adjacent side sec 5 hypotenuse Evaluate the six trigonometric functions of the angle.

Size: px
Start display at page:

Download "adjacent side sec 5 hypotenuse Evaluate the six trigonometric functions of the angle."

Transcription

1 A Trigonometric Fnctions (pp 8 ) Rtios of the sides of right tringle re sed to define the si trigonometric fnctions These trigonometric fnctions, in trn, re sed to help find nknown side lengths nd ngle mesres of tringles A Trigonometric Fnctions Voclr The si trigonometric fnctions nd their revitions re: sine, or sin cosine, or cos tngent, or tn cotngent, or cot secnt, or sec cosecnt, or csc Evlte Trigonometric Fnctions In right tringle with cte ngle, the si trigonometric fnctions re defined s follows: sin 5 opposite side djcent side opposite side } cos 5 } tn 5 } hpotense hpotense djcent side csc 5 } hpotense sec 5 hpotense djcent side } cot 5 } opposite side djcent side opposite side Evlte the si trigonometric fnctions of the ngle 7 hpotense 4 From the Pthgoren Theorem, the hpotense hs length Ï } Ï } sin 5 opp } hp 5 4 } 5 cos 5 dj } hp 5 7 } 5 tn 5 opp } dj 5 4 } 7 csc 5 hp } opp 5 5 } 4 sec 5 hp } dj 5 5 } 7 cot 5 dj } opp 5 7 } 4 Evlte the si trigonometric fnctions of the ngle Copright McDogl Littell, division of Hoghton Mifflin Compn 8 Benchmrk 6 Chpters nd 4

2 Solve Right Tringle B Solve ABC A nd B re complementr ngles, so B C c A csc 5 } sin sec 5 } cos cot 5 } tn tn 60 5 opp } dj sec 60 5 hp } dj Write trigonometric eqtion tn 60 5 } 4 sec 60 5 c } 4 Sstitte 5 4(tn 60 ) c 5 4(sec 60 ) 5 4 } cos 60 Solve for the vrile 5 4Ï } c 5 4 } } Solve ABC sing the digrm nd given mesrements Evlte trigonometric fnctions 4 B 5 45, c 5 Ï } 5 A 5 0, 5 6 B C c A A Trignometric Fnctions Drw Generl Angles Copright McDogl Littell, division of Hoghton Mifflin Compn Voclr The mesre of n ngle is positive if the rottion of the terminl side is conterclockwise If the rottion is clockwise, the ngle mesre is negtive Initil side The fied r of n ngle on coordinte plne Terminl side The r of n ngle tht is rotted ot the verte on coordinte plne Stndrd position The loction of n ngle whose verte is t the origin nd initil side lies on the positive -is 80 terminl side verte Drw n ngle with the given mesre in stndrd position c 5 Since 0 is 0 more thn 80, the terminl side is 0 conterclockwise pst the negtive -is initil side 0 60 Benchmrk 6 Chpters nd 4 9

3 Since 480 is 0 more thn 60, the terminl side mkes one whole revoltion conterclockwise pls 0 more A Trigonometric Fnctions Voclr To convert degrees to rdins, mltipl degrees rdins } 80 To convert rdins to degrees, mltipl rdins 80 } rdins c Since 5 is negtive, the terminl side is 5 clockwise from the positive -is Drw n ngle with the given mesre in stndrd position Convert Between Degrees nd Rdins 5 Rdin The mesre of n ngle in stndrd position whose terminl side intercepts n rc of length r Convert () 5 to rdins nd () to degrees rdins } 80 5 } 5 4 rdins, or simpl } 5 4 } 5 } 80 rdins } 5 0 rdins Convert the degree mesre to rdins or the rdin mesre to degrees } } 6 9 r r Copright McDogl Littell, division of Hoghton Mifflin Compn 0 Benchmrk 6 Chpters nd 4

4 For n ngle in stndrd position whose terminl side intersects circle with rdis r, the trigonometric fnctions re defined s follows: sin 5 } r cos 5 } r tn 5 }, 0 Voclr 5 Evlte Trigonometric Fnctions Given Point The point (, 4) is point on the terminl side of n ngle in stndrd position Evlte the si trigonometric fnctions of B the Pthgoren Theorem, r 5 Ï } 5 Ï } () 4 5 Ï } For 5, 5 4, nd r 5 5, (, 4) r sin 5 } r 5 4 } 5 cos 5 } r 5 } 5 tn 5 } 5 4 } csc 5 } sin 5 5 } 4 sec 5 } cos 5 5 } cot 5 } tn 5 } 4 Evlte the si trigonometric fnctions with the given point on the terminl side of ngle (6, 8) (8, 5) 4 (, 5) 6 Use Reference Angles Reference ngle The cte ngle 9 formed the terminl side of n ngle in the stndrd position nd the -is A Trignometric Fnctions Copright McDogl Littell, division of Hoghton Mifflin Compn Signs of trigonometric fnctions in ech qdrnt re s follows: Qdrnt I sin 5 cos 5 tn 5 Qdrnt II sin 5 cos 5 tn 5 Qdrnt III sin 5 cos 5 tn 5 Qdrnt IV sin 5 cos 5 tn Use reference ngles to evlte () tn (5 ) nd () cos 8 The ngle 5 is coterminl with 5 The reference ngle is The tngent fnction is negtive in Qdrnt II, so tn (5 ) 5tn Benchmrk 6 Chpters nd 4

