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1 Table f Cntents: Backgrund Infrmatin Pthagrean Therem Similar Triangles Angles Standard Psitin...3 Measurement f an Angle..3 Cnverting Measures.4 Angles and Arc Length..4 The Trignmetric Functins...5 Right Triangles and Acute Angles.5 General Angles...6 Cnventinal Ntatin 6 Cmmn Angles.7 Signs 7 The Unit Circle 8 Graphs f Trignmetric Functins 9 Inverse Trignmetric Functins... Trignmetric Identities.5 Cnsequences f Definitin...5 Fundamental Identities 5 Additin Identities..6 Subtractin Identities..6 Duble Angle Identities..7 Half Angle Identities..7 Prduct Identities 7 Trignmetric Prblems.7 The Law f Sines 9 The Law f Csines 9 Etra Prblems Answers t Etra Prblems.3 Page f 3

2 Trignmetr is the stud f hw the sides and angles f a triangle relate t each ther. Backgrund Infrmatin: ) The Pthagrean Therem: Let Δ ABC be a right triangle (See Figure ) B Let the length f AB c (hptenuse) Let the length f BC a (leg) c a Let the length f AC b (leg) C b Figure A Then a + b c. In ther wrds, the sum f the squares f the legs f a right triangle equals the square f the hptenuse. ) Similar Triangles Definitin: Tw triangles are similar if crrespnding angles are cngruent (f equal measure). Fr a triangle, it is enugh fr tw crrespnding angles t be cngruent, because that wuld frce the third set t be cngruent. It is als enugh fr the crrespnding sides t be prprtinal: a B b d E e Figure A c C D f F If Δ ABC is similar t Δ DEF, then A D, B E, C F and a d b e c f Frm this, we als btain that the rati f an tw sides f ne triangle is equal t the rati f the crrespnding sides f the similar triangle. a d Fr eample: b e Page f 3

3 Angles: ) Standard Psitin: Angle Terminal Side Figure 3 Verte Initial Side The Standard Psitin f an angle ccurs when the verte f the angle is placed at the rigin f a crdinate sstem and its initial side is n the psitive -ais. Angles in Standard Psitin Figure 4 Figure 5 - A psitive angle is btained b rtating the initial side cunterclckwise until it cincides with the terminal side (Figure 4). - A negative angle is btained b rtating the initial side clckwise until it cincides with the terminal side (Figure 5). ) Measurement f Angles: Angles can be measured in degrees r radians (abbreviated rad). A cmplete revlutin is 360 r rad. Thus: rad rad 57.3 r rad rad Page 3 f 3

4 3) Cnverting frm degrees t radians and frm radians t degrees Eample : a) Find the radian measure f 90 Slutin: 90( ) 90 rad 90 rad 80 7 b) Epress rad 6 in degrees 7 6 Slutin: ( ) rad rad The fllwing table gives sme cnversins between radians and degrees f sme cmmn angles: Degrees Radians Figure 6 shws sme angles in standard psitin: Figure ) Angles and Arc Length: Let O be a circle with radius r (Figure 7) Let be a central angle. r Let a be the length f the arc subtended b a O Figure 7 Page 4 f 3

5 The length f the arc is prprtinal t the measure f the angle and since the entire circle a has circumference r and central angle measure r a a a r, r, r Nte: These are nl valid when is measured in radians. Eample : a) If the radius f a circle is 7 cm, what angle is subtended b an arc f cm? a Slutin: 3 rad r 7 b) A circle has a radius f 4 cm. What is the length f the arc subtended b an angle f 90? Slutin: Step : Angle must be in radians 90 rad Step : a r 4 cm The Trignmetric Functins: ) Right Triangles and Acute Angles Let be an acute angle 0 < hptenuse The trignmetric functins are defined as ratis f ppsite lengths f sides f a right triangle adjacent Nte: An right triangle with angle is similar t an ther right triangle with angle. Since the ratis f sides in similar triangles are equal, then the trig functins will remain the same regardless f the lengths f the sides. sin cs pp hp adj hp csc sec hp pp hp adj A tl t remember SOHCAHTOA tan pp adj ct adj pp NOTE: These ratis nl appl if 0 <. Page 5 f 3

