1999 by CRC Press LLC

Transcript

1 Plarikas A. D. Trignmetric and Hyperblic Fnctins The Handbk f Frmlas and Tables fr Signal Prcessing. Ed. Aleander D. Plarikas Bca Ratn: CRC Press LLC, by CRC Press LLC

2 43 Trignmetry and Hyperblic Trignmetry 43. Trignmetry Angle Relatins f the Fnctins Fndamental Identities 43. Hyperblic Trignmetry Hyperblic Fnctins 43. Trignmetry 43.. Angle Radian π π radians; radians; radian degrees 80 π Right Angle An angle f Trignmetric fnctins f an arbitary angle (see Figre 43.) sin y/ r csc r/ y cs / r sec r/ tan y / ct ctn / y esec sec cvers sin vers cs hav vers ia cis cs + isin e, in radians, i 999 by CRC Press LLC

3 y y r FIGURE Relatins f the Fnctins Relatins sin csc csc sin cs sec sec cs sin tan sin + cs ct cs cs ct + tan sec tan sin *sin ± cs + ct csc *tan ± sec *cs ± sin *ct ± csc * sec ± tan + sin cs( 90 ) sin( 80 ) *csc ± ct + cs sin( 90 ) cs( 80 ) tan ct( 90 ) tan( 80 ) ct tan( 90 ) ct( 80 ) csc ct ct * The sign in frnt f the radical depends n the qadrant in which falls Fndamental Identities Fndamental Identities Where a dble sign appears in the fllwing, the chice f sign depends pn the qadrant in which the angle terminates. 999 by CRC Press LLC

4 Reciprcal Relatins Prdct Relatins Qtient Relatins sin, cs, tan csc sec ct csc sin, sec, ct cs tan sin tan cs, cs ct sin tan sin sec, ct cs csc sec csc tan, csc sec ct tan ct sin sin, cs, tan sec csc cs sec csc tan, sec csc cs, ct ct sin Pythagrian Relatins sin + cs, + tan sec, + ct csc Angle-Sm and Angle-Difference Relatins sin( + β) sincs β+ cssinβ sin( β) sincsβ cssinβ cs( + β) cscsβ sinsinβ cs( β) cscsβ+ sinsinβ tan+ tanβ tan( + β) tantanβ tan tanβ tan( β) + tantanβ ctβct ct( + β) ctβ+ ct ctβct+ ct( β) ctβ ct 999 by CRC Press LLC

5 sin( + β)sin( β) sin sin β cs β cs cs( + β)cs( β cs sin β cs β sin Dble-Angle Relatins sin sincs Mltiple-Angle Relatins tan tan tan cs cs sin cs sin + tan tan ct tan, ct tan ct 3 sin3 3sin 4sin 3 cs3 4cs 3cs 3 sin 4 4sincs 8sin cs 4 cs4 8cs 8cs sin5 5sin 0sin + 6sin 5 3 cs5 6cs 0cs + 5cs 5 3 sin6 3cs sin 3cs sin+ 6cssin 6 4 cs6 3cs 48cs + 8cs sinn sin( n ) cs sin( n ) csn cs( n ) cs cs( n ) 3 3tan tan tan3 3tan 3 4tan 4tan tan 4 4 6tan + tan tan( n ) + tan tann tan( n ) tan Fnctin-Prdct Relatins sinsinβ cs( β) cs( + β) cscsβ cs( β) + cs( + β) 999 by CRC Press LLC

6 sincsβ sin( + β) + sin( β) cssinβ sin( + β) sin( β) Fnctin-Sm and Fnctin-Difference Relatins sin + sin sin β ( + β)cs ( β) sin sinβ cs ( + β)sin ( β) cs+ csβ cs ( + β)cs ( β) cs csβ sin ( + β)sin ( β) sin( + β) tan tan cs cs, tan tan sin( β) + β β β cscsβ sin( + β) sin( β ) ct+ ctβ, ct ctβ sinsinβ sinsinβ tan ( + β) sin+ sinβ sin sinβ tan ( β) sin+ sinβ tan ( + β). cs+ csβ sin+ sinβ ct ( β ) cs csβ sin sinβ tan ( β) cs+ csβ Half-Angle Relatins cs + cs sin ±, cs ± tan ± ct ± cs cs sin + cs sin + cs + cs + cs sin cs sin cs Pwer Relatins sin 3 ( cs ), sin ( 4 3 sin sin 3 ) 4 sin ( cs cs 4 ) 999 by CRC Press LLC

