Trigonometric Functions This section reviews radian measure and the basic trigonometric functions. C ' θ r s ' ngles ngles are measured in degrees or radians. The number of radians in the central angle C within a circle of radius r is defined as the number of radius units contained in the arc s subtended b that central angle. If we denote this central angle b u when measured in radians, this means that u = s>r (Figure ), or Unit circle Circle of r radius s = ru (u in radians). () FIGURE The radian measure of the central angle C is the number u = s>r. For a unit circle of radius r =, u is the length of arc that central angle C cuts from the unit circle. If the circle is a unit circle having radius r =, then from Figure.38 and Equation (), we see that the central angle u measured in radians is just the length of the arc that the angle cuts from the unit circle. Since one complete revolution of the unit circle is 360 or p radians, we have p radians = 80 () and radian = 80 p p ( L 57.3) degrees or degree = ( L 0.07) radians. 80 Table shows the equivalence between degree and radian measures for some basic angles. TLE ngles measured in degrees and radians Degrees 80 35 90 5 0 30 5 60 90 0 35 50 80 70 360 3p p p p p p p p 3p 5p 3p U (radians) p 0 p p 6 3 3 6
n angle in the -plane is said to be in standard position if its verte lies at the origin and its initial ra lies along the positive -ais (Figure ). ngles measured counterclockwise from the positive -ais are assigned positive measures; angles measured clockwise are assigned negative measures. Terminal ra Initial ra Positive measure Initial ra Terminal ra Negative measure FIGURE ngles in standard position in the -plane. ngles describing counterclockwise rotations can go arbitraril far beond p radians or 360. Similarl, angles describing clockwise rotations can have negative measures of all sizes (Figure 3). hpotenuse opposite 9 3 5 3 FIGURE 3 Nonzero radian measures can be positive or negative and can go beond p. opp sin hp adj cos hp opp tan adj adjacent hp csc opp hp sec adj adj cot opp FIGURE Trigonometric ratios of an acute angle. P(, ) r O FIGURE 5 The trigonometric functions of a general angle u are defined in terms of,, and r. r ngle 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 eplicitl. When we talk about the angle p>3, we mean p>3 radians (which is 60 ), not p>3 degrees. We use radians because it simplifies man 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 Si asic Trigonometric Functions You are probabl familiar with defining the trigonometric functions of an acute angle in terms of the sides of a right triangle (Figure ). We etend this definition to obtuse and negative angles b first placing the angle in standard position in a circle of radius r. We then define the trigonometric functions in terms of the coordinates of the point P(, ) where the angle s terminal ra intersects the circle (Figure 5). sine: cosine: tangent: cosecant: secant: cotangent: These etended definitions agree with the right-triangle definitions when the angle is acute. Notice also that whenever the quotients are defined, tan u = sin u sec u = sin u = r = r tan u = cot u = tan u csc u = sin u csc u = r sec u = r cot u =
Chapter : Functions 6 3 3 FIGURE 6 Radian angles and side lengths of two common triangles. s ou can see, tan u and sec u are not defined if = = 0. This means the are not defined if u is ;p>, ;3p>, Á. Similarl, cot u and csc u are not defined for values of u for which = 0, namel u = 0, ;p, ;p, Á. The eact values of these trigonometric ratios for some angles can be read from the triangles in Figure 6. For instance, sin p = cos p = tan p = sin p 6 = cos p 6 = 3 tan p 6 = 3 sin p 3 = 3 cos p 3 = tan p 3 = 3 The CST rule (Figure 7) is useful for remembering when the basic trigonometric functions are positive or negative. For instance, from the triangle in Figure 8, we see that sin p 3 = 3, cos p 3 = -, tan p 3 = - 3. S sin pos all pos cos 3, sin 3, 3 P T tan pos C cos pos 3 3 FIGURE 7 The CST rule, remembered b the statement Calculus ctivates Student Thinking, tells which trigonometric functions are positive in each quadrant. FIGURE 8 The triangle for calculating the sine and cosine of p>3 radians. The side lengths come from the geometr of right triangles. Using a similar method we determined the values of sin u,, and tan u shown in Table. TLE Values of sin u,, and tan u for selected values of u Degrees 80 35 90 5 0 30 5 60 90 0 35 50 80 70 360 3p p p p p p p p 3p 5p 3p u (radians) p 0 p p 6 3 3 6 sin u tan u - - 3 3 0-0 0-0 - - 3 - - 3 0 0 - - 0 3-3 0-0 3-3 - 0 0 3 3