6 ) General Angles Let be an angle in standard psitin Let P (, ) be an pint n the terminal side f Let r be the distance frm the rigin t P (See Figure 9) P (, ) Then, sin r csc r r cs r sec r O Figure 9 tan ct Nte: If 0, then csc and ct are nt defined. If 0, then tan and sec are nt defined. These definitins are cnsistent with the previus definitin if is an acute angle. 3) Cnventinal Ntatin If is a number, then b cnventin f the angle whse radian measure is. where as sin( 5 ) S, sin sin (r an trig functin) means the sine When using ur calculatr t cmpute trignmetric functins, u need t make sure that ur calculatr is set t the crrect mde. If u are cmputing radian measures, u need t have ur calculatr in radians, and in degrees if u are cmputing degree measures. 4) Sme Cmmn Angles: Here is a table f cmmn angles and the trig functins cmputed at them. 0 sin 0 cs tan Und. 3 Page 6 f 3

7 5) Signs The signs f the trig functins fr angles in each quadrant can be remembered b the fllwing saing: All Students Take Calculus Quad I: All ratis are psitive S A Quad II: Sine and csecant are psitive Rest are negative Quad III: Tangent and ctangent are psitive Rest are negative T C Quad IV: Csine and secant are psitive Rest are negative Figure 0 3 Eample 3: Let P (, ) be a pint n the terminal side f the angle. Find the 4 eact trignmetric ratis fr. Slutin: Since P (, ) is a pint n the terminal side f the angle ( ) + () r - P sin csc cs sec tan ct Page 7 f 3

8 Nte: Yu culd have als frmed the right triangle with and made the apprpriate 4 sign changes fr the quadrant u were in. Eample 4: Let sin 5 3 where < <. Find the values f the ther trignmetric 6 functins. Slutin: 5 6 We knw that is in the 3 rd Quadrant. Thus, nl tangent and ctangent have psitive signs. Ever thing else will be negative. Fr nw we will just drp the signs t frm a right triangle with hp 6 and pp 5 since sine is -5/6 B the Pthagrean Therem, we knw that S, we get the fllwing ratis fr the ther 5 trig functins cs tan csc sec ct The Unit Circle: The unit circle is a circle centered at the rigin with radius, hence the name unit circle. It allws us t easil find the values f sine and csine, and thus the rest f the trignmetric functins quickl and easil. There is a picture f the unit circle n the fllwing page (Figure ). The -crdinates are the values f csine and the - crdinates are the values f sine. Page 8 f 3

9 Figure : The Unit Circle Graphs f the Trignmetric Functins: ) f ( ) sin Dmain f f: All real numbers -intercepts: Let 0, sin 0 n where n is an integer. -intercept: Let 0, sin 0 0 Figure : f ( ) sin Page 9 f 3

10 Range f f: sin The functin is dd (smmetric with respect t the rigin) The perid is ) f ( ) cs Dmain f f: All real numbers n -intercepts: Let 0, cs 0 where n is an dd integer -intercept: Let 0, cs 0 Figure 3: f ( ) cs Range f f: cs The functin is even (smmetric with respect t -ais) The perid is The graphs f the remaining trignmetric functins are as fllws. It is left t the reader t determine the characteristics f the graphs. (-scale is units) Figure 4: f ( ) tan Figure 5: f ( ) csc Figure 6: f ( ) sec Page 0 f 3

11 Figure 7: f ( ) ct Inverse Trignmetric Functins The trignmetric functins allw us t find the rati f sides f a triangle if we knw a given angle. What happens if we knw the rati and want t find the angle? Fr instance, we knw that sin and we want t find This is similar t when u want t knw the value f when 5. In that case, u use an inverse functin namel, the square rt functin which undes the peratin f the squaring functin. In the same wa we want t build an inverse trignmetric functin which undes the peratin f the trignmetric functin. Nw, in rder fr a functin t have an inverse functin, the riginal functin must pass the Hrizntal Line Test. This means that an hrizntal line drawn intersects the graph in at mst ne place. This is a prblem fr the trignmetric functins since the are peridic. In rder t get inverse functins, we must restrict ur dmains s that we are able t define a unique inverse. We must meet the fllwing criteria fr the trig functins t have an inverse functin:. Each value f the range is taken n nl nce s that we can pass the hrizntal line test.. The range f the functin with its restricted dmain is the same as the range f the riginal functin. 3. The dmain includes the mst cmmnl used numbers (r angles) 0 < < ) Inverse Sine The range f sine is, s this will becme the dmain f ur inverse sine functin. Nw we need t find ur restricted dmain. If we lk at the graph f sine, we see that the range is satisfied and we pass the hrizntal line test if we take nl the values. This will becme the range f ur inverse sine functin. We nw define ur inverse sine functin as fllws: If, then f ( ) sin arcsin if and nl if sin f ( ) and f ( ). Page f 3