5 The ngle 8 } is coterminl with } The reference ngle is } 5 } 95 The cosine fnction is negtive in Qdrnt II, so cos 8 } 5cos } = } 5 8 Evlte the trigonometric fnction withot sing clcltor A Trigonometric Fnctions For sin 5, the inverse sine is sin 5, For cos 5, the inverse cosine is cos 5, 0 80 For tn 5, the inverse tngent is tn 5, cos 00 6 csc 0 7 tn 7 Evlte Inverse Trigonometric Fnctions, Solve Trigonometric Eqtion Evlte () sin 05 nd () tn When 90 90, the ngle whose sine is 05 is: 5 sin When 90 90, the ngle whose tngent is Ï } is: 5 tn Ï } } 6 A 7-meter rmp hs horizontl length of 5 meters Wht is the ngle of the rmp? Drw tringle tht represents the rmp Write trigonometric eqtion tht involves the rmp s length nd horizontl length cos 5 } 5 7 Use clcltor to find the mesre of 5 5 cos } 7 < 8 Evlte the epression 7 m 5 m 8 tn 9 cos 05 0 sin Ï } A cle wire is ttched to the top of 0-foot pole 6 feet from the se of the pole Wht is the ngle the wire mkes with the grond? Copright McDogl Littell, division of Hoghton Mifflin Compn Benchmrk 6 Chpters nd 4

6 Qiz Evlte the si trigonometric fnctions of the ngle Solve ABC sing the digrm nd given mesrements A 5 45, 5 4 B 5 60, c 5 5 Convert the degree mesre to rdins or the rdin mesre to degrees } 7 50 Evlte the si trigonometric fnctions with the given point on the terminl side of ngle 8 (0, 5) 9 (4, 7) 0 (, ) B C c A A Trignometric Fnctions Copright McDogl Littell, division of Hoghton Mifflin Compn Evlte withot sing clcltor tn (0 ) cos 5 } cos Ï} } 4 sin Ï} } 5 An irplne egins its descent for lnding t n ltitde of 9,000 feet At this time, the irplne is 50 miles from the rnw At wht ngle does the irplne descend? Benchmrk 6 Chpters nd 4

7 B Lw of Sines nd Lw of Cosines (pp 4 6) The si trigonometric rtios cn e sed to solve right tringles When tringle contins no right ngles, formls relting to sine nd cosine cn e sed to solve the tringle B Lw of Sines nd Cosines Voclr For nabc with opposite sides,, nd c, the lw of sines is: sin A } 5 } sin B 5 } sin C c Use the Lw of Sines Lw of sines A method for solving tringle when two ngles nd side re known (AAS or ASA cses) or when the lengths of two sides nd n ngle opposite one of those sides re known (SSA cse) Solve ABC with A 5, C 5 7, nd c 5 7 First find the third ngle: B B the lw of sines, sin sin 4 sin 7 } 5 } 5 } 7 sin sin 7 } 5 } Write two eqtions with one vrile 7 sin 4 sin 7 } 5 } 7 7 sin 5 } sin 7 Solve for ech vrile 7 sin 4 5 } sin 7 < 4 Use clcltor < 0 Solve ABC A 5 45, B 5 6, c 5 0 C 5 05, B 5 0, 5 6 Voclr In the digrm t right, h 5 sin A Emine SSA Tringles SSA cse When the lengths of two sides nd n ngle opposite one of those sides re known, reslting in either no tringle, one tringle, or two different tringles A A A is otse # No tringle One tringle A A h h h h,, Two tringles A is cte A h h 5 One tringle h One tringle Determine the nmer of tringles tht cn e formed Solve the tringle if onl one tringle cn e formed A 5 4, 5, 5 8 A 5 0, 5 6, 5 0 Copright McDogl Littell, division of Hoghton Mifflin Compn c A 5 55, 5, Benchmrk 6 Chpters nd 4

8 Since A is otse nd the side opposite A is longer thn the given djcent side, onl one tringle cn e formed Use the lw of sines to solve the tringle A 4 8 B C Copright McDogl Littell, division of Hoghton Mifflin Compn Voclr For ABC with opposite sides,, nd c, thelw of cosines is: 5 c c cos A 5 c c cos B c 5 cos C sin 4 } 5 } sin B Lw of sines 8 sin B 5 B < 07 8 sin 4 } < 050 Mltipl ech side 8 Use inverse sine fnction C < sin 4 sin 5 } 5 } c Lw of sines c 5 sin 5 } sin 4 < 67 Cross mltipl Since A is otse nd the side opposite A is shorter thn the given djcent side, it is not possile to drw the indicted tringle No tringle eists with these given sides nd ngle c Since sin A 5 4 sin 55 < 5, nd 5,, 4 (h,, ), two tringles cn e formed Determine the nmer of tringles tht cn e formed 4 C A B A Tringle Tringle A 5 60, 5 6, A 5 8, 5 0, A 5 4, 5 8, 5 5 Find n Unknown Side with the Lw of Cosines Lw of cosines A method for solving tringle when two sides nd the inclded ngle re known (SAS cse) or when ll three sides re known (SSS cse) Find the nknown side in ABC when 5 6, c 5, nd A c c cos A Lw of cosines 5 6 (6)() cos 75 Sstitte for, c, nd A < 006 < Ï } 006 < 7 Simplif Tke positive sqre root C Benchmrk 6 Chpters nd 4 5 B B Lw of Sines nd Cosines

9 Find the nknown side in ABC 6 5 7, c 5 5, nd B , 5 6, nd C Find Unknown Angles with the Lw of Cosines Solve ABC with 5 9, 5, nd c 5 0 B Lw of Sines nd Cosines Find the ngle opposite the longest side sing the lw of cosines 5 c c cos B Lw of cosines (9)(0) cos B Sstitte 9 0 }} (9)(0) 5 cos B Solve for cos B 0056 < cos B Simplif B < 78 Use inverse cosine Now se the lw of sines sin A } 5 } sin B Lw of sines sin A sin 78 } 9 5 } Sstitte sin A 5 9 sin 78 } < 079 Mltipl ech side 9 nd simplif A < 47 Use inverse sine The third ngle of the tringle is C < Solve nabc with 5 6, 5 7, nd c 5 Qiz Determine the nmer of tringles tht cn e formed A 5 4, 5 6, 5 A 5 54, 5 4, 5 7 A 5 99, 5 8, 5 8 Solve ABC 4 B 5 50, A 5 8, A 5 7, C 5 97, , c 5 8, A , 5 5, C , 5, c , 5 4, c 5 Copright McDogl Littell, division of Hoghton Mifflin Compn 6 Benchmrk 6 Chpters nd 4

