2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES


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1 . TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an angle α is said to be in standad position if its vete is at the oigin O and its initial side coincides with the positive ais (Figue.). An angle is said to be in a cetain quadant if, when the angle is in standad position, the teminal side lies in that quadant. Fo instance, a 6 angle lies in quadant I o simpl that it is a quadant I angle. As Figue b shows, an angle of 8 is a quadant II angle. If the teminal side of an angle in standad position lies along eithe the ais o the ais, then the angle is called quadantal. Fo eample, 60, 0, 80, 90, 0, 90, 80, 0, 60 ae quadantal angles. Evidentl, an angle is quadantal if and onl π if its measue is an intege multiple of 90 ( o adians). Figue. α is in standad position teminal side α initial side Figue. (a) quadant I angle (b) quadant II angle teminal side 6 teminal side 8 9
2 Definition.: Tigonometic Functions of a Geneal Angle Let θ be an angle in standad position and suppose that (, ) is an point othe than ( 0, 0 ) on the teminal side of θ (Figue.). If + is the distance between (, ) and ( 0, 0 ), then the si tigonometic functions of θ ae defined b Figue. sin θ cos θ tan θ csc θ sec θ cot θ.(, ) θ povided that the denominatos ae not zeo. O Using simila tiangles, ou can see that the values of the si tigonometic functions in Definition. depend onl on the angle θ and not on the choice of the point (, ) on the teminal side of θ. Eample Evaluate the si tigonometic functions of the angle θ in standad position if the teminal side of θ contains the point (, ) (, ). Hee,, , and Thus, + + ( ). sin θ cos θ tan θ csc θ sec θ cot θ. You can detemine the algebaic signs of the tigonometic functions fo angles in the vaious quadants b ecalling the algebaic signs of and in these quadants and 0
3 emembeing that is alwas positive. Fo instance, as Figue. shows, is positive in quadants I and II (whee both and ae positive), and it is negative in quadants III and IV (whee is negative and is positive). B poceeding in a simila wa, ou can detemine the signs of the emaining tigonometic functions in the vaious quadants and thus confim the esults in Table.. Figue. > 0 > 0 > 0 Quadant II > 0 Quadant I < 0 O < 0 < 0 Quadant III < 0 Quadant IV Table. Quadant Containing θ I II III IV Positive Functions All, cscθ tanθ, cotθ, secθ Negative Functions None, secθ, tanθ, cotθ, cscθ,, secθ, cscθ, tanθ, cotθ Eample Find the quadant in which θ lies if tanθ > 0 and < 0. This eample can be woked b using Table.; howeve, athe than eling on the table, we pefe to eason as follows: Let (, ) be a point othe than the oigin on the teminal side of θ (in standad position). Because tanθ > 0, we see that and have the same algebaic sign. Futhemoe, since < 0, it follows that < 0. Because < 0 and < 0, the angle is in quadant III.
4 Recipocal Identities If θ is an angle fo which the functions ae defined, then: (i) cscθ (ii) secθ (iii) cotθ tanθ. Quotient Identities If θ is an angle fo which the functions ae defined, then: tanθ and cotθ. Eample If and of θ. tanθ secθ cotθ cscθ tanθ, find the values of the othe fou tigonometic functions. B using the ecipocal and quotient identities, ou can quickl ecall the algebaic signs of the secant, cosecant, tangent, and cotangent in the fou quadants (Table ), if ou know the algebaic signs of the sine and cosine in these quadants. Anothe impotant identit is deived as follows: Again suppose that θ is an angle in standad position and that (, ) is a point othe than the oigin on the teminal side of θ (Figue 9). Because +, we have +, so (cos θ ) + ( ) + + The elationship: ( ) + ( ) is called the fundamental Pthagoean identit because its deivation involves the fact that +, which is a consequence of the Pthagoean theoem..