12 Figure 8: f ( ) sin ) Inverse Csine If, then f ( ) cs arccs 0 f ( ). if and nl if cs f ( ) and Figure 9: f ( ) cs 3) Inverse Tangent If is an real number, then and f ( ). f ( ) tan arctan if and nl if tan f ( ) Figure 0: f ( ) tan Page f 3

13 4) Inverse Csecant If, then f ( ) csc arc csc if and nl if csc f ( ) and f, f ( ) 0 ( ) Figure : f ( ) csc 5) Inverse Secant If, then f ( ) sec arcsec if and nl if sec f ( ) and 0 f ( ), f ) ( Figure : f ( ) sec 6) Inverse Ctangent If is an real number, then f ( ) ct arcct if and nl if ct f ( ) and 0 < f ( ) < Figure 3: f ( ) ct Page 3 f 3

14 Nw, back t ur prblem, If sin, then what is? One wa t cme up with the slutin is t lk at the unit circle and find ut which pints have a -crdinate f. Yu will see that this ccurs when and 6 5 when. Since sine is peridic, we knw that we have infinite slutins t ur 6 prblem, namel + n and 5 + n. 6 6 Nw, sa instead that we wanted t find ( sin ). This will nl give us ne answer, but which ne will it be? Well, b ur restrictins, we need t find such that sin and. Lking at all f ur pssible slutins, nl ne wrks, ( ) sin 6 In general, there are infinitel man slutins t prblems where u knw the value f the trignmetric functin but nt the angle. Using inverse trignmetric functins gives us what is called the principal value f the relatin. The principal value f the relatin is the angle that is a slutin t the prblem and that has the smallest abslute value. If psitive and negative values bth satisf, then we use the psitive angle. Eample: Find tan Slutin: We want t find such that tan and. If we use the 5 unit circle, tan when cs sin, which ccurs when,. Onl ne f these 4 4 values is in ur interval, s tan. 4. Trignmetric Identities: ) Cnsequences f the Definitins ) csc ) sec 3) sin cs ct tan 4) sin tan 5) cs cs ct sin ) Sme fundamental identities 6) cs + sin Prf: Refer back t Figure 9. The distance frmula gives that + r + r. Thus, cs + sin + + r r r r r r Page 4 f 3

15 7) tan + sec Prf: Take sin + cs and divide b cs sin cs + tan + sec cs cs cs 8) + ct csc Prf: Take sin + cs and divide b sin 9a) sin( ) sin (sine is an dd functin) 9b) cs( ) cs (csine is an even functin) Prf: Yu can refer t figures and 3, r B definitin: (, ) sin r sin( ) sin r cs r (, -) cs( ) 0a) sin( + ) sin 0b) cs( + ) cs True because + This implies that sine and csine are r cs peridic 3) Additin Frmulas a) sin( + ) sin cs + sin cs b) cs( + ) cs cs sin sin Prf: Refer t Figure 5 B l( OB) l( AE) d l( BA) a l( DA) c l( OC) f l( DC) d l( BD) g l( OA) b l( BC) e l( CE) c O D F A C E OAD AEC OCB BDC are right angles m AFO 90 m DFC m DBC 90 m DFC 90 (90 ) Figure 5(Nt t Scale) Page 5 f 3

