3.1 Review of Sine, Cosine and Tangent for Right Angles

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "3.1 Review of Sine, Cosine and Tangent for Right Angles"

Transcription

1 Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon, which mens tringle, nd metri, which mens mesurement. Trigonometry hs been used for 2500 yers. Its first uses were in surveying, nvigtion nd stronomy. Tody, it is used extensively in business, engineering, surveying, nvigtion, stronomy, nd physicl nd socil sciences. The trigonometry covered in this chpter is bsed on tringles other thn right tringles, but first we must review right tringles nd the use of clcultor to find length nd ngles of right tringles. Nming the Sides of Right Tringle right tringle hs one right ngle (90 ). The side opposite the right ngle is the hypotenuse. This is the longest side. Hypotenuse Opposite djcent If one of the cute ngles is θ, you must be ble to identify the side opposite θ, nd the side djcent to θ. Hypotenuse djcent Opposite Rtios in Right Tringles onsider the rtio opposite hypotenuse for the four similr tringles shown below. The vlues for this rtio re 1 2, 2 4, 3 6, nd 4 8. Since ll the rtios reduce to 1, the rtios re ll the sme. If we chose different ngle, the rtio vlue 2 would be different (not 1 ), but it would be the sme for ny size of right tringle tht hs the sme ngle The three fundmentl rtios of trigonometry re: sine of θ (sin θ ), cosine of θ (cos θ ), nd tngent of θ (tn θ ). Ech of these is rtio of two sides of right tringle. opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

2 126 hpter 3 Non-Right ngle Tringles Foundtions of Mth 11 Trigonometric Rtios for Right Tringles If θ is n cute ngle in right tringle, then sinθ = opposite hypotenuse,cosθ = djcent opposite,tnθ = hypotenuse djcent Where opposite is the length of the side opposite θ, djcent is the length of the side djcent to θ nd hypotenuse is the length of the hypotenuse of the right tringle. word for remembering these three rtios is SOH/H/TO. Sin = O H,os= H,Tn= O For this chpter, mke sure your clcultor is in degree mode (DEG). Exmple 1 Use clcultor to write the following rtios in deciml form. ) sin 65 b) cos 65 c) tn 65 Use the sin, cos nd tn keys. Mke sure the clcultor is in degree mode (DEG). ) sin 65 = b) cos 65 = c) tn 65 = Exmple 2 Given the deciml vlue for ech trigonometric rtio, solve for θ to one deciml plce. ) sinθ = b) cosθ = c) tnθ = Use the sin 1, cos 1 nd tn 1 keys. Mke sure the clcultor is in degree mode (DEG). ) θ = sin 1 (0.6358) = 39.5 b) θ = cos 1 (0.6358) = 50.5 c) θ = tn 1 (0.6358) = 32.4 opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

3 Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 127 Exmple 3 Solve the right tringle. (This mens clculte ll missing ngles nd sides. ) 12 y 35 sinθ = cosθ = x opposite hypotenuse sin 35 = y 12 y = 12sin35 =6.88 djcent hypotenuse cos35 = x 12 x = 12cos35 =9.83 α = =55 Exmple 4 Solve the right tringle x sin β = opposite hypotenuse = 9 25 β = 9 sin 1 25 = 21.1 Therefore, α = =68.9 x cn be found using Pythgors theorem: x = 25 2 x 2 = x = x = 23.3 Exmple 5 Solve the right tringle. z 8 35 sinθ = x opposite hypotenuse sin 35 =8 z z = 8 sin 35 = tnθ = opposite djcent α = =55 tn 35 = 8 x x = 8 tn 35 = opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

4 128 hpter 3 Non-Right ngle Tringles Foundtions of Mth Exercise Set 1. Find ech rtio to four deciml plces using clcultor. ) sin 35 b) cos 42 c) tn 81 d) sin 69 e) cos 77 f) tn 9 g) sin 0 h) cos 0 i) tn 0 j) sin 90 k) cos 90 l) tn 90 m) tn n) sin 45 o) cos 45 p) tn 45 q) sin 60 r) cos 30 opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

5 Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles Find the mesure of ngle θ to one deciml plce. ) sinθ = b) cosθ = c) tnθ = d) sinθ = e) cosθ = f) tnθ = g) sinθ = h) cosθ = i) tnθ = 1 j) sinθ = 0 k) cosθ = 0 l) tnθ = 0 m) sinθ = 1 n) cosθ = 1 opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

6 130 hpter 3 Non-Right ngle Tringles Foundtions of Mth Solve the following tringles to one deciml plce. ) b) 41 z x α = x = z = z 9 5 α = β = z = c) d) x y ρ = 11 α = x = y = 4 x θ = x = e) y z 66 5 β = y = z = f) 7 α = β = 5 x x = g) α = h) α = x θ = 6 x β = 57 8 x = 62 y 42 y = opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

