1.6. Trigonometric Functions. 48 Chapter 1: Preliminaries. Radian Measure

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1 48 Chapte : Peliminaies.6 Tigonometic Functions Cicle B' B θ C A Unit of cicle adius FIGURE.63 The adian measue of angle ACB is the length u of ac AB on the unit cicle centeed at C. The value of u can be found fom an othe cicle, howeve, as the atio s>. Thus s = u is the length of ac on a cicle of adius when u is measued in adians. Convesion Fomulas degee = p s L.d adians 8 Degees to adians: multipl b p 8 adian = 8 p s L57d degees Radians to degees: multipl b 8 p s A' This section eviews the basic tigonometic functions. The tigonometic functions ae impotant because the ae peiodic, o epeating, and theefoe model man natuall occuing peiodic pocesses. Radian Measue In navigation and astonom, angles ae measued in degees, but in calculus it is best to use units called adians because of the wa the simplif late calculations. The adian measue of the angle ACB at the cente of the unit cicle (Figue.63) equals the length of the ac that ACB cuts fom the unit cicle. Figue.63 shows that s = u is the length of ac cut fom a cicle of adius when the subtending angle u poducing the ac is measued in adians. Since the cicumfeence of the cicle is p and one complete evolution of a cicle is 36, the elation between adians and degees is given b Fo eample, 45 in adian measue is and p>6 adians is p adians = # p 8 = p 4 ad, p # 8 6 p = 3. Figue.64 shows the angles of two common tiangles in both measues. An angle in the -plane is said to be in standad position if its vete lies at the oigin and its initial a lies along the positive -ais (Figue.65). Angles measued counteclockwise fom the positive -ais ae assigned positive measues; angles measued clockwise ae assigned negative measues.

2 .6 Tigonometic Functions 49 Degees Radians Teminal a Positive measue Initial a Teminal a Initial a Negative measue FIGURE.64 The angles of two common tiangles, in degees and adians. FIGURE.65 Angles in standad position in the -plane. When angles ae used to descibe counteclockwise otations, ou measuements can go abitail fa beond p adians o 36. Similal, angles descibing clockwise otations can have negative measues of all sizes (Figue.66) FIGURE.66 Nonzeo adian measues can be positive o negative and can go beond p.

3 5 Chapte : Peliminaies hpotenuse opp sin hp adj cos hp opp tan adj adjacent hp csc opp hp sec adj adj cot opp opposite FIGURE.67 Tigonometic atios of an acute angle. Angle Convention: Use Radians Fom now on in this book it is assumed that all angles ae measued in adians unless degees o some othe unit is stated eplicitl. When we talk about the angle p>3, we mean p>3 adians (which is 6 ), not p>3 degees. When ou do calculus, keep ou calculato in adian mode. The Si Basic Tigonometic Functions You ae pobabl familia with defining the tigonometic functions of an acute angle in tems of the sides of a ight tiangle (Figue.67). We etend this definition to obtuse and negative angles b fist placing the angle in standad position in a cicle of adius. We then define the tigonometic functions in tems of the coodinates of the point P(, ) whee the angle s teminal a intesects the cicle (Figue.68). sine: sin u = cosecant: csc u = cosine: cos u = secant: sec u = P(, ) FIGURE.68 The tigonometic functions of a geneal angle u ae defined in tems of,, and. tangent: tan u = cotangent: cot u = These etended definitions agee with the ight-tiangle definitions when the angle is acute (Figue.69). Notice also the following definitions, wheneve the quotients ae defined. tan u = sin u cos u sec u = cos u cot u = tan u csc u = sin u As ou can see, tan u and sec u ae not defined if =. This means the ae not defined if u is ;p>, ;3p>, Á. Similal, cot u and csc u ae not defined fo values of u fo which =, namel u =, ;p, ;p, Á. The eact values of these tigonometic atios fo some angles can be ead fom the tiangles in Figue.64. Fo instance, hpotenuse adjacent P(, ) opposite sin p 4 = cos p 4 = sin p 6 = cos p 6 = 3 sin p 3 = 3 cos p 3 = tan p tan p tan p 6 = 4 = 3 3 = 3 The CAST ule (Figue.7) is useful fo emembeing when the basic tigonometic functions ae positive o negative. Fo instance, fom the tiangle in Figue.7, we see that sin p 3 = 3, cos p 3 =-, tan p 3 = -3. FIGURE.69 The new and old definitions agee fo acute angles. Using a simila method we detemined the values of sin u, cos u, and tan u shown in Table.4.

