Transcription of Cosecant, Secant, and Cotangent
1 Cosecant, Secant, and CotangentIn this chapter we ll introduced three more trigonometric functions: thecosecant, thesecant, and thecotangent. These functions are written as csc( ),sec( ), and cot( ) respectively. They are the functions defined by the formulasbelow:csc( ) =1sin( )sec( ) =1cos( )cot( ) =cos( )sin( )Graphs of cosecant, secant, and cotangent27400 PeriodsCosecant, secant, and Cotangent are periodic functions. Cosecant and se-cant have the same period as sine and cosine do, namely 2 . Cotangenthas period , just as tangent does. In terms of formulas, the previous twosentences mean thatcsc( + 2 ) = csc( )sec( + 2 ) = sec( )cot( + ) = cot( )It s easy to check why these functions have the periods that they do. Forexample, because sine has period 2 that is, because sin( + 2 ) = sin( ) we can check thatcsc( + 2 ) =1sin( + 2 )=1sin( )= csc( )Similarly, the secant function has the same period, 2 , as the function usedto define it, and oddRecall that an even function is a functionf(x) with the property thatf( x) =f(x).
2 Examples includex2,x4,x6, and can add secant to the list of functions that we know are even is, sec( ) = sec( ). The reason secant is even is that cosine is even:sec( ) =1cos( )=1cos( )= sec( )An odd function is a functionf(x) with the property thatf( x) = f(x).Examples includex3,x5,x7, sine, and and Cotangent are odd functions, meaning that csc( ) = csc( )and cot( ) = cot( ). We can check that these identities are true by usingthat sine is an odd function and that cosine is even:csc( ) =1sin( )=1 sin( )= csc( )cot( ) =cos( )sin( )=cos( ) sin( )= cot( )275 Cofunction identitiesSine and cosine, secant and cosecant, tangent and Cotangent ; these pairs offunctions satisfy a common identity that is sometimes called thecofunctionidentity:sin( 2 )= cos( )sec( 2 )= csc( )tan( 2 )= cot( )These identities also go the other way :cos( 2 )= sin( )csc( 2 )= sec( )cot( 2 )= tan( )Let s check one of these six identities, the identity cos( 2 )= sin( ).
3 Inorder to see that this identity is true, we ll start with cos( 2 )and we lluse that cosine is an even function, socos( 2 )= cos( [ 2 ])= cos( 2)Now we can use the identity cos( 2)= sin( ) (which is Lemma 9 fromthe Sine and Cosine chapter) so that we havecos( 2 )= cos( 2)= sin( )as we had the cofunction identity that we just examined, sin( ) = cos( 2 ),we can check that the first cofunction identity from the list above is true:sin( 2 )= cos( 2 [ 2 ])= cos( )276 ExercisesFor #1-12, use the chart on page 227 in the chapter Sine and Cosine andthat csc( ) =1sin( ), sec( ) =1cos( ), and cot( ) =cos( )sin( )to find the given ) csc( 6)2.) csc( 4)3.) csc( 3)4.) csc( 2)5.) sec(0)6.) sec( 6)7.) sec( 4)8.) sec( 3)9.) cot( 6)10.) cot( 4)11.) cot( 3)12.) cot( 2)Find the solutions of the following equations in one ) loge(x) = loge(12) loge(x+ 1)14.
4 (x 4)2= 3615.)e3x 2= 4277 Match the numbered piecewise defined functions with their lettered )f(x) ={csc(x)if 0< x < 2or 2< x < 2ifx= 217.)g(x) ={csc(x)if 0< x < 2or 2< x < 2ifx= 218.)h(x) ={cot(x)if 0< x < 2or 2< x < 2ifx= 219.)p(x) ={cot(x)if 0< x < 2or 2< x < 2ifx= 2A.)B.)C.)D.)278 IIIIIIII}}}}