16 Frm Frm Frm Frm Δ AOB, we get that sin( + ) a and cs( + ) b Δ OCB, we get that sin e and cs f c Δ OCE, we get that sin c cs sin and f b + d cs d cs cs cs( + ) f d g Δ BDC, we get that sin d sin sin and cs g cs sin e e Putting everthing tgether, we get that d sin sin cs cs cs( + ) cs( + ) cs cs sin sin And Since a g + c, then sin( + ) sin cs + sin cs 4) Subtractin Frmulas a) sin( ) sin cs sin cs b) cs( ) cs cs + sin sin Prf: Replace b in Additin Frmulas. 5) Additin and Subtractin with Tangent tan + tan 3a) tan( + ) tan tan tan tan 3b) tan( ) + tan tan sin( + ) sin cs + cs sin Prf f 3a: Left-hand side is cs( + ) cs cs sin sin tan( + ). Divide the tp and bttm f the right-hand side b cs cs t get sin cs cs sin + cs cs cs cs tan + tan tan( + ) cs cs sin sin tan tan cs cs cs cs 6) Duble Angle Frmulas 4a) sin sin cs 4b) cs cs sin Prf f 4a) sin sin( + ) sin cs + sin cs sin cs Page 6 f 3

17 7) Alternate frms f Duble Angle Frmulas fr Csine 5a) cs cs 5b) cs sin Prf f 5a: Use sin + cs cs cs sin cs ( cs ) cs 8) Half Angle Frmulas cs 6a) cs + cs 6b) sin Prf f 6a: cs cs cs + cs cs + cs 9) Prduct Frmulas 7a) sin cs [ sin( + ) + sin( )] 7b) cs cs [ cs( + ) + cs( )] 7c) sin sin [ cs( ) cs( + )] Prf f 7a: sin( + ) + sin( ) sin cs + sin cs + sin cs sin cs sin cs sin cs [ ] [ ] [ ] Trignmetric Prblems Eample 5: Find all values f in the interval [ 0, ] such that cs cs. Slutin: We knw that, cs cs, s we get cs cs cs cs 0 Let u cs u u 0 (u + )( u ) 0 u cs u cs Page 7 f 3

18 Eample 6: Find the perid f the functin 7 sin(9t + ) Slutin: We knw that the perid f sine is. sin(bt ) is ging thrugh the radians b times faster, s the values f sin(bt ) repeat b times sner, r in b th the time. This gives us that the perid is. Fr ) b sin( bt + c, the c des nt change hw quickl we g thrugh the radians, it just affects where we start, s it has n impact n the perid (It prduces a hrizntal shift, s it just mves the entire graph right r left, but des nt change the shape, which wuld change the perid). Thus the perid f 7 sin(9t + ) is 9. Eample 7: The wheels f a biccle have radius 55 cm and are rtating at 60 rpm. Find the speed f the biccle in km/hur. Rund ur answer t the nearest tenth. Slutin: In ne minute, the angle changes b 60 0 radians In ne hur, the angle changes b radians Hence, the linear speed in cm/hr is S in km/hr we get. 4 km/hr 00,000 Eample 8: The angle f elevatin f tree 30 feet awa is degrees. Find the number f feet in the height f the tree. Rund t the nearest integer. Slutin: If we lk at this prblem as a right triangle, then we are lking fr the length f the side ppsite the degree angle. We knw that the adjacent side is 30 feet. Thus, tan( ) 30 tan( ) 49 feet tall. 30 Eample 9: The trignmetric functin is equal t which f the cs(7 ) + cs(7) fllwing? a) csc(7 )ct(7) b) sec(7 )ct(7) c) ct(4 )csc(7) d) sec(7 csc(7) Slutin: We will start b getting a cmmn denminatr [ + cs(7) ] [ cs(7) ] + cs(7) + cs(7) cs(7) cs(7) [ cs(7) ][ + cs(7) ] cs (7) sin (7) sin(7) sin(7) ct(7 )csc(7) a 5 Eample 0: If sin p, cs 3 4 p, sin q and cs sin p + q Slutin: sin( p + q) sin p csq + sin qcs p q, find ( ) Eample : What are the slutin(s) f sin 4sin + 3 0,? Slutin: Let u sin u 4u ( u 3)( u ) 0 u 3, sin 3 Nte: sin will never be 3 because the range f sine is sin in the interval ( ) Page 8 f 3

