MAT 1275: Introduction to Mathematical Analysis

Transcription

1 MAT 7: Intductin t Mathematical Analysis D A Rzenblyum Tignmetic Functins f Abitay Angles Unit Cicle In the pevius sectin we defined tig functins f acute angles: we cnstucted ight tiangle with given angle, and defined tig functins as atis f sides f this tiangle This appach cannt be used f angles that ae nt acute like btuse negative angles: thee ae n ight tiangles with such angles Nevetheless, it is pssible t define tig functins f abitay angles T d this we will use a special tl that allws t efmulate definitin f tig functin f acute angles in such a way that a new definitin can be used f abitay angles This tl is called the unit cicle in the system f cdinates This is just a cicle with the adius f and the cente at the igin: y x In this figue, we will daw angles in standad psitin It means that thei vetices ae in the igin, and the initial sides ges alng the psitive pat f x-axis Hee is an example f such angle in the st quadant (ie acute angle): y A x Angle is uniquely defined by the pint A n the cicle at which teminal side intesects the cicle We will call pint A cespnding t angle Let (a, b) be cdinates f the pint A (we als use the ntatin A(a, b)):

2 y A B a x b Ntice that A = (adius f the unit cicle) Then fm ight tiangle AB, we have sin AB b = = = b A, B a cs = = = A a We see that f acute angles, sine and csine ae cdinates f the cespnding pints n the unit cicle: sine is the secnd cdinate (y-cdinate), and csine is the fist cdinate (x-cdinate) We ve gt the efmulatin (ie a new definitin) f sine and csine f acute angles: they ae cdinates f pints n unit cicle We can use this efmulatin as a geneal definitin f abitay angles Definitin Let be an abitay angle in standad psitin, and A(a, b) be the cespnding pint n unit cicle Then, by definitin, sin = b, cs = a, tan = sin cs = b a Nte T memize which f the tig functins sine csine, is the fist cdinate and which is the secnd, yu may use the alphabetical de f the fist lettes in the wds sine and csine (c is befe s, s csine is the fist cdinate, and sine is the secnd) Othe thee tig functins can be defined as ecipcal t the basics: ct =, tan sec =, cs csc = sin Because sine and csine ae cdinates, tig functins may take bth psitive and negative values depending in what quadant the angle lies The fllwing figues shw the signs f basic tig functins Signs f sine Signs f csine Signs f tangent

3 3 Nte The fllwing phase may help t memize which f these functins is psitive in each quadant: All Students Take Calculus This phase hints that in the fist quadant all thee ae psitive, in the secnd nly sine, in the thid nly tangent, and in the futh nly csine Example Calculate basic tig functins f quadant angles f and 3, 9, 8, 7, Slutin ) F and 3 angles the cespnding pint n unit cicle has cdinates (, ) Theefe, sin = sin 3 =, cs = cs3 =, tan = tan3 = ) F 9 angle the cespnding pint has cdinates (, ) Theefe, sin sin 9 =, cs 9 = By definitin, tan = Because cs 9 =, cs tan 9 is undefined (we can nt devide by ze) 3) F 8 angle the cespnding pint has cdinates (, ) Theefe, sin 8 =, cs8 =, tan 8 = 4) F 7 angle the cespnding pint has cdinates (, ) Theefe, sin 7 =, cs 7 =, tan 7 is undefined We summaize the esults f example in the fllwing table Angle sin cs tan undefined undefined We see that maximal and minimal values f sine and csine f the quadant angles ae and espectively F all the angles sine and csine ae between and In geneal, f any angle, sin, cs Thee is n estictin f tangent Reductin Fmulas (The Head Rule) In example we calculated sine, csine and tangent f quadant angles, 9, 8, 7, and 3 Hee we discibe the way hw t simplify sine and csine f angles when we add ( subtact) quadant angles t angle In the wds we will simplify the fllwing expessins: ( ± ) ( ± ) ( ± ) ( ± ) sin 9, cs 8, sin 7, sin 3

