26 2

sin2 6 = I - cos 26 2

If a, b, and c are sides of a triangle ABC and if 6 is the angle opposite c, then

TIris equation is called the law of cosioes.

(6)

(7)

(8)

y

B(a cos 8, a sin 8)

------~~--~~~~~x C b A(b,O)

FIGURE 1.48 The square of the distance betweeo A and B gives the law of cosines.


We can see why the law holds if we introduce coordinate axes with the origin at C and the positive x-axis along one side of the triangle, as in Figure 1,48. The coordinates of A are (b, 0); the coordinates of B are (a cos II, a sin II). The square of the distance between A and B is therefore

c2 = (acosll- b)2 + (a sin 11)2

= a 2(cos211 + sin211) + b 2 - Zabcosll

I

= a 2 + b 2 - Zabcosll.

The law of cosines generalizes the Pythagorean theorem. If II = 7T /Z, then cos II = 0 andc2 = a2 + b2

•

Transfonnations of Trigonometric Graphs

The rules for shifting, stretching, compressing, and reflecting the graph of a function summarized in the following diagram apply to the trigonometric functions we have discussed in this section.

Vertical stretch or compression; ~ / Vertica1 shift reflection about x-axis ifnegative ~

y = af(b(x + c)) + d

Horizontal stretch or compression; / ~ Horizontal shift reflection about y-axis if negative

The transformation rules applied to the sine function give the general sine function or sinusoid formula

f(x) = A Sin(Z; (x - C») + D,

where I A I is the amplitude, I B I is the period, C is the horizontal shift, and D is the vertical shift. A graphical interpretation of the various terms is revealing and given below.

y

y = A sin (2;<x - C») + D

o

Two Spedallnequalities

For any angle II measured in radians,

-1111 :5 sinll:5 1111 and -1111 :5 I - cos II :5 1111.


y

p

To establish these inequalities, we picture fJ as a nonzero angle in standard position (Figure 1.49). The circle in the figure is a unit circle, sollil equals the length of the circular arc AP. The length of line segmentAP is therefore less than IfJl.

cos 6 1-cosO

Triangle APQ is a right triangle with sides of length

QP = IsinfJl, AQ = I - cosfJ.

From the Pythagorean theorem and the fact that AP < I fJ I, we get

sin2 fJ + (I - cos fJ)2 = (AP)2 :5 6'. (9)

The terms on the left-band side of Equation (9) are both positive, so each is smaller than their sum and hence is less than or equal to 6':

and (I - cos 11)2 :5 6'. FIGURE 1.49 From the geometry of this figure, drawn for By taking square roots, this is equivalent to saying that

8 > 0, we get the inequality sin28 + (I - cos 8)2 :5 0'.

so

and 11- cosfJl

-161 :5 sinfJ:5 161 and -161 :5 I - cosll :5 161. These inequalities will be useful in the next chapter.

Exercises 1.3

Radians and Degrees 1. On a circle of radius 10m, how long is an arc that subtends a cen

tral angle of (8) 4'1f/5 radiaos? (b) 110'?

2. A central aogle in a circle of radius 8 is subtended by ao arc of length 101r. Find the aogle's radian and degree measures.

3. You waot to ma1re ao 80' aog1e by marking ao arc on the perimeter of. 12-in.-diarneter disk and drawing lines from the ends of the arc to the disk's ceoter. To the nearest tenth of an incb, how long should the arc be?

4. If you roll a I-m-diaroeter wheel forward 30 em over level grouod, through what aogle will the wheel turn? Aoswer in radiaos (to the nearest tenth) aod degrees (to the nearest degree).

Evaluating Trigonometric Functions S. Copy aod complete the following table of function values. If the

function is undefmed at a given angle, enter "UND." Do not use a calculator or tables.

(} -". -2"./3 o "./2 3"./4

sinO cosO tan 8 cot 8 sec 0 esc 8

6. Copy and complete the following table of function values. If the function is undefmed at a given angle, enter "UND." Do not use a calculator or tables.

fJ

sinO cos 8 tan 8 cot 8 sec 0 esc 8

-3"./2 -"./3 -"./6 "./4 S1r/6

In Exercises 7-12, one of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval.