Periods of Trigonometric Functions Period P : tans + pd = tan cots + pd = cot Period P : sins + pd = sin coss + pd = cos secs + pd = sec Even coss -d = cos secs -d = sec Odd cscs + pd = csc sins -d = -sin tans -d = -tan cscs -d = -csc cots -d = -cot Periodicit and Graphs of the Trigonometric Functions When an angle of measure u and an angle of measure u + p are in standard position, their terminal ras coincide. The two angles therefore have the same trigonometric function values: sin(u + p) = sin u, tan(u + p) = tan u, and so on. Similarl, cos(u - p) =, sin(u - p) = sin u, and so on. We describe this repeating behavior b saing that the si basic trigonometric functions are periodic. DEFINITION function ƒ() is periodic if there is a positive number p such that ƒ( + p) = ƒ() for ever value of. The smallest such value of p is the period of ƒ. When we graph trigonometric functions in the coordinate plane, we usuall denote the independent variable b instead of u. Figure 9 shows that the tangent and cotangent functions have period p = p, and the other four functions have period p. lso, the smmetries 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). 0 3 0 3 sin Domain: Domain: Domain: 3,,... Range: Range: Range: Period: Period: (a) (b) Period: (c) 3 0 cos sec csc cot 3 Domain:, 3,... Range: or Period: (d) (e) sin 0 3 Domain: 0,,,... Range: or Period: 3 (f) 0 tan 3 0 3 Domain: 0,,,... Range: Period: FIGURE 9 Graphs of the si basic trigonometric functions using radian measure. The shading for each trigonometric function indicates its periodicit. P(cos, sin ) sin cos O Trigonometric Identities The coordinates of an point P(, ) in the plane can be epressed in terms of the point s distance r from the origin and the angle u that ra OP makes with the positive -ais (Figure 5). Since >r = and >r = sin u, we have = r, = r sin u. When r = we can appl the Pthagorean theorem to the reference right triangle in Figure 0 and obtain the equation FIGURE 0 The reference triangle for a general angle u. cos u + sin u =. (3)
This equation, true for all values of u, is the most frequentl used identit in trigonometr. Dividing this identit in turn b cos u and sin u gives + tan u = sec u + cot u = csc u The following formulas hold for all angles and. ddition Formulas coss + d = cos cos - sin sin sins + d = sin cos + cos sin () Double-ngle Formulas = cos u - sin u sin u = sin u (5) dditional formulas come from combining the equations cos u + sin u =, cos u - sin u =. We add the two equations to get cos u = + and subtract the second from the first to get sin u = -. This results in the following identities, which are useful in integral calculus. Half-ngle Formulas cos u = sin u = + - (6) (7) The Law of Cosines If a, b, and c are sides of a triangle C and if u is the angle opposite c, then c = a + b - ab. (8) This equation is called the law of cosines.
(a cos, a sin ) c a C b (b, 0) FIGURE The square of the distance between and gives the law of cosines. We can see wh the law holds if we introduce coordinate aes with the origin at C and the positive -ais along one side of the triangle, as in Figure. The coordinates of are (b, 0); the coordinates of are sa, a sin ud. The square of the distance between and is therefore c = sa - bd + sa sin ud = a scos u + sin ud + b - ab ('')''* = a + b - ab. The law of cosines generalizes the Pthagorean theorem. If u = p>, then = 0 and c = a + b. Transformations of Trigonometric Graphs The rules for shifting, stretching, compressing, and reflecting the graph of a function summarized in the following diagram appl to the trigonometric functions we have discussed in this section. Vertical stretch or compression; reflection about -ais if negative Vertical shift = aƒ(bs + cdd + d Horizontal stretch or compression; reflection about -ais if negative Horizontal shift The transformation rules applied to the sine function give the general sine function or sinusoid formula ƒ() = sin a p ( - C)b + D, where ƒ ƒ is the amplitude, ƒ ƒ is the period, C is the horizontal shift, and D is the vertical shift. graphical interpretation of the various terms is revealing and given below. D D Horizontal shift (C) mplitude () ( ) sin ( C) D This ais is the line D. D 0 Vertical shift (D) This distance is the period (). Two Special Inequalities For an angle u measured in radians, - ƒ u ƒ sin u ƒ u ƒ and - ƒ u ƒ - ƒ u ƒ.