7 Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles Find the length, to the nerest tenth. ) b) 36 6 mm 28 D m D 30 m E c) E d) D cm 65 6 ft 46 e) f) 22 yd 24 cm 18 cm yd 70 cm g) 4 cm 7 cm h) 1 9 m opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

8 132 hpter 3 Non-Right ngle Tringles Foundtions of Mth Lw of Sines n oblique tringle is tringle tht does not contin right ngle. To solve n oblique tringle, three pieces of informtion must be given. This informtion cn be ctegorized in four wys. t lest one of the pieces of informtion must be side of the tringle. 1. Two ngles nd one side of tringle. S (ngle side ngle) S (ngle ngle side) Note: If you know two ngles of tringle, you cn solve for the third ngle becuse the sum of the interior ngles of tringle equl Two sides nd their included ngle. SS (side ngle side) 3. Three sides. SSS (side side side) 4. Two sides nd n ngle opposite one of the sides. SS (ngle side side) Note: SS is referred to s the Donkey Theorem becuse the tringle is not lwys unique. opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

9 Foundtions of Mth 11 Section 3.2 Lw of Sines 133 Lw of Sines The Lw of Sines llows you to solve oblique tringles of the type S, S nd SS. The Lw of Sines If Δ is tringle with sides, b nd c, then: sin = sin b = sin c The Lw of Sines is used to find the missing side of n S or S tringle, or the missing ngle of n SS tringle. Δ cn be n cute or obtuse oblique tringle. b h h b cute tringle: ll ngles less thn 90 obtuse tringle: one ngle greter thn 90 c c Let h be the ltitude of either tringle Then sin = h b h = bsin sin = h h = sin Equting the two vlues of h gives bsin = sin or sin = sin b. In similr mnner, by constructing n ltitude from vertex to side, it cn be shown sin = sin c. Exmple 1 Solve Δ, given = 29, = 5 nd b = =180 = 46 sin sin = sin5 c = sin 46 c = 30sin5 = 59.8 sin 29 30sin 46 = = 44.5 sin c 29 opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

10 134 hpter 3 Non-Right ngle Tringles Foundtions of Mth 11 The mbiguous se (SS) Given, b nd in Δ efore demonstrting n exmple of n SS tringle problem, the mbiguous cse of SS must be estblished. se 1 < h with h = b sin (No Tringle Solution) b h Possible tringles: 0 Exmple 2 Given Δ, with = 30, = 4, b =, find. sin 30 4 = sin sin = sin30 4 = 1.25 Since sin cnnot be greter thn 1, there is no such ngle. Therefore, no tringle cn be mde with the given condition. se 2 = h (Right Tringle Solution) b = h Possible tringles: 1 Exmple 3 Given Δ, with = 30, = 5, b =, find. sin 30 5 = sin sin = sin30 5 = 1 = 90 se 3 b (Isosceles or Oblique Tringle Solution) b h Possible tringles: 1 Exmple 4 Given Δ, with = 30, = 12, b =, find. sin = sin sin = sin30 12 = 5 12 = 24.6 opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

11 Foundtions of Mth 11 Section 3.2 Lw of Sines 135 se 4 h < < b (The mbiguous se) b h Possible tringles: 2 ' Note: Δ nd Δ' re two different tringles with the sme SS informtion. Exmple 5 Given Δ, with = 30, = 7, b =, find. sin 30 7 = sin sin = sin30 7 = 5 7 = 45.6 However, since sine is positive in qudrnts I nd II, nother nswer is ' = = se 5 b (No Tringle Solution) b Possible tringles: 0 Exmple 6 Given Δ, with = 120, = 8, b =, find. sin120 8 = sin sin = sin120 8 = 1.08 Since sin cnnot be greter thn 1, there is no such ngle. Therefore, no tringle cn be mde with the given condition. se 6 Obtuse nd > b (Obtuse Tringle Solution) b Possible tringles: 1 Exmple 7 Given Δ, with = 120, = 12, b =, find. sin = sin sin = sin = 0.72 = 46.2 opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

12 136 hpter 3 Non-Right ngle Tringles Foundtions of Mth 11 Exmple 8 The distnce from the Sun (S) to Erth (E) nd to Venus (V) ws km nd km respectively when VES mesured 28. Find the possible distnces from Erth to Venus S h E V' V sin 28 = h h = sin 28 = km Since h < e < v < < 1.5 8, there re two solutions. sin = sinv sinv = sin V = 39.8 or140.2 ESV = =112.2 ESV' = =11.8 sin112.2 EV or sin11.8 EV' = = sin 28 EV = sin112.2 = km sin 28 sin 28 EV' = sin11.8 = km sin 28 Therefore, the two possible distnces from Erth to Venus re km nd km. Note: For the ske of ccurcy, do not round off during intermedite steps. Wit until the finl clcultion before rounding to the desired mount. opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