7. 3

• smx = 5'

I 9. cosx = 3'

XE [f, 'If] XE[-f,o]

8. tanx = 2, XE [o,f] 10. cosx = -1

53' XE [f,'lT]

12. sinx = -t, XE ['If, 3;]

Graphing Trigonometric Functions Graph the functions in Exercises 13-22. What is the period of each function?

13. sin 2x 14. sin (x/2)

15. cos 71'X

17 . 'If X

• - 8m T

19. cos(X - f)

'If X 16. cosT

18. - cos 2 '1TX

21. Sin(X - -;f) + I 22. COS(X + 2:) - 2

Graph the functioos in Exercises 23-26 in the Is-plane (t·axis horizoo· tal, s-axis vertical). What is the period of each function? What sym· metries do the graphs have?

23. s = cot2t 24. s = -tao 'ITt

25. s = sec(~t) 26. s = csc(f)

D 27. a. Graph y = cos x and y = sec x together for - 3'IT /2 ,;; x ,;; 3'IT /2. Cornmeot on the behavior of sec x in relatioo to the signs and values of cos x.

b. Grapby = sin x andy = c8Cxtogetherfor-w ~ x ~ 2'17". Comment on the behavior of esc x in relation to the signs and values of sin x.

D 28. Graph y = taox aod y = cotx together for -7 ,;; x ,;; 7. Comment on the behavior of cot x in relation to the signs and values of taox.

29. Graph y = sinx aod y = l sin x J together. What are the domain and raoge of l sin x J ?

30. Graph y = sinx aod y = r sin x 1 together. What are the domain and raoge of r sin x 1 ?

Using the Addition Formulas Use the addition formulas to derive the ideotities in Exercises 31-36.

31. cos(x - I-) = sinx 32. cos (x + 1-) = -sinx

33. Sin(X + f) = cosx 34. Sin(X - -I-) = -cosx

35. cos(A - B) = cosA cosB + sinA sinB (Exercise 57 provides a different derivation.)

36. sin(A - B) = sinA cosB - cosA sinB

37. What happeos if you take B = A in the trigoooroetric ideotity cos(A - B) = cosA cosB + sinA sinB? Does the result agree with something you already know?

38. What happeos if you take B = 2'IT in the addition formulas? Do the results agree with something you already know?

In Exercises 39-42, express the giveo quaotity in terms of sin x aod cosx.

39. cos('IT + x)

41. Sin(3; - x) 40. sin(2'IT - x)

42. cose; + x) 43 E al . 7'IT . ('IT 'IT) . v uatesffi12assm 4+3.

44. Evaluate cos If; as cos (-;f + 2:).

45. Evaluate cos ~ . 46. Evaluate sin ;; .

Using the Double-Angle Formulas Find the functioo values in Exercises 47-50.

47. cos2 '!! 8

48 2 5'IT • cos 12

49. . 2 'IT SO. sin231T sm 12 8


Solving Trigonometric Equations For Exercises 51-54, solve for the angle 0, where 0 ,;; 0 ,;; 2'IT.

51. sin' 0 = t 52. sin' 0 = cos' 0

53. sin20 - cosO = 0 54. cos20 + cosO = 0

Theory and Examples 55. The tangent snm formula The staodard formula for the tao

goot of the smn of two aogles is

taoA+taoB tao(A + B) = I - taoA taoB'

Derive the formula.

56. (ContinUIJtion of Exercise 55.) Derive a formula for tao (A - B).

57. Apply the law of cosines to the triaogle in the accoropaoying figure to derive the formula for cos (A - B).

y

1

----4---~~~~-l----+_--~x o

58. a. Apply the formula for cos(A - B) to the ideotity sinO =

cos(I - 0) to obtain the addition formula for sin(A + B).

b. Derive the formula for cos (A + B) by substituting - B for B in the formula for cos (A - B) from Exercise 35.

59. A triaogle has sides a = 2 aod b = 3 aod angle C = 60°. Find the leogtil of side c.

60. A triaogle has sides a = 2 aod b = 3 aod aogle C = 40°. Find the leogtil ofside c.

61. The law of sine. The law of sines says that if a, b, aod c are the sides opposite the angles A, B, and C in a triangle, then

sinA sinB sin C abe .