19 The Law f Sines In an triangle ABC with sides a, b, and c (as in Figure 6), a b c sin A sin B sinc Prf: B h Frm Δ ADB, sin A h csin A c h a Frm Δ CDB, sin C h asinc c h a Thus, csin A asinc a c A D b C sin A sinc Figure 6 Other ratis can be fund b cnstructing the perpendiculars frm ther vertices. The Law f Sines is used when we knw mre angles than sides. Eample : Slve Δ ABC if A 3, C 8.8, and a 4. 9 cm Slutin: We want t find the measure f angle B and the length f sides b and c. Step : We knw tw f the three angles. The sum f angles in a triangle is 80. S m B Step : We will use the Law f Sines t cmpute the sides. 4.9 b 4.9 c and sin(3 ) sin(66. ) sin(3 ) sin(8.8 ) With sme cmputatin, we find that b 74. cm and c 80. cm The Law f Csines In an triangle ABC with sides a, b, and c, a b + c bccs A b a + c accs B c a + b abcsc Prf: Let (, ) be the crdinates f A A sin B and cs B c c b c csin B and ccs B A ( ccs B, csin B) Als, C (a,0 D ) and l ( AC B ) b. a B the distance C Frmula, b ( ccs B a) + ( csin B) Figure 7 Page 9 f 3

20 B squaring bth sides f the equatin, we get b ( ccs B a) + ( csin B) c cs B accs B + a + c sin B a + c (cs B + sin B) accs B a + c accs B The ther equatins ma be btained b placing the ther vertices at the rigin. The Law f Csines is used when we knw mre sides than angles. Eample 3: A surver wishes t find the distance between tw inaccessible pints A and B n ppsite sides f a lake. While standing at pint C, she finds that l ( AC) 59 meters, l ( BC) 43 meters, and m ACB 5. Find the distance frm A t B. See Figure 8. Slutin: The Law f Csines can be used here since A we knw the length f tw sides and the measure f the included angle. B [ l ( AB) ] (59)(43) cs(5 ) l( AB) meters C Figure 8 Page 0 f 3

21 Etra Prblems. A circle has a radius f r cm. If 3 rad, and the arc a 33 cm, then what is the radius f the circle, r?. The angle f elevatin f a tree 9 feet awa is 7 degrees. Find the number f feet in the height f the tree. Give ur answer t the nearest integer. 3. In a right triangle, a 39 cm, b 36 cm and c 5 cm. Find ct. c a b 4. Using the same right triangle frm prblem 3, but nw with a 37 cm, b 35cm, and c cm, find sec. 5. The trignmetric functin sin(3 ) + sin(3) is equal t which f the fllwing: a) sec(3 )ct(3) b) ct(6 )csc(3) c) sec(3 )csc(3) d) sec( 3 ) tan(3) e) sec(3 ) tan(3) + sin(6) sin(6) 6. The trignmetric functin is equal t which f the sin(6) + sin(6) fllwing: a) sec(6 ) tan(6) b) 4ct( )csc(6) c) 4sec(6 )ct(6) d) sec( 6 ) tan(6) e) 4 tan(6 )sec(6) If sin p, cs p, sin q, and cs q. Find sin( q p). Epress ur answer as a cmmn fractin. 8. Tw straight railwas crss each ther at angle 33. Tw trains A and B leave the intersectin O at the same time. The velcit f train A is 9 mph and the velcit f train B is 57 mph. Find the distance between the tw trains after 5 hurs. Give ur answer t the nearest integer. The picture is nt drawn t scale. A O B Page f 3

22 5 9. If cs p and sin p < 0. Find ct p If cs p and sin p < 0. Find csc p 7. The perid f the functin 5sec(7t) is?. The perid f the functin 3 cs(t + 5) is? 3. The radius f the wheels f ur car measure 7 cm. Suppse that each wheel is rtating at 964 rpm. Find the speed f ur car in km/h. Rund ur answer t the nearest tenth. 4. The rectangle belw has a diagnal AC 3cm. The angle 7. Hw man square centimeters are in the area f the rectangle. Picture nt t scale. D C A B 5 5. If sin p, find sin( p) What are the slutins f sin + sin a 0 7. What are the slutins f sin b 9sinb + 4 0? a in the interval (,0)? 8. Find 3 sin. 9. Find sec (). 0. Erin wishes t measure the distance acrss Eagle River. She find that C 0, A 5 and b 345 feet. Find the required distance. Rund ur answer t the nearest ft. A b B a C Page f 3

23 Answers t Etra Prblems. cm. 67 feet 3. cm cm 5. e 6. e miles km/hr square centimeters a,, b + n, + n feet Page 3 f 3

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