10 C Grph Trigonometric Fnctions (pp 7 9) Yo lerned how to se sine, cosine, nd tngent fnctions to solve right tringles Here o will lern how to grph these fnctions on the coordinte plne Copright McDogl Littell, division of Hoghton Mifflin Compn Voclr Given nonzero rel nmers nd in fnctions 5 sin nd 5 cos, the mplitde of ech is nd the period of ech is } Given nonzero rel nmers nd in the fnction 5 tn, the period is } The verticl smptotes re odd mltiples of } There re ( ) no mimm or minimm vles, so there is no mplitde Grph Sine nd Cosine Fnctions Amplitde Hlf the difference etween fnction s mimm M nd minimm m Periodic fnction A fnction with repeting pttern Ccle The shortest repeting portion of grph Period The horizontl length of ech ccle Grph () 5 sin nd () 5 cos The mplitde is 5 nd the period is } 5 } 5 Intercepts: (0, 0); }, 0 5 (, 0); (, 0) Mimm: } 4, 5 }, Minimm: } 4, 5 }, The mplitde is 5 nd the period is } 5 } Intercepts: } 4 }, 0 5 } 6, 0 ; } 4 }, 0 5 }, 0 Mimms: (0, ); }, Minimm: } }, 5 }, Grph the fnction cos 5 sin 5 sin 4 Grph Tngent Fnction Grph one period of the fnction 5 tn The period is } 5 } Intercept: (0, 0) Asmptotes: 5 } 5 } or } 4 ; 5 } 5 } or } 4 Hlfw points: } 8, nd } 8, Benchmrk 6 Chpters nd C Grph Trigonometrics

11 Voclr Grph one period of the fnction tn 5 5 tn 6 5 tn } Grph Trnsltions nd Reflections Trnsltion of trigonometric fnction A horizontl shift h nits, verticl shift k nits, or comintion of oth horizontl shift nd verticl shift in the grph of fnction Reflection of trigonometric fnction A flip cross horizontl line eqidistnt from the mimm nd minimm points on fnction s grph Midline The horizontl line fnction is reflected cross C Grph Trigonometrics The grphs of 5 sin ( h) k nd 5 cos ( h) k re shifted horizontll h nits nd verticll k nits from their prent fnctions Grph 5 sin } ( ) Step : Identif the mplitde, period, horizontl shift, nd verticl shift Amplitde: 5 5 ; Period: } } 5 } 5 4 Horizontl shift: h 5; Verticl shift: k 5 0 Step : Drw the midline of the grph Since k 5 0, the midline is the -is Step : Find five ke points of 5 sin } ( ) On the midline 5 k: (0, 0) 5 (, 0); (, 0) 5 (, 0); (4, 0) 5 (, 0) Mimm: (, ) 5 (0, ) Minimm: (, ) 5 (, ) Step 4: Reflect the grph Since 0, the grph is reflected in the midline 5 0 So, (0, ) ecomes (0, ) nd (, ) ecomes (, ) Step 5: Drw the grph throgh the ke points Grph the fnction 7 5cos } ( ) 8 5 sin } 4 4 Copright McDogl Littell, division of Hoghton Mifflin Compn 8 Benchmrk 6 Chpters nd 4

12 Qiz Grph one period of the fnction 5 6 sin 5 cos 5 tn } cos sin } 6 5 tn 7 5 sin } cos } 9 5tn } 4 ( ) 0 5 sin } ( ) C Grph Trigonometrics Copright McDogl Littell, division of Hoghton Mifflin Compn Benchmrk 6 Chpters nd 4 9

13 D Trigonometric Identities (pp 0 ) Fndmentl trigonometric identities tht cn e sed to simplif epressions, evlte fnctions, nd help solve eqtions Severl trigonometric identities re descried nd pplied elow D Trigonometric Identities Voclr Trigonometric cofnction identities inclde: sin } 5 cos cos } 5 sin sec } 5 csc csc } 5 sec tn } 5 cot cot } 5 tn Trigonometric Pthgoren identities inclde: sin cos 5 tn 5 sec cot 5 csc sin ( ) Þ sin sin sin ( ) Þ sin sin The sme is tre for the other trigonometric fnctions Use Fndmentl Trigonometric Identities Trigonometric identit An epression or eqtion involving trigonometric fnctions tht is tre for ll vles of the vrile cos Simplif the epression }} tn } cos } } tn 5 } sin tn Sstitte sin for cos } Simplif the epression 5 } sin Sstitte sin } cos 5 cos Simplif sin } cos for tn cot sec } csc csc sin csc cot (sec ) Use Sm nd Difference Formls Trigonometric fnctions relting to the sm nd difference of two ngles re s follows: sin ( ) 5 sin cos cos sin sin ( ) 5 sin cos cos sin cos ( ) 5 cos cos sin sin cos ( ) 5 cos cos sin sin tn ( ) 5 tn tn }} tn tn tn ( ) 5 tn tn }} tn tn Copright McDogl Littell, division of Hoghton Mifflin Compn 0 Benchmrk 6 Chpters nd 4