7 cs ( + cs ), cs ( cs+ cs ) cs 4 ( + cs + cs ) tan cs, ct + cs cs + cs Epnential Relatins (a in radians) e ia cs+ isin, i e e e + e sin a, csa i e tan a i e ia ia ia ia ia ia e + e ia ia e i e ia ia Relatins f Trignmetric Fnctins Fnctin sin cs tan ct sec csc sin sin ± cs tan ± + tan ± +ct ± sec a sec csc cs ± sin cs ± +tan ct ± + ct sec ± csc csc tan sin ± sin ± cs cs tan ct ± sec ± csc ct ± sin sin cs ± cs tan ct ± sec ± csc sec ± sin cs ± + tan ± + ct ct sec csc ± csc csc sin ± cs ± + tan tan ± + ct sec ± sec csc Nte: The chice f sign depends pn the qadrant in which the angle terminates Identities Invlving Principal Vales If Arcsin, then Arcsin + Arccs π/ Arctan + Arcct π/ sin, cs, tan csc, sec, ct 999 by CRC Press LLC

8 If Arccs, then sin, cs, tan csc, sec, ct If Arctan, then sin, cs + +, tan csc + sec +, ct Plane Triangle Frmlae In the fllwing, A, B, and C dente the angles f any plane triangle, a, b, c, the crrespnding ppsite sides, and s a+ b+ c ( ). Radis f inscribed circle: r ( s a)( s b)( s c) s Radis f circmscribed circle: a b c R sin Α sin B sinc Law f sines: a b c sin A sin B sin C Law f csines: a b c bc A A b + c + cs, cs a bc b c a ca B B c + a + cs, cs b ca c a b ab C C a + b + cs, cs c ab 999 by CRC Press LLC

9 Law f tangents: tan ( B C) b c, b+ c tan ( B+ C) tan ( C A) c a c+ a tan ( C+ A) tan ( A B) a b a+ b tan ( A+ B) Half-angle frmlae: tan A r r B s a, tan r C s b, tan s c sin A sin B sin C ( s b)( s c) ss ( a), cs A bc bc ( s c)( s a) ss ( b), cs B ca ca ( s a)( s b) ss ( c), cs C ab ab Area: K bcsin A casin B absinc K a B C sin sin b sincsin A c sin Asin B sin A sin B sinc abc K s( s a)( s b)( s c) rs 4R Mllweide s frmlae: sin ( B C) b c, a cs A sin ( C A) c a b cs B sin ( A B) a b c cs C 999 by CRC Press LLC

10 Newtn s frmlae: cs ( B C) b+ c, a sin A cs ( C A) c+ a b sin B cs ( A B) a+ b c sin C Sltin f Right Triangles a) Given acte angle A and ppsite leg a. B 90 A, b a/ tan A act A, c a/ sin A acsc A b) Given acte angle A and adjacent leg b. B 90 A, a b tan A, c b/ cs A bsec A c) Given acte angle A and hyptense c. B 90 A, a csin A, b c cs A d) Given legs a and b. c a + b, tan A a/ b, B 90 A e) Given hyptense c and leg a. b ( c+ a)( c a), sin A a/ c, B 90 A Sltin f Obliqe Triangles a) Given sides b and c and inclded angle A. Nnlgarithmic sltin a b + c bccs A, cs B ( c + a b )/ ca, Lgarithmic sltin cs C ( a + b c )/ ab ( B C) A, tan ( B C) b c + b c tan 90 ( B C + ). B ( B+ C) + ( B C), C ( B+ C) ( B C). 999 by CRC Press LLC