cos O Q P sin cos (, 0) FIGURE From the geometr of this figure, drawn for u 7 0, we get the inequalit sin u + ( - ) u. To establish these inequalities, we picture u as a nonzero angle in standard position (Figure ). The circle in the figure is a unit circle, so equals the length of the circular ƒ u ƒ arc P. The length of line segment P is therefore less than. ƒ u ƒ Triangle PQ is a right triangle with sides of length QP = ƒ sin u ƒ, Q = -. From the Pthagorean theorem and the fact that P 6 ƒ u ƒ, we get sin u + ( - ) = (P) u. (9) The terms on the left-hand side of Equation (9) are both positive, so each is smaller than their sum and hence is less than or equal to u : sin u u and ( - ) u. taking square roots, this is equivalent to saing that ƒ sin u ƒ ƒ u ƒ and ƒ - ƒ ƒ u ƒ, so - ƒ u ƒ sin u ƒ u ƒ and - ƒ u ƒ - ƒ u ƒ. These inequalities will be useful in the net chapter. Eercises.3 8 Chapter : Functions Radians and Degrees. On a circle of radius 0 m, how long is an arc that subtends a central angle of (a) p> 5 radians? (b) 0?. central angle in a circle of radius 8 is subtended b an arc of length 0p. Find the angle s radian and degree measures. 3. You want to make an 80 angle b marking an arc on the perimeter of a -in.-diameter disk and drawing lines from the ends of the arc to the disk s center. To the nearest tenth of an inch, how long should the arc be?. If ou roll a -m-diameter wheel forward 30 cm over level ground, through what angle will the wheel turn? nswer in radians (to the nearest tenth) and degrees (to the nearest degree). Evaluating Trigonometric Functions 5. Cop and complete the following table of function values. If the function is undefined at a given angle, enter UND. Do not use a calculator or tables. U sin u tan u cot u sec u csc u P P>3 6. Cop and complete the following table of function values. If the function is undefined at a given angle, enter UND. Do not use a calculator or tables. 0 P> 3P> U sin u tan u cot u sec u csc u 3P> P>3 In Eercises 7, one of sin, cos, and tan is given. Find the other two if lies in the specified interval. 9. cos = 0. 3, H c- p, 0 d P>6 7. sin = 3 8. tan =, H c0, p 5, H cp, p d d. tan =. sin = -, H cp, 3p, H cp, 3p d d Graphing Trigonometric Functions Graph the functions in Eercises 3. What is the period of each function? 3. sin. sin ( >) 5. cos p 6. cos p 7. -sin p 3 8. -cos p 9. cos a - p 0. sin a + p b 6 b P> 5P>6 cos = - 5 3, H cp, p d
. sin a - p. cos a + p b + 3 b - Graph the functions in Eercises 3 6 in the ts-plane (t-ais horizontal, s-ais vertical). What is the period of each function? What smmetries do the graphs have? 3. s = cot t. s = -tan pt 5. s = sec apt 6. s = csc a t b b T 7. a. Graph = cos and = sec together for -3p> 3p>. Comment on the behavior of sec in relation to the signs and values of cos. b. Graph = sin and = csc together for -p p. Comment on the behavior of csc in relation to the signs and values of sin. T 8. Graph = tan and = cot together for -7 7. Comment on the behavior of cot in relation to the signs and values of tan. 9. Graph = sin and = :sin ; together. What are the domain and range of :sin ;? 30. Graph = sin and = <sin = together. What are the domain and range of <sin =? Using the ddition Formulas Use the addition formulas to derive the identities in Eercises 3 36. 3. cos a - p 3. cos a + p b = sin b = -sin Solving Trigonometric Equations For Eercises 5 5, solve for the angle u, where 0 u p. 5. sin u = 3 5. sin u = cos u 53. sin u - = 0 5. + = 0 Theor and Eamples 55. The tangent sum formula The standard formula for the tangent of the sum of two angles is Derive the formula. tans + d = 56. (Continuation of Eercise 55.) Derive a formula for tans - d. 57. ppl the law of cosines to the triangle in the accompaning figure to derive the formula for coss - d. tan + tan - tan tan. 0 33. sin a + p 3. b = cos sin a - p b = -cos 35. coss - d = cos cos + sin sin (Eercise 57 provides a different derivation.) 