14 Find the ect vle of tn } tn } 5 tn } 4 } 6 Sstitte 4 6 for tn } 4 tn } 6 5 }} tn } 4 tn } Difference forml for tngent 6 Copright McDogl Littell, division of Hoghton Mifflin Compn The sine of n ngle in Qdrnt II is positive The cosine of n ngle in Qdrnt II is negtive 5 Ï} } } Evlte Ï} } 5 Ï } Simplif Find sin ( ) given tht cos 5 } 5 with } nd sin 5 8 } 7 with 0 } Using Pthgoren identit nd qdrnt signs gives sin 5 4 } 5 nd cos 5 5 } 7 sin ( ) 5 sin cos cos sin 5 4 } 5 5 } 7 } 5 8 } 7 Sstitte 5 84 } 85 Simplif Find the ect vle of the epression Difference forml for sine 4 cos 75 5 tn 05 6 sin 7 Find cos ( ) given tht sin 5 } with } nd sin 5 4 } 5 with } Use Dole-Angle nd Hlf-Angle Formls Dole-ngle formls: sin 5 sin cos Hlf-ngle formls: sin } cos } 56 Ï } cos 5 cos sin } cos cos } 56 Ï } 5 cos tn } 5 } cos sin } D Trigonometric Identities 5 sin 5 sin } cos tn 5 tn } tn Benchmrk 6 Chpters nd 4

15 Becse 5 is in Qdrnt I, the vle of the sine is positive Find the ect vle of sin 5 } sin 5 5 sin } } cos 0 (0 ) 5 Ï } 5 Ï Ï} } } 5 Ï} Ï } } D Trigonometric Identities Give sin 5 } 5 with }, find sin nd sin } Using Pthgoren identit nd qdrnt signs gives cos 5 4 } 5 sin 5 sin cos 5 } 5 4 } sin } } 5 } cos } } 5 Ï } 5 Ï 4 } 5 9}0 } 5 Ï 5 Ï} 0 } 0 Given cos 5 } 5 with p 8 cos 9 tn Qiz Simplif the epression cos tn cos cos } p, find ech vle } 0 cos } cos F cot } G sec Find the ect vle of the epression 4 sin 05 5 cos } 6 tn 55 7 Find cos ( ) given tht sin 5 4 } 5 with } nd cos 5 4 } 5 with } Simplif the epression 8 sin ( ) 9 tn ( ) 0 cos } Copright McDogl Littell, division of Hoghton Mifflin Compn Benchmrk 6 Chpters nd 4

16 E Solve Trigonometric Eqtions (pp 5) When trigonometric fnction is tre for ll vles of the vrile, the fnction is n identit When fnction is tre onl for some vles of the vrile, the fnction is n eqtion The methods for finding the tre vles of the vrile in trigonometric eqtion re presented elow Solve Trigonometric Eqtion Solve sin 0 First isolte sin on one side of the eqtion sin 5 0 Write originl eqtion sin 5 Add to ech side sin 5 } Divide ech side One soltion of sin 5 } over the intervl 0 is 5 sin } 5 } 6 The other soltion in this intervl is 5 } 6 5 } 5 6 Since 5 sin is periodic, there re infinitel mn soltions Using the two soltions ove, the generl soltion is written s follows: E Solving Trigonometrics 5 } 6 n or 5 5 } 6 n, for n integer n Copright McDogl Littell, division of Hoghton Mifflin Compn It is helpfl to memorize the trigonometric vles for specil ngles 0, 0, 45, 60, 90, nd 80 Solve 8 cos in the intervl 0 8 cos 5 Write originl eqtion 8 cos 5 Strct from ech side cos 5 } Divide ech side 8 4 cos 56 } Tke sqre roots of ech side One soltion of cos 56 } is 5 cos } 5 } Another soltion is 5 cos } 5 } Over the intervl 0, the soltions re: 5 } 5 } 5 } 5 } 5 } 4 5 } 5 } 5 Find the generl soltion of the eqtion tn sin 5 5 Solve the eqtion in the intervl 0 cos 5 4 tn 5 8 Benchmrk 6 Chpters nd 4

17 Solve n Eqtion with Etrneos Soltions Solve sin cos in the intervl 0 sin 5 cos Write originl eqtion ( sin ) 5 (cos ) Sqre oth sides sin sin 5 cos Mltipl sin sin 5 sin Pthgoren identit sin sin 5 0 Qdrtic form E Solving Trigonometrics It is good prctice to lws check soltions in trigonometric eqtions since it is possile for these eqtions to contin etrneos soltion Voclr In the sinsoid = sin ( h) + k, is the mplitde, is sed to find the period, h is the horizontl shift, nd k is the verticl shift sin (sin ) 5 0 Fctor ot sin sin 5 0 or sin 5 0 Zero prodct propert sin 5 0 or sin 5 Solve for sin Over the intervl 0, sin 5 0 hs two soltions: 5 0 or 5 Over the intervl 0, sin 5 hs one soltion: 5 } Therefore, sin 5 cos hs three possile soltions: 5 0, }, nd Check Sstitte the possile soltions into the originl eqtion nd simplif 5 0: sin 0 5 cos Soltion checks 5 } : sin } 5 cos } Soltion checks 5 : sin 5 cos 0 5 Soltion is etrneos The onl soltions to sin 5 cos in the intervl 0 re 5 0 nd 5 } Solve the eqtion in the intervl 0 5 cos 5 sin cos 6 tn sin tn 5 0 Write Sinsoid Sinsoid A grph of sine or cosine fnction Write fnction for the sinsoid Step : Find the mimm vle M nd the minimm vle m From the grph, M 5 nd m 5 Step : Identif the verticl shift, k This vle is the men of M nd m So, k Copright McDogl Littell, division of Hoghton Mifflin Compn 4 Benchmrk 6 Chpters nd 4

