. 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 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 is simpl said to be a quadant I angle. As Figue. b shows, an angle of measue 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 all quadantal angles. Evidentl, an angle is quadantal if and onl if its measue is an intege multiple of 90 ( o adians). Figue. is in standad position Teminal side i Initial side Figue. (a) quadant I angle (b) quadant II angle Teminal side 6 Teminal side 8 8
Definition.: Tigonometic Functions of a Geneal Angle Let be an angle in standad position and suppose that (, ) is an point othe than ( 0, 0 ) on the teminal side of (Figue.). If is the distance between (, ) and ( 0, 0 ), then the si tigonometic functions of ae defined b Figue. sin cos tan csc sec cot 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 and not on the choice of the point (, ) on the teminal side of. Eample. ---------------------------- ------------------------------------------------------------ Evaluate the si tigonometic functions of the angle in standad position if the teminal side of contains the point (, ) (, ). Hee,,, and Thus, ( ). sin cos tan csc sec cot. 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 9
emembeing that is alwas positive. Fo instance, as Figue. shows, sin 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 Z > 0 Z sin θ > 0 in Quadant II sin θ > 0 in Quadant I < 0 Z O < 0 Z sin θ < 0 in Quadant III sin θ < 0 in Quadant IV Table. Quadant Containing I II III IV Positive Functions All sin, csc tan, cot cos, sec Negative Functions None cos, sec, tan, cot sin, csc, cos, sec sin, csc, tan, cot Eample. ---------------------------- ------------------------------------------------------------ Find the quadant in which lies if tan > 0 and sin < 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 (in standad position). Because tan > 0, we see that and have the same algebaic sign. Futhemoe, since sin < 0, it follows that < 0. Because < 0 and < 0, the angle is in quadant III. 0
Recipocal Identities If is an angle fo which the functions ae defined, then: (i) csc sin (ii) sec cos (iii) cot tan. Quotient Identities If is an angle fo which the functions ae defined, then: sin cos tan and cot. cos sin Eample. ---------------------------- ------------------------------------------------------------ If sin and cos, find the values of the othe fou tigonometic functions of. tan sec cot csc sin cos cos tan sin. 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 is an angle in standad position and that (, ) is a point othe than the oigin on the teminal side of (Figue 9). Because, we have (cos ) (sin ) + +, so + The elationship: (cos ) (sin ) is called the fundamental Pthagoean identit because its deivation involves the fact that +, which is a consequence of the Pthagoean theoem..
The fundamental Pthagoean identit is used quite often, and it would be bothesome to wite the paentheses each time fo (cos ) and (sin ) ; et, if the paentheses wee simpl omitted, the esulting epessions would be misundestood. (Fo instance, cos is usuall undestood to mean the cosine of the squae of.) Theefoe, it is customa to wite cos and sin to mean (cos ) and (sin ). Simila notation is used fo the emaining tigonometic functions and fo powes othe than. Thus, cot means ( cot ), n n sec means ( sec ), and so foth. With this notation, the fundamental Pthagoean identit becomes cos + sin. Actuall, thee ae thee Pthagoean identities the fundamental identit and two othes deived fom it. Pthagoean Identities If is an angle fo which the functions ae defined, then: (i) cos + sin (ii) tan sec (iii) cot csc We alead poved (i). To pove (ii), we divide both sides of (i) b + sin cos sin o + cos povided that cos 0. Since sin tan and cos cos cos cos sec,, cos to obtain we have that + tan sec. Identit (iii) is poved b dividing both sides of (i) b sin. Eample. ---------------------------- ------------------------------------------------------------ The value of one of the tigonometic functions of an angle is given along with the infomation about the quadant in which lies, Find the values of the othe five tigonometic functions of :
( a ) sin, in quadant II. B the fundamental Pthagoean identit, cos sin cos + 69 sin, so 69 69 Theefoe, cos. 69 Because is in quadant II, we know that cos is negative; hence, cos. It follows that sin tan cos sec cos cot tan csc sin. 69-8 ( b ) tan and sin < 0. Because tan < 0 onl in quadants II and IV, and sin < 0 onl in quadants III and IV, it follows that must be in quadant IV. B pat (ii) sec tan, so sec tan - 8 6 9 Since is in quadant IV, sec > 0; hence, sec. Because sec it follows that cos sec Now, tan, cos sin cos 8 so sin (tan )(cos ) Finall, csc and cot tan sin - 8-8. 8. 8 9 8..
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 and in standad position whose teminal side contains the given point. Aange it so that is positive, 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.. 60.... 6 6.. 8. 0 6 In poblems 6 to 8, evaluate the si tigonometic functions of the angle in standad position if the teminal side of 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. 9. (, ) 0. (, ). (, ). (, ). (, ). (, ). (, ) 6. (, ). (, ) 8. ( 0, ) 9. Is thee an angle fo which sin? Eplain.
0. Using simila tiangles, show that the values of the si tigonometic functions in Definition depend onl on the angle and not on the choice of the point (, ) on the teminal side of.. In each case, assume that is an angle in standad position and find the quadant in which it lies. (a) tan > 0 and sec > 0 (b) sin > 0 and sec < 0 (c) sin > 0 and cos < 0 (d) sec > 0 and tan < 0 (e) tan > 0 and csc < 0 (f) cos < 0 and csc < 0 (g) sec > 0 and cot < 0 (h) cot > 0 and sin > 0. Is thee an angle fo which sin > 0 and csc < 0? Eplain.. Give the algebaic sign of each of the following. (a) cos 6 (b) sin o (c) sec (d) tan 8 (e) cot (f) csc 8 (g) sec. If is an angle fo which the functions ae defined, show that sec ( sin )( tan ) cos.. If sin and cos, use the ecipocal and quotient identities to find (a) sec (b) csc (c) tan (d) cot.
6. If sec and csc, use the ecipocal and quotient identities to find (a) sin (b) cos (c) tan (d) cot. In Poblems to 8, the values of one of the tigonometic functions of an angle is given along with infomation about the quadant (Q) in which lies. Find the values of the othe five tigonometic functions of.. sin, in Q I 8. cos, in Q IV 9. sin, in Q III 0. sin, not in Q I. cos, sin < 0. cos, not in Q I. csc, in Q I. sec, in Q III. tan, in Q I 6. tan, sin < 0. cot, csc > 0 8. csc, sec < 0 6