Use the accoropaoying figures aod the identity sin('IT - 0) = sin 0 , if required, to derive the law.

A A

B "---"'. - -'-----' C B £..----:. ,----.J

62. A triaogle has sides a = 2 aod b = 3 and aogle C = 60° (as in Exercise 59). Find the sine of aogle B using the law of sines.


63. A triangle has side c = 2 and angles A = 1f/4 and B = 1f/3. Find the length a of the side opposite A.

D 64. The appnDimation sin x '" x It is often useful to know that, whenx is measured in radians, sin x ~ x for n'lDDerically small values of x. In Sectioo 3.9, we will see why the approximation holds. The approximatioo error is less than 1 in 5000 if Ixl < 0.1.

L With your grapher in radian mode, graph y = sinx and y = x together in a viewiog wiodow about the otigin. What do you see happening as x nears the origin?

b. With your grapher in degree mode, graph y = sinx and y = x together about the origin again. How is the picture different froru the ooe obtained with radian mode?

General Sine Curves For

f(x) = A Sine: (x - C») + D, identify A, B, C, and D for the sine functions in Exercises 65-<i8 aod sketch their graphs.

65. Y = 2 sin (x + 1f) - 1

67. y=-%Sin(ft) +,\0

COMPUTER EXPLORATIONS

66. y = kSin(1fX - 1f) + k

68. y = ~sin 2L1ft, L > 0 2"

In Exercises 6'J-72, you will explore graphically the general sine functioo

f(x) = A Sine: (x - C)) + D

as you change the values of the constants A, B, C, andD. Use a CAS or coruputer grapher to perform the steps in the exercises.

69. TheperiodB Set the constants A = 3, C = D = O.

a. Plot f(x) for the values B = I, 3, 21f, 5" over the interval -41f :5 x :5 41f. Describe what happens to the graph of the general sine function as the period increases.

b. What happens to the graph for negative values of B? Try it withB = -3 andB = -21f.

70. The horizontal shift C Set the constants A = 3, B = 6, D = O.

a. Plot f(x) for the values C = 0, I, and 2 over the interval -41f :5 x :5 41f. Describe what happens to the graph of the general sine function as C increases through positive values.

b. What happens to the graph for negative values of C?

c. What smallest positive value should be assigned to C so the graph exhibits no horizontal shift? Confum your answer with a plot.

71. The vertical shift D Set the constants A = 3, B = 6, C = O.

a. Plot f(x) for the values D = 0, I, and 3 over the interval -41f :5 x :5 41f. Describe what happens to the graph of the general sine function as D increases through positive values.

b. What happens to the graph for negative values of D?

72. The amplitude A Set the constants B = 6, C = D = O.

a. Describe what happens to the graph of the general sine function as A increases through positive values. Confirm your answer by plotting f(x) for the values A = I, 5, and 9.

h. What happens to the graph for negative values of A?

1.4 Graphing with Calculators and Computers

A graphing calculator or a computer with graphing software enables us to graph very complicated functions with high precision. Many of these functions could not otherwise be easily graphed. However, care must be taken when using such devices for graphing purposes, and in this section we address some of the issues involved. In Chapter 4 we will see how calculus helps us determine that we are accurately viewing all the important features of a function's graph.

Graphing Windows

When using a graphing calculator or computer as a graphing tool, a portion of the graph is displayed in a rectangular display or viewing window. Often the default window gives an incomplete or misleading picture of the graph. We use the tenD square window when the units or scales on both axes are the same. Ibis tenD does not mean that the display window itself is square (usually it is rectangular), but ioslead it means that the x-unit is the same as the y-unit.

When a graph is displayed io the default window, the x-unit may differ from the y-unit of scaling in order to fit the graph in the window. The viewing window is set by specifying an ioterval [a, b] for the x-values and an ioterval [c, d] for the y-values. The machine selects equally spaced x-values in [a, b] and then plots the points (x, f(x». A poiot is plotted if and

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