5 The fundamental Pthagoean identit is used quite often, and it would be bothesome to wite the paentheses each time fo ( ) and ( ) ; et, if the paentheses wee simpl omitted, the esulting epessions would be misundestood. (Fo instance, is usuall undestood to mean the cosine of the squae of θ.) Theefoe, it is customa to wite cos θ and sin θ to mean ( ) and ( ). Simila notation is used fo the emaining tigonometic functions and fo powes othe than. Thus, cot θ means ( cot θ ), n n sec θ means ( sec θ ), and so foth. With this notation, the fundamental Pthagoean identit becomes cos θ + sin θ. Actuall, thee ae thee Pthagoean identities the fundamental identit and two othes deived fom it. Pthagoean Identities If θ is an angle fo which the functions ae defined, then: (i) θ cos + sin θ (ii) + tan θ sec θ (iii) + cot θ csc θ We alead poved (i). To pove (ii), we divide both sides of (i) b sin θ + cos θ cos θ o +, povided that cos θ 0. Since tanθ and secθ, cos θ to obtain we have that + tan θ θ sec. Identit (iii) is poved b dividing both sides of (i) b sin θ. Eample The value of one of the tigonometic functions of an angle θ is given along with the infomation about the quadant in which θ lies, Find the values of the othe five tigonometic functions of θ :
6 ( a ), θ in quadant II. B the fundamental Pthagoean identit, cos θ + sin θ, so cos θ θ sin Theefoe, cos θ ± ±. 69 Because θ is in quadant II, we know that is negative; hence, . It follows that tanθ secθ cotθ tanθ cscθ ( b ) tanθ and < 0. Because tanθ < 0 onl in quadants II and IV, and < 0 onl in quadants III and IV, it follows that θ must be in quadant IV. B pat (ii) θ sec + tan θ, so sec θ ± + tan θ ± + ( 8 ) ± Since θ is in quadant IV, secθ > 0; hence, secθ. Because secθ it follows that secθ, θ θ sin Now, tanθ cos 8 so (tanθ )( ) Finall, cscθ and cotθ tanθ ± 9 8. ±.
7 In the applications of tigonomet, and especiall in calculus, it is often necessa to make tigonometic calculations, as we have done in this section, without the use of calculatos o tables. Section Poblems In poblems to 0, sketch two coteminal angles α and β in standad position whose teminal side contains the given point. Aange it so that α is positive, β is negative, and neithe angle eceeds one evolution. In each case, name the quadant in which the angle lies, o indicate that the angle is quadantal.. (, ). (, ), ). (, 0 ). (. (, ) 6. ( 0, ). (, ) 8. (, 0 ) 9. (, ) 0. ( 0, ) In poblems to 8, specif and sketch thee angles that ae coteminal with the given angle in standad position π π.. π π In poblems 6 to 8, evaluate the si tigonometic functions of the angle θ in standad position if the teminal side of θ contains the given point (, ). [Do not use a calculato leave all answes in the fom of a faction o an intege.] In each case, sketch one of the coteminal angles θ. 9. (, ) 0. (, ). (, ). (, ). (, ). (, ). (, ) 6. (, ). (, ) 8. ( 0, ) 9. Is thee an angle θ fo which? Eplain.
8 0. Using simila tiangles, show that the values of the si tigonometic functions in Definition. depend onl on the angle θ and not on the choice of the point (, ) on the teminal side of θ.. In each case, assume that θ is an angle in standad position and find the quadant in which it lies. (a) tanθ > 0 and secθ > 0 (b) > 0 and secθ < 0 (c) > 0 and < 0 (d) secθ > 0 and tanθ < 0 (e) tanθ > 0 and cscθ < 0 (f) < 0 and cscθ < 0 (g) secθ > 0 and cotθ < 0 (h) cotθ > 0 and > 0. Is thee an angle θ fo which > 0 and cscθ < 0? Eplain.. Give the algebaic sign of each of the following. (a) cos 6 (b) sin o π (d) tan (c) sec ( ) (e) cot (g) sec 8π 8π (f) csc π. If θ is an angle fo which the functions ae defined, show that secθ ( )( tanθ ).. If and cos θ, use the ecipocal and quotient identities to find (a) secθ (b) cscθ (c) tanθ (d) cotθ. 6
9 6. If secθ and cscθ, use the ecipocal and quotient identities to find (a) (b) (c) tanθ (d) cotθ. In Poblems to 8, the values of one of the tigonometic functions of an angle θ is given along with infomation about the quadant (Q) in which θ lies. Find the values of the othe five tigonometic functions of θ.., θ in Q I 8., θ in Q IV 9., θ in Q III 0., θ not in Q I., < 0., θ not in Q I. cscθ, θ in Q I. secθ, θ in Q III. tanθ, θ in Q I 6. tanθ, < 0. cotθ, cscθ > 0 8. cscθ, secθ < 0
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