18 Step : Decide whether the grph models sine or cosine fnction Since the grph crosses the midline 5 on the -is, the grph is sine crve with no horizontl shift So, h 5 0 Step 4: Find the mplitde nd period The period } 5 } so 5 The mplitde 5 } M m 5 } () 5 The grph is not reflection, so 5 The fnction is 5 sin Write fnction for the sinsoid 7 Qiz 4 4 Find the generl soltion of the eqtion sin cos 5 7 tn 5 E Solving Trigonometrics Copright McDogl Littell, division of Hoghton Mifflin Compn Solve the eqtion in the intervl sin cos cot 5 sin 6 sin Ï } cos 5 Write fnction for the sinsoid Benchmrk 6 Chpters nd 4 5

9.5 Start Thinking. 9.5 Warm Up. 9.5 Cumulative Review Warm Up

9.5 Start Thinking. 9.5 Warm Up. 9.5 Cumulative Review Warm Up 9.5 Strt Thinking In Lesson 9.4, we discussed the tngent rtio which involves the two legs of right tringle. In this lesson, we will discuss the sine nd cosine rtios, which re trigonometric rtios for cute

More information

Physics I Math Assessment with Answers

Physics I Math Assessment with Answers Physics I Mth Assessment with Answers The prpose of the following 10 qestions is to ssess some mth skills tht yo will need in Physics I These qestions will help yo identify some mth res tht yo my wnt to

More information

Section 13.1 Right Triangles

Section 13.1 Right Triangles Section 13.1 Right Tringles Ojectives: 1. To find vlues of trigonometric functions for cute ngles. 2. To solve tringles involving right ngles. Review - - 1. SOH sin = Reciprocl csc = 2. H cos = Reciprocl

More information

Date Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( )

Date Lesson Text TOPIC Homework. Solving for Obtuse Angles QUIZ ( ) More Trig Word Problems QUIZ ( ) UNIT 5 TRIGONOMETRI RTIOS Dte Lesson Text TOPI Homework pr. 4 5.1 (48) Trigonometry Review WS 5.1 # 3 5, 9 11, (1, 13)doso pr. 6 5. (49) Relted ngles omplete lesson shell & WS 5. pr. 30 5.3 (50) 5.3 5.4

More information

KEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a

KEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the -is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider

More information

spring from 1 cm to 2 cm is given by

spring from 1 cm to 2 cm is given by Problem [8 pts] Tre or Flse. Give brief explntion or exmple to jstify yor nswer. ) [ pts] Given solid generted by revolving region bot the line x, if we re sing the shell method to compte its volme, then

More information

P 1 (x 1, y 1 ) is given by,.

P 1 (x 1, y 1 ) is given by,. MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce

More information

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions ) - TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the

More information

Precalculus Due Tuesday/Wednesday, Sept. 12/13th Mr. Zawolo with questions.

Precalculus Due Tuesday/Wednesday, Sept. 12/13th  Mr. Zawolo with questions. Preclculus Due Tuesd/Wednesd, Sept. /th Emil Mr. Zwolo (isc.zwolo@psv.us) with questions. 6 Sketch the grph of f : 7! nd its inverse function f (). FUNCTIONS (Chpter ) 6 7 Show tht f : 7! hs n inverse

More information

Trigonometric Functions

Trigonometric Functions Exercise. Degrees nd Rdins Chpter Trigonometric Functions EXERCISE. Degrees nd Rdins 4. Since 45 corresponds to rdin mesure of π/4 rd, we hve: 90 = 45 corresponds to π/4 or π/ rd. 5 = 7 45 corresponds

More information

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1

More information

Chapter 6 Techniques of Integration

Chapter 6 Techniques of Integration MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln

More information

Loudoun Valley High School Calculus Summertime Fun Packet

Loudoun Valley High School Calculus Summertime Fun Packet Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!

More information

Prerequisite Knowledge Required from O Level Add Math. d n a = c and b = d

Prerequisite Knowledge Required from O Level Add Math. d n a = c and b = d Prerequisite Knowledge Required from O Level Add Mth ) Surds, Indices & Logrithms Rules for Surds. b= b =. 3. 4. b = b = ( ) = = = 5. + b n = c+ d n = c nd b = d Cution: + +, - Rtionlising the Denomintor

More information

Review Exercises for Chapter 4

Review Exercises for Chapter 4 _R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite www.mthgrphs.com... In Eercises

More information

TO: Next Year s AP Calculus Students

TO: Next Year s AP Calculus Students TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC

More information

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2 SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the

More information

Polynomials and Division Theory

Polynomials and Division Theory Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

More information

Chapters Five Notes SN AA U1C5

Chapters Five Notes SN AA U1C5 Chpters Five Notes SN AA U1C5 Nme Period Section 5-: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles

More information

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of

More information

Lesson 8.1 Graphing Parametric Equations

Lesson 8.1 Graphing Parametric Equations Lesson 8.1 Grphing Prmetric Equtions 1. rete tle for ech pir of prmetric equtions with the given vlues of t.. x t 5. x t 3 c. x t 1 y t 1 y t 3 y t t t {, 1, 0, 1, } t {4,, 0,, 4} t {4, 0,, 4, 8}. Find

More information

y b y y sx 2 y 2 z CHANGE OF VARIABLES IN MULTIPLE INTEGRALS

y b y y sx 2 y 2 z CHANGE OF VARIABLES IN MULTIPLE INTEGRALS ECION.8 CHANGE OF VAIABLE IN MULIPLE INEGAL 73 CA tive -is psses throgh the point where the prime meridin (the meridin throgh Greenwich, Englnd) intersects the eqtor. hen the ltitde of P is nd the longitde

More information

8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1

8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1 8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.

More information

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81 FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first

More information

SECTION 9-4 Translation of Axes

SECTION 9-4 Translation of Axes 9-4 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.

More information

CET MATHEMATICS 2013

CET MATHEMATICS 2013 CET MATHEMATICS VERSION CODE: C. If sin is the cute ngle between the curves + nd + 8 t (, ), then () () () Ans: () Slope of first curve m ; slope of second curve m - therefore ngle is o A sin o (). The

More information

Mathematics. Area under Curve.