4 .6 Tigonometic Functions 5 cos 3, sin 3, 3 P S sin pos T tan pos A all pos C cos pos 3 3 FIGURE.7 The CAST ule, emembeed b the statement All Students Take Calculus, tells which tigonometic functions ae positive in each quadant. FIGURE.7 The tiangle fo calculating the sine and cosine of p>3 adians. The side lengths come fom the geomet of ight tiangles. Most calculatos and computes eadil povide values of the tigonometic functions fo angles given in eithe adians o degees. TABLE.4 Values of sin u, cos u, and tan u fo selected values of u Degees p p p p p p p p 3p 5p 3p u (adians) -p p p sin u cos u tan u EXAMPLE Finding Tigonometic Function Values If tan u = 3> and 6 u 6 p>, find the five othe tigonometic functions of u. Solution Fom tan u = 3>, we constuct the ight tiangle of height 3 (opposite) and base (adjacent) in Figue.7. The Pthagoean theoem gives the length of the hpotenuse, = 3. Fom the tiangle we wite the values of the othe five tigonometic functions: cos u = 3, sin u = 3 3 3, sec u =, csc u = 3 3, cot u = 3

5 5 Chapte : Peliminaies Peiodicit and Gaphs of the Tigonometic Functions 3 3 When an angle of measue u and an angle of measue u + p ae in standad position, thei teminal as coincide. The two angles theefoe have the same tigonometic function values: cossu + pd = cos u secsu + pd = sec u sinsu + pd = sin u cscsu + pd = csc u tansu + pd = tan u cotsu + pd = cot u Similal, cos su - pd = cos u, sin su - pd = sin u, and so on. We descibe this epeating behavio b saing that the si basic tigonometic functions ae peiodic. FIGURE.7 The tiangle fo calculating the tigonometic functions in Eample. DEFINITION Peiodic Function A function ƒ() is peiodic if thee is a positive numbe p such that ƒs + pd = ƒsd fo eve value of. The smallest such value of p is the peiod of ƒ. When we gaph tigonometic functions in the coodinate plane, we usuall denote the independent vaiable b instead of u. See Figue.73. tan cos sin 3 3 sin Domain: Domain: Domain: 3,,... Range: Range: Range: Peiod: Peiod: (a) (b) Peiod: (c) 3 sec csc cot Domain:, 3,... Range: and Peiod: (d) Domain:,,,... Range: and Peiod: (e) Domain:,,,... Range: Peiod: (f) FIGURE.73 Gaphs of the (a) cosine, (b) sine, (c) tangent, (d) secant, (e) cosecant, and (f) cotangent functions using adian measue. The shading fo each tigonometic function indicates its peiodicit.

6 .6 Tigonometic Functions 53 Peiods of Tigonometic Functions Peiod P : tans + pd = tan cots + pd = cot Peiod P : sins + pd = sin coss + pd = cos secs + pd = sec cscs + pd = csc As we can see in Figue.73, the tangent and cotangent functions have peiod p = p. The othe fou functions have peiod p. Peiodic functions ae impotant because man behavios studied in science ae appoimatel peiodic. A theoem fom advanced calculus sas that eve peiodic function we want to use in mathematical modeling can be witten as an algebaic combination of sines and cosines. We show how to do this in Section.. The smmeties in the gaphs in Figue.73 eveal that the cosine and secant functions ae even and the othe fou functions ae odd: Even coss -d = cos secs -d = sec Odd sins -d = -sin tans -d = -tan cscs -d = -csc cots -d = -cot Identities P(cos, sin ) sin cos The coodinates of an point P(, ) in the plane can be epessed in tems of the point s distance fom the oigin and the angle that a OP makes with the positive -ais (Figue.69). Since > = cos u and > = sin u, we have = cos u, = sin u. When = we can appl the Pthagoean theoem to the efeence ight tiangle in Figue.74 and obtain the equation FIGURE.74 The efeence tiangle fo a geneal angle u. cos u + sin u =. () This equation, tue fo all values of u, is the most fequentl used identit in tigonomet. Dividing this identit in tun b cos u and sin u gives + tan u = sec u. + cot u = csc u. The following fomulas hold fo all angles A and B (Eecises 53 and 54). Addition Fomulas cossa + Bd = cos A cos B - sin A sin B sinsa + Bd = sin A cos B + cos A sin B ()