36. sins - d = sin cos - cos sin 37. What happens if ou take = in the trigonometric identit coss - d = cos cos + sin sin? Does the result agree with something ou alread know? 38. What happens if ou take = p in the addition formulas? Do the results agree with something ou alread know? In Eercises 39, epress the given quantit in terms of sin and cos. 39. cossp + d 0. sinsp - d. sin a3p. cos a 3p - b 7p 3. Evaluate sin as sin a p + p 3 b. p. Evaluate cos as cos a p + p 3 b. 5. Evaluate cos p 6. Evaluate. Using the Half-ngle Formulas Find the function values in Eercises 7 50. 7. 8. cos 5p cos p 8 9. 50. sin 3p sin p 8 + b sin 5p. 58. a. ppl the formula for coss - d to the identit sin u = cos a p to obtain the addition formula for sins + d. - ub b. Derive the formula for coss + d b substituting - for in the formula for coss - d from Eercise 35. 59. triangle has sides a = and b = 3 and angle C = 60. Find the length of side c. 60. triangle has sides a = and b = 3 and angle C = 0. Find the length of side c. 6. The law of sines The law of sines sas that if a, b, and c are the sides opposite the angles,, and C in a triangle, then Use the accompaning figures and the identit sinsp - ud = sin u, if required, to derive the law. c a sin a h = sin b b C = sin C c. 6. triangle has sides a = and b = 3 and angle C = 60 (as in Eercise 59). Find the sine of angle using the law of sines. a c C b h
ƒ ƒ 63. triangle has side c = and angles = p> and = p>3. 69. The period Set the constants = 3, C = D = 0. Find the length a of the side opposite. a. Plot ƒ() for the values =, 3, p, 5p over the interval T 6. The approimation sin L It is often useful to know that, -p p. Describe what happens to the graph of the when is measured in radians, sin L for numericall small values general sine function as the period increases. of. In Section 3., we will see wh the approimation holds. b. What happens to the graph for negative values of? Tr it The approimation error is less than in 5000 if 6 0.. with = -3 and = -p. a. With our grapher in radian mode, graph = sin and = 70. The horizontal shift C Set the constants = 3, = 6, D = 0. together in a viewing window about the origin. What do ou a. Plot ƒ() for the values C = 0,, and over the interval see happening as nears the origin? -p p. Describe what happens to the graph of the b. With our grapher in degree mode, graph = sin and general sine function as C increases through positive values. = together about the origin again. How is the picture different from the one obtained with radian mode? b. What happens to the graph for negative values of C? c. What smallest positive value should be assigned to C so the General Sine Curves For graph ehibits no horizontal shift? Confirm our answer with a plot. ƒsd = sin a p identif,, C, and D for the sine functions in Eercises 65 68 and sketch their graphs. 65. = sins + pd - 66. 67. = - p sin ap 68. = tb + p s - Cdb + D, = sinsp - pd + L pt sin p L, L 7 0 COMPUTER EXPLORTIONS In Eercises 69 7, ou will eplore graphicall the general sine function 7. The vertical shift D Set the constants = 3, = 6, C = 0. a. Plot ƒ() for the values D = 0,, and 3 over the interval -p p. 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? 7. The amplitude Set the constants = 6, C = D = 0. a. Describe what happens to the graph of the general sine function as increases through positive values. Confirm our answer b plotting ƒ() for the values =, 5, and 9. b. What happens to the graph for negative values of? ƒsd = sin a p s - Cdb + D as ou change the values of the constants,, C, and D. Use a CS or computer grapher to perform the steps in the eercises.