Mathematics. Area under Curve. Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

More information

at its center, then the measure of this angle in radians (abbreviated rad) is the length of the arc that subtends the angle.

at its center, then the measure of this angle in radians (abbreviated rad) is the length of the arc that subtends the angle. Notes 6 ngle Mesure Definition of Rdin If circle of rdius is drwn with the vertex of n ngle Mesure: t its center, then the mesure of this ngle in rdins (revited rd) is the length of the rc tht sutends

More information

Elementary Linear Algebra

Elementary Linear Algebra Elementry Liner Algebr Anton & Rorres, 9 th Edition Lectre Set Chpter : Vectors in -Spce nd -Spce Chpter Content Introdction to Vectors (Geometric Norm of Vector; Vector Arithmetic Dot Prodct; Projections

More information

ES.182A Topic 32 Notes Jeremy Orloff

ES.182A Topic 32 Notes Jeremy Orloff ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In

More information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

More information

Level I MAML Olympiad 2001 Page 1 of 6 (A) 90 (B) 92 (C) 94 (D) 96 (E) 98 (A) 48 (B) 54 (C) 60 (D) 66 (E) 72 (A) 9 (B) 13 (C) 17 (D) 25 (E) 38

Level I MAML Olympiad 2001 Page 1 of 6 (A) 90 (B) 92 (C) 94 (D) 96 (E) 98 (A) 48 (B) 54 (C) 60 (D) 66 (E) 72 (A) 9 (B) 13 (C) 17 (D) 25 (E) 38 Level I MAML Olympid 00 Pge of 6. Si students in smll clss took n em on the scheduled dte. The verge of their grdes ws 75. The seventh student in the clss ws ill tht dy nd took the em lte. When her score

More information

JEE Advnced Mths Assignment Onl One Correct Answer Tpe. The locus of the orthocenter of the tringle formed the lines (+P) P + P(+P) = 0, (+q) q+q(+q) = 0 nd = 0, where p q, is () hperol prol n ellipse

More information

MTH 4-16a Trigonometry

MTH 4-16a Trigonometry MTH 4-16 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on right-ngled tringle. Identify the opposite, djcent nd hypotenuse in the following right-ngled

More information

10.2 The Ellipse and the Hyperbola

10.2 The Ellipse and the Hyperbola CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point

More information

f ) AVERAGE RATE OF CHANGE p. 87 DEFINITION OF DERIVATIVE p. 99

f ) AVERAGE RATE OF CHANGE p. 87 DEFINITION OF DERIVATIVE p. 99 AVERAGE RATE OF CHANGE p. 87 The verge rte of chnge of fnction over n intervl is the mont of chnge ivie by the length of the intervl. DEFINITION OF DERIVATIVE p. 99 f ( h) f () f () lim h0 h Averge rte

More information

1 cos. cos cos cos cos MAT 126H Solutions Take-Home Exam 4. Problem 1

1 cos. cos cos cos cos MAT 126H Solutions Take-Home Exam 4. Problem 1 MAT 16H Solutions Tke-Home Exm 4 Problem 1 ) & b) Using the hlf-ngle formul for cosine, we get: 1 cos 1 4 4 cos cos 8 4 nd 1 8 cos cos 16 4 c) Using the hlf-ngle formul for tngent, we get: cot ( 3π 1 )

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

Objective: Use the Pythagorean Theorem and its converse to solve right triangle problems. CA Geometry Standard: 12, 14, 15

Objective: Use the Pythagorean Theorem and its converse to solve right triangle problems. CA Geometry Standard: 12, 14, 15 Geometry CP Lesson 8.2 Pythgoren Theorem nd its Converse Pge 1 of 2 Ojective: Use the Pythgoren Theorem nd its converse to solve right tringle prolems. CA Geometry Stndrd: 12, 14, 15 Historicl Bckground

More information

= = 6 radians 15. θ = s 30 feet (2π) = 13 radians. rad θ R, we have: 180 = 60

= = 6 radians 15. θ = s 30 feet (2π) = 13 radians. rad θ R, we have: 180 = 60 SECTION - 7 CHAPTER Section. A positive ngle is produced y counterclockwise rottion from the initil side to the terminl side, negtive ngle y clockwise rottion.. Answers will vry. 5. Answers will vry. 7.

More information

= = 6 radians 15. θ = s 30 feet (2π) = 13 radians. rad θ R, we have: 180 = 60

= = 6 radians 15. θ = s 30 feet (2π) = 13 radians. rad θ R, we have: 180 = 60 SECTION - 7 CHAPTER Section. A positive ngle is produced y counterclockwise rottion from the initil side to the terminl side, negtive ngle y clockwise rottion.. Answers will vry. 5. Answers will vry. 7.

More information

CONIC SECTIONS. Chapter 11

CONIC SECTIONS. Chapter 11 CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round

More information

Ch AP Problems

Ch AP Problems Ch. 7.-7. AP Prolems. Willy nd his friends decided to rce ech other one fternoon. Willy volunteered to rce first. His position is descried y the function f(t). Joe, his friend from school, rced ginst him,

More information

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

More information

Calculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION

Calculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION lculus Section I Prt LULTOR MY NOT US ON THIS PRT OF TH XMINTION In this test: Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers for which f () is rel numer..

More information

Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Partial List. January 27, 2017 Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

More information

, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF

, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs

More information

Algebra II Notes Unit Ten: Conic Sections

Algebra II Notes Unit Ten: Conic Sections Syllus Ojective: 10.1 The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting

More information

Advanced Algebra & Trigonometry Midterm Review Packet

Advanced Algebra & Trigonometry Midterm Review Packet Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.