4 4 Fmulas t simplify these expessins ae called eductin fmulas F example, it is sin 9 = cs cs 9 = sin (sine and csine nt difficult t get that ( ) and ( ) f cmpemental angles ae equal) Anthe example is cs( 8 ) = cs We can analize each f such expessins sepaately, and get a lt f eductin fmulas Instead, we suggest a simple ule t get such fmulas We call this ule the head ule Head ule wks like this We assume that angle is acute, and we need t answe tw questins t get the eductin fmula: ) Shuld we put minus sign n the ight side f the fmula? ) Shuld we change sine t csine and/ vice vesa? T answe the fist questin, detemine the quadant in which angle unde cnsideatin lies Based n the quadant, detemine the sign f tig functin (as discibed abve) T answe the secnd qustin, mve yu head alng the axis n which the quadtant angle lies In ding this yu autmatically get answe yes n Example Get eductin fmulas f sin( 9 ), cs ( 8 ), sin ( 7 ) Slutin sin 9 F ( ) : ) Angle 9 lies in nd quadant Hee sine is psitive, s minus sign is nt needed ) Mve yu head alng vetical axis (whee 9 angle is lcated) and yu get the sin 9 = cs answe yes, s change sine t csine Final answe: ( ) F cs( 8 ): ) Angle 8 lies in nd quadant Hee csine is negative, s minus sign is needed ) Mving yu head alng hizntal axis (whee 8 angle is lcated) yu get the cs 8 = cs answe n, s d nt change csine t sine Final answe: ( ) F sin( 7 ): ) Angle 7 lies in 4 th quadant Hee sine is negative, s minus sign is needed ) Mving yu head alng vetical axis (whee 7 angle is lcated) yu get the sin 7 = cs answe yes, s change sine t csine Final answe: ( ) Special cases f eductin fmulas (when quadant angle is ) ae Refeence Angle sin cs ( ) ( ) = sin (dd ppety f sine) = cs (even ppety f csine) This is a useful tl t educe calculatin f tig functins f abitay angles t acute angles Definitin Let be an abitay angle in standad psitin Angle is called the efeence angle t, if it satisfies thee cnditins:

5 ) Teminal side f cincides with the teminal side f ) Initial side f is hizntal (it cincides with eithe the psitive negative pats f the x-axis) 3) Angle is acute angle Let s see hw efeence angle lks like depending n the quadant in which iginal angle is lcated ) Angle is lcated in the fist quadant Then cincides with ) Angle is lcated in the secnd quadant Then = 8 : 8 3) Angle is lcated in the thid quadant Then = 8 : 8 4) Angle is lcated in the futh quadant Then = 3 : 3 Refeence angle is useful because up t sign, the values f tig functins f cinside with the value f the same tig functin f the efeence angle and is always acute angle Yu can check this using eductin fmulas descibed abve Main Ppety f Refeence Angle: The value f any tig functin f the efeence angle is equal t the abslute value f the same tig functin f the iginal angle Hence, t calculate the value f a tig functin, it is enugh t find the sign f the functin and calculate the value f tig functin f the efeence angle

6 Example Calculate cs Slutin Angle is lcated in the nd quadant, s cs < This is the case ) abve Refeence angle = 8 = We have cs = Theefe, cs = Example 3 Calculate sin Slutin Angle is lcated in the 3 d quadant, s sin < This is the case 3) abve Refeence angle = 8 = 4 We have cs 4 = Theefe, Example 4 Calculate tan 33 sin = Slutin Angle 33 is lcated in the 4 th quadant, s tan 33 < This is the case 4) abve Refeence angle = 3 33 = 3 3 We have tan 3 = Theefe, 3 3 tan 33 = 3 Example Find the values f the five tig functins, if cs = and tan > Slutin F efeence angle, cs = Let s daw a ight tiangle, using definitin f cs as ati f adjacent side t hyptenuse: By the Pythagean theem, vetical leg f this tiangle is = Fm hee, sin = and tan = Since cs < and tan >, angle lies in the 3 d quadant Theefe, sin = and tan = Othe thee tig functins ae: ct = = =, sec = =, csc = = = tan cs sin

7 7 It is pssible t define tig functin using a cicle with abitay adius (nt nly equal t ) Namely, sine, csine and tangent f any angle (in a standat psitin), which has pint A(a, b) n its teminal side ae: b a b a sin =, cs =, tan =, = a b Nte In the abve fmulas, adius is the distance fm pint A(a, b) t the igin Example Find the value f the six tig functins f the angle if pint (, 3) lies n the teminal side f angle, and is in standad psitin Slutin We have a =, b = 3 Using the abve fmulas, = a b = 3 ( ) = 3, sin = b = 3 3 = 3 3 3, cs = a = 3 = 3 3, tan = b a = 3 = 3 Othe thee tig functin ae 3 3 csc = =, sec = =, ct = = sin 3 cs tan 3