7 54 Chapte : Peliminaies Thee ae simila fomulas fo cossa - Bd and sinsa - Bd (Eecises 35 and 36). All the tigonometic identities needed in this book deive fom Equations () and (). Fo eample, substituting u fo both A and B in the addition fomulas gives Double-Angle Fomulas cos u = cos u - sin u sin u = sin u cos u (3) Additional fomulas come fom combining the equations cos u + sin u =, cos u - sin u = cos u. We add the two equations to get cos u = + cos u and subtact the second fom the fist to get sin u = - cos u. This esults in the following identities, which ae useful in integal calculus. Half-Angle Fomulas cos u = sin u = + cos u - cos u (4) (5) The Law of Cosines If a, b, and c ae sides of a tiangle ABC and if u is the angle opposite c, then c = a + b - ab cos u. (6) B(a cos, a sin ) c a C b A(b, ) FIGURE.75 The squae of the distance between A and B gives the law of cosines. This equation is called the law of cosines. We can see wh the law holds if we intoduce coodinate aes with the oigin at C and the positive -ais along one side of the tiangle, as in Figue.75. The coodinates of A ae (b, ); the coodinates of B ae sa cos u, a sin ud. The squae of the distance between A and B is theefoe c = sa cos u - bd + sa sin ud = a scos u + sin ud + b - ab cos u ('')''* = a + b - ab cos u. The law of cosines genealizes the Pthagoean theoem. If u = p>, then cos u = and c = a + b.

8 .6 Tigonometic Functions 55 Tansfomations of Tigonometic Gaphs The ules fo shifting, stetching, compessing, and eflecting the gaph of a function appl to the tigonometic functions. The following diagam will emind ou of the contolling paametes. Vetical stetch o compession; eflection about -ais if negative Vetical shift = aƒ(bs + cdd + d Hoizontal stetch o compession; eflection about -ais if negative Hoizontal shift EXAMPLE Modeling Tempeatue in Alaska The buildes of the Tans-Alaska Pipeline used insulated pads to keep the pipeline heat fom melting the pemanentl fozen soil beneath. To design the pads, it was necessa to take into account the vaiation in ai tempeatue thoughout the ea. The vaiation was epesented in the calculations b a geneal sine function o sinusoid of the fom ƒsd = A sin c p B s - Cd d + D, whee ƒ A ƒ is the amplitude, ƒ B ƒ is the peiod, C is the hoizontal shift, and D is the vetical shift (Figue.76). D A D Hoizontal shift (C) Amplitude (A) ( ) A sin ( C) D B This ais is the line D. D A Vetical shift (D) This distance is the peiod (B). FIGURE.76 The geneal sine cuve = A sin [sp>bds - Cd] + D, shown fo A, B, C, and D positive (Eample ). Figue.77 shows how to use such a function to epesent tempeatue data. The data points in the figue ae plots of the mean dail ai tempeatues fo Faibanks, Alaska, based on ecods of the National Weathe Sevice fom 94 to 97. The sine function used to fit the data is ƒsd = 37 sin c p s - d d + 5, 365

9 56 Chapte : Peliminaies whee ƒ is tempeatue in degees Fahenheit and is the numbe of the da counting fom the beginning of the ea. The fit, obtained b using the sinusoidal egession featue on a calculato o compute, as we discuss in the net section, is ve good at captuing the tend of the data. 6 Tempeatue ( F) 4 Jan Feb Ma Ap Ma Jun Jul Aug Sep Oct Nov Dec Jan Feb Ma FIGURE.77 Nomal mean ai tempeatues fo Faibanks, Alaska, plotted as data points (ed). The appoimating sine function (blue) is ƒsd 37 sin [sp 365ds d] 5.