More information

A toolbox. Objectives. Defining sine, cosine and tangent. 1.1 Circular functions

A toolbox. Objectives. Defining sine, cosine and tangent. 1.1 Circular functions C H P T E R 1 toolo Ojectives To revise the properties of sine, cosine nd tngent To revise methods for solving right-ngled tringles To revise the sine rule nd cosine rule To revise sic tringle, prllel

More information

Algebra & Functions (Maths ) opposite side

Algebra & Functions (Maths ) opposite side Instructor: Dr. R.A.G. Seel Trigonometr Algebr & Functions (Mths 0 0) 0th Prctice Assignment hpotenuse hpotenuse side opposite side sin = opposite hpotenuse tn = opposite. Find sin, cos nd tn in 9 sin

More information

Answers for Lesson 3-1, pp Exercises

Answers for Lesson 3-1, pp Exercises Answers for Lesson -, pp. Eercises * ) PQ * ) PS * ) PS * ) PS * ) SR * ) QR * ) QR * ) QR. nd with trnsversl ; lt. int. '. nd with trnsversl ; lt. int. '. nd with trnsversl ; sme-side int. '. nd with

More information

[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves

[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C.

More information

Log1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1?

Log1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1? 008 009 Log1 Contest Round Thet Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing

More information

CHAPTER 6 Introduction to Vectors

CHAPTER 6 Introduction to Vectors CHAPTER 6 Introduction to Vectors Review of Prerequisite Skills, p. 73 "3 ".. e. "3. "3 d. f.. Find BC using the Pthgoren theorem, AC AB BC. BC AC AB 6 64 BC 8 Net, use the rtio tn A opposite tn A BC djcent.

More information

Sect 10.2 Trigonometric Ratios

Sect 10.2 Trigonometric Ratios 86 Sect 0. Trigonometric Rtios Objective : Understnding djcent, Hypotenuse, nd Opposite sides of n cute ngle in right tringle. In right tringle, the otenuse is lwys the longest side; it is the side opposite

More information

SAMPLE. Vectors. e.g., length: 30 cm is the length of the page of a particular book time: 10 s is the time for one athlete to run 100 m

SAMPLE. Vectors. e.g., length: 30 cm is the length of the page of a particular book time: 10 s is the time for one athlete to run 100 m jectives H P T E R 5 Vectors To nderstnd the concept of vector To ppl sic opertions to vectors To nderstnd the zero vector To se the nit vectors i nd j to represent vectors in two dimensions To se the

More information

Introduction. Definition of Hyperbola

Introduction. Definition of Hyperbola Section 10.4 Hperbols 751 10.4 HYPERBOLAS Wht ou should lern Write equtions of hperbols in stndrd form. Find smptotes of nd grph hperbols. Use properties of hperbols to solve rel-life problems. Clssif

More information

Exploring parametric representation with the TI-84 Plus CE graphing calculator

Exploring parametric representation with the TI-84 Plus CE graphing calculator Exploring prmetric representtion with the TI-84 Plus CE grphing clcultor Richrd Prr Executive Director Rice University School Mthemtics Project rprr@rice.edu Alice Fisher Director of Director of Technology

More information

We are looking for ways to compute the integral of a function f(x), f(x)dx.

We are looking for ways to compute the integral of a function f(x), f(x)dx. INTEGRATION TECHNIQUES Introdction We re looking for wys to compte the integrl of fnction f(x), f(x)dx. To pt it simply, wht we need to do is find fnction F (x) sch tht F (x) = f(x). Then if the integrl

More information

Calculus AB. For a function f(x), the derivative would be f '(

Calculus AB. For a function f(x), the derivative would be f '( lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:

More information

AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions

AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic

More information

A LEVEL TOPIC REVIEW. factor and remainder theorems

A LEVEL TOPIC REVIEW. factor and remainder theorems A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division

More information

Mathematics Extension 1

Mathematics Extension 1 04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen

More information

MCR 3U Exam Review. 1. Determine which of the following equations represent functions. Explain. Include a graph. 2. y x

MCR 3U Exam Review. 1. Determine which of the following equations represent functions. Explain. Include a graph. 2. y x MCR U MCR U Em Review Introduction to Functions. Determine which of the following equtions represent functions. Eplin. Include grph. ) b) c) d) 0. Stte the domin nd rnge for ech reltion in question.. If

More information

MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations

MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll

More information

Introduction to Algebra - Part 2

Introduction to Algebra - Part 2 Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite

More information

Year 12 Trial Examination Mathematics Extension 1. Question One 12 marks (Start on a new page) Marks

Year 12 Trial Examination Mathematics Extension 1. Question One 12 marks (Start on a new page) Marks THGS Mthemtics etension Tril 00 Yer Tril Emintion Mthemtics Etension Question One mrks (Strt on new pge) Mrks ) If P is the point (-, 5) nd Q is the point (, -), find the co-ordintes of the point R which

More information

Solution Set 2. y z. + j. u + j

Solution Set 2. y z. + j. u + j Soltion Set 2. Review of Div, Grd nd Crl. Prove:. () ( A) =, where A is ny three dimensionl vector field. i j k ( Az A = y z = i A A y A z y A ) ( y A + j z z A ) ( z Ay + k A ) y ( A) = ( Az y A ) y +

More information

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O 1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

More information

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

More information

Lecture 7: 3.2 Norm, Dot Product, and Distance in R n

Lecture 7: 3.2 Norm, Dot Product, and Distance in R n Lectre 7: 3. Norm, Dot Prodct, nd Distnce in R n Wei-T Ch 010/10/08 Annoncement Office hors: Qiz Mondy, Tesdy, nd Fridy fternoon TA: R301B, 王星翰, 蔡雅如 Sec. 1.5~1.7 8:45.m., Oct. 13, 010 Definitions Let nd