11 a ( bsin A)/ sin B, K bcsin A Check. A + B + C 80, r se Newtn s frmla r law f sines. b) Given angles B and C and inclded side a. A 80 ( B+ C), b ( asin B)/ sin A, c a C A K a sin B sin ( sin )/ sin, C sin A Check. a bcsc + ccsb, r se Newtn s frmla r law f tangents. c) Given sides a and c and ppsite angle A. Check. a bcsc + ccsb, r se Newtn s frmla r law f tangents. Nte. In this case there may be tw sltins, fr C may have tw vales: C < 90 and C 80 C > 90. If A + C > 80, se nly C. d) Given the three sides a,b,c. Nnlgarithmic sltin Lgarithmic sltin sin C ( csin A)/ a, B 80 - (A + C), b (asinb)/sina, K acsinb cs A ( b + c a )/ bc, cs B ( c + a b )/ ca, cs C ( a + b c )/ ab Check. A + B + C 80. s ( a+ b+ c), r 43. Hyperblic Trignmetry 43.. Hyperblic Fnctins Gemetrical Defintins (see Figre 43.) ( s a)( s b)( s c), s r r r tan A, tan B, tan C, s a s b s c K s( s a)( s b)( s c) Let O be the center, A the verte, and P any pint f the branch B AB f a rectanglar hyperbla. Set OM, MP y, OA a, and 999 by CRC Press LLC

12 FIGURE 43. area OPAP a. Then hyperblic sine f sinh y/a, hyperblic csine f csh /a Epnential Defintins hyperblic sine f sinh ( e e ) hyperblic csine f csh ( e + e ) hyperblic tangent f sinh e tanh csh e e + e csch h sinh csh tanh Fndamental Identities sinh( ) sinh, csc h( ) csch csh( ) csh, sec h( ) sech tanh( ) tanh, cth( ) cth csh sinh tanh + sech cth csch csch sech csch sech 999 by CRC Press LLC

13 sinh( + v) sinhcshv+ cshsinhv sinh( v) sinhcshv cshsinhv csh( + v) cshcshv+ sinhsinhv csh( v) cshcshv sinhsinhv tanh+ tanhv tanh( + v) + tanhtanhv tanh tanhv tanh( v) tanhtanhv sinh( + v)sinh( v) sinh sinh v csh csh v csh( + v)csh( v) sinh + csh v csh + sinh v sinhcshv sinh( + v) + sinh( v) cshsinhv sinh( + v) sinh( v) cshcshv csh( + v) + sinh( v) sinhsinhv csh( + v) csh( v) sinh+ sinhv sinh ( + v)csh ( v) sinh sinhv csh ( + v)sinh ( v) cshh + cshv csh ( + v)csh ( v) cshh cshv sinh ( + v)sinh ( v) tanh sinh tanh + tanh csh tanh tanh tanh tanh 999 by CRC Press LLC

14 + tanh sinh+ cs tanh ( ) sinh + v tanh+ tanhv cshcshv ( ) sinh v tanh tanhv cshcshv ( ) sinh + v cth+ cthv sinhsinhv ( ) sinh v cth cthv sinhsinhv sinh sihcsh csh csh + sinh csh + sinh tanh tanh + tanh 3 sinh3 3sinh+ 4sinh 3 csh3 4csh 3csh 3 3tanh+ tanh tanh3 + 3tanh sinh ± (csh ) csh (csh + ) csh sinh tan sinh csh Inverse Hyperblic Fnctins * sinh lg ( + + ) e csh lg e( ± ),. The pls sign is sed fr the principal vale. 999 by CRC Press LLC

15 tanh + lg,, < ± + csch lg e. The pls sign is sed if > 0, the mins sign if < 0. ± sech lg e, 0 <. The pls sign is sed fr the principal vales. lg + cth e, > * sinh, csh. etc., are smetimes replaced by arg sinh, arg csh, etc., i.e., sinh arg sinh Relatins with Circlar Fnctins sinhi isin, sinh isini cshi cs, csh csi tanhi i tan, tanh i tani sinh( + iv) sinhcsv+ icshsinv sinh( iv) sinhcsv icshsinv csh( + iv) cshcsv+ isinhsinv csh( iv) cshcsv isinhsinv tanh( + iv) sinh + i sin v csh+ csv sinh isinv tanh( iv) csh+ csv sinh isinv cth( + iv) csh csv sinh+ isinv cth( iv) csh csv sinh + i icsh, csh i isinh + π π sinh( + πi) sinh, csh( + πi) csh sinh( + πi) sinh, csh( + πi) csh e csh+ sinh, e csh sinh i e cs+ isin, i e cs isin 999 by CRC Press LLC

16 Special Vales f Hyperblic Fnctins π 3π 0 i πi i sinh 0 i 0 i csh 0 0 tanh 0 i 0 i csch i i 0 sech 0 cth by CRC Press LLC