More information

APPM 1360 Exam 2 Spring 2016

APPM 1360 Exam 2 Spring 2016 APPM 6 Em Spring 6. 8 pts, 7 pts ech For ech of the following prts, let f + nd g 4. For prts, b, nd c, set up, but do not evlute, the integrl needed to find the requested informtion. The volume of the

More information

1Preliminary topics FINAL PAGES. Chapter 1. Objectives

1Preliminary topics FINAL PAGES. Chapter 1. Objectives 1Preliminr topics jectives To revise the properties of sine, cosine nd tngent. To revise the sine rule nd the cosine rule. To revise geometr in the plne, including prllel lines, tringles nd circles. To

More information

Year 12 Mathematics Extension 2 HSC Trial Examination 2014

Year 12 Mathematics Extension 2 HSC Trial Examination 2014 Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bord-pproved clcultors my be used A tble of

More information

Chapter 1: Fundamentals

Chapter 1: Fundamentals Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,

More information

4.6 Numerical Integration

4.6 Numerical Integration .6 Numericl Integrtion 5.6 Numericl Integrtion Approimte definite integrl using the Trpezoidl Rule. Approimte definite integrl using Simpson s Rule. Anlze the pproimte errors in the Trpezoidl Rule nd Simpson

More information

Lesson-5 ELLIPSE 2 1 = 0

Lesson-5 ELLIPSE 2 1 = 0 Lesson-5 ELLIPSE. An ellipse is the locus of point which moves in plne such tht its distnce from fied point (known s the focus) is e (< ), times its distnce from fied stright line (known s the directri).

More information

50. Use symmetry to evaluate xx D is the region bounded by the square with vertices 5, Prove Property 11. y y CAS

50. Use symmetry to evaluate xx D is the region bounded by the square with vertices 5, Prove Property 11. y y CAS 68 CHAPTE MULTIPLE INTEGALS 46. e da, 49. Evlute tn 3 4 da, where,. [Hint: Eploit the fct tht is the disk with center the origin nd rdius is smmetric with respect to both es.] 5. Use smmetr to evlute 3

More information

LINEAR ALGEBRA APPLIED

LINEAR ALGEBRA APPLIED 5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order

More information

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =. Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos( - 1 2 ) = rcsin( 1 2 ) = rcsin( - 1 2 ) = Cn you do similr problems? Review of Bsic Concepts

More information

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) = Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos( - 1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin( - 1 ) = -π 2 6 2 6 Cn you do similr problems?

More information

Identify graphs of linear inequalities on a number line.

Identify graphs of linear inequalities on a number line. COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line. - When grphing first-degree eqution, solve for the vrible. The grph of this solution will be single point

More information

Bridging the gap: GCSE AS Level

Bridging the gap: GCSE AS Level Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions

More information

CHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx

CHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twice-differentile function of x, then t

More information

Lesson 1: Quadratic Equations

Lesson 1: Quadratic Equations Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring

More information

Before we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!

Before we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!! Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble

More information

Alg 3 Ch 7.2, 8 1. C 2) If A = 30, and C = 45, a = 1 find b and c A

Alg 3 Ch 7.2, 8 1. C 2) If A = 30, and C = 45, a = 1 find b and c A lg 3 h 7.2, 8 1 7.2 Right Tringle Trig ) Use of clcultor sin 10 = sin x =.4741 c ) rete right tringles π 1) If = nd = 25, find 6 c 2) If = 30, nd = 45, = 1 find nd c 3) If in right, with right ngle t,

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

More information

PROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by

PROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the

More information

PHYS 1114, Lecture 1, January 18 Contents:

PHYS 1114, Lecture 1, January 18 Contents: PHYS 1114, Lecture 1, Jnury 18 Contents: 1 Discussed Syllus (four pges). The syllus is the most importnt document. You should purchse the ExpertTA Access Code nd the L Mnul soon! 2 Reviewed Alger nd Strted

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

3.1 Review of Sine, Cosine and Tangent for Right Angles

3.1 Review of Sine, Cosine and Tangent for Right Angles Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,

More information

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time) HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time llowed Two hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions

More information

Individual Events I3 a 10 I4. d 90 angle 57 d Group Events. d 220 Probability

Individual Events I3 a 10 I4. d 90 angle 57 d Group Events. d 220 Probability Answers: (98-8 HKMO Finl Events) Creted by: Mr. Frncis Hung Lst updted: 8 Jnury 08 I 800 I Individul Events I 0 I4 no. of routes 6 I5 + + b b 0 b b c *8 missing c 0 c c See the remrk 600 d d 90 ngle 57

More information

Calculus AB Bible. (2nd most important book in the world) (Written and compiled by Doug Graham)

Calculus AB Bible. (2nd most important book in the world) (Written and compiled by Doug Graham) PG. Clculus AB Bile (nd most importnt ook in the world) (Written nd compiled y Doug Grhm) Topic Limits Continuity 6 Derivtive y Definition 7 8 Derivtive Formuls Relted Rtes Properties of Derivtives Applictions

More information

S56 (5.3) Vectors.notebook January 29, 2016

S56 (5.3) Vectors.notebook January 29, 2016 Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution

More information

Pre-Calculus TMTA Test 2018

Pre-Calculus TMTA Test 2018 . For the function f ( x) ( x )( x )( x 4) find the verge rte of chnge from x to x. ) 70 4 8.4 8.4 4 7 logb 8. If logb.07, logb 4.96, nd logb.60, then ).08..867.9.48. For, ) sec (sin ) is equivlent to

More information

( β ) touches the x-axis if = 1

( β ) touches the x-axis if = 1 Generl Certificte of Eduction (dv. Level) Emintion, ugust Comined Mthemtics I - Prt B Model nswers. () Let f k k, where k is rel constnt. i. Epress f in the form( ) Find the turning point of f without

More information

5.2 Volumes: Disks and Washers

5.2 Volumes: Disks and Washers 4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of cross-section or slice. In this section, we restrict

More information