13 Foundtions of Mth 11 Section 3.2 Lw of Sines Exercise Set 1. Explin why no tringle is possible with the given informtion in Δ. ) = 38 = 12 = 69 b = 14 = 73 c = 13 b) = 42 = 7 = 65 b = 11 = 70 c = 12 c) = 39 = 46 = 95 d) = 120 = 20 = 40 = 5 b = 6 c = 12 = 12 b = 6 c = Find the sine ngle equivlent to the following, 0 θ 180. ) sin b) sin 30 c) sin 42 d) sin 71 e) sin 121 f) sin 137 opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

14 138 hpter 3 Non-Right ngle Tringles Foundtions of Mth Determine if the set of dt leds to 0, 1 or 2 tringles. drwing my be helpful. ) = 60, = 6 3, b = 12 b) = 60, = 11, b = 12 c) = 60, =, b = 12 d) = 60, = 12, b = 12 e) = 1, = 16, b = 12 f) = 1, = 12, b = 12 opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

15 Foundtions of Mth 11 Section 3.2 Lw of Sines Given nd side b, determine the lengths for side tht llow 0, 1 or 2 tringles to be formed. ) = 30, b = 12 b) = 45, b = 4 2 c) = 60, b = 6 3 d) = 120, b = 12 opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

16 140 hpter 3 Non-Right ngle Tringles Foundtions of Mth Solve for the unknown ngle, if possible, then determine if second ngle, 0 <θ < 180, exists tht will stisfy the proportion. If this second ngle is not solution for the tringle in the proportion, write no. ) sin = sin b) sin 200 = sin c) sin = sin 30 5 d) sin 40 = sin57 53 e) sin 3 = sin125 5 f) sin 7.3 = sin opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

17 Foundtions of Mth 11 Section 3.2 Lw of Sines Solve ech tringle using the Lw of Sines. If two tringles exist, solve both completely. drwing is very helpful. ) = 140, = 25, = 20 b) = 38, b = 8, = 6 c) = 27, = 46, = 120 d) = 1, = 24, b = 25 e) = 60, b = 4 3, = 8 f) = 41, c = 9, = 9 g) = 74, = 7, b = 8.1 h) = 58, = 48, b = 30.5 opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

18 142 hpter 3 Non-Right ngle Tringles Foundtions of Mth i) = 43, = 38, c = 17.2 j) = 33, = 27.2, b = 12.4 k) = 30, = 8, b = l) = 58, = 9, b = m) =, = 60, = 4.5 n) =, = 135, c = 60 o) = 52, c = 8.5, b = 12.4 p) = 40, b = 55, c = 80 opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

19 Foundtions of Mth 11 Section 3.3 Lw of osines Lw of osines There re four cses in which it is possible to solve generl tringle. The Lw of Sines is used for two of the cses (S or S), the Lw of osines is used for the remining cses (SS nd SSS). The Lw of osines For ny tringle with corresponding sides, b nd c: orollry 2 = b 2 + c 2 2bccos os = b2 + c 2 2 2bc b 2 = 2 + c 2 2ccos os = 2 + c 2 b 2 2c c 2 = 2 + b 2 2bcos os = 2 + b 2 c 2 2b Note: If = 0, os = 0 2 = b 2 + c 2, which is Pythgoren Theorem for right tringle. Derivtion of the Lw of osines onsider the oblique tringle. b h Length c is divided into two prts, x nd c x. cos = x b x = bcos 50 x y the Pythgoren theorem: b 2 = h 2 + x 2 h 2 = b 2 x 2 2 = h 2 + (c x) 2 h 2 = 2 (c x) 2 Equting these vlues for h 2 : 2 (c x) 2 = b 2 x 2 2 = b 2 x 2 + (c x) 2 2 = b 2 x 2 + c 2 2cx + x 2 2 = b 2 + c 2 2cx 2 = b 2 + c 2 2bccos bcos is substituted for x Just s esily n ltitude cn be drwn from ngle to side b, nd from ngle to side, to show the remining two cses of the Lw of osines. opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

20 144 hpter 3 Non-Right ngle Tringles Foundtions of Mth 11 Using the Lw of osines for SSS lwys find the lrgest ngle first in n SSS problem. This will gurntee the other two ngles re cute. The Lw of osines never hs the mbiguous cse since unique ngle is lwys obtined between 0 nd 180. Exmple 1 Solve Δ, given = 5, b = 7ndc =. 5 7 c 2 = 2 + b 2 2bcos 2 = cos 0 = cos 70cos = cos = 26 cos = = cos = The Lw of Sines or the Lw of osines cn be used to find or. The Lw of Sines is esier to use. sin = sin c sin 5 = sin111.8 sin = 5sin111.8 = 27.7 Therefore, =180 = 40.5 Note: If ws solved for first, it would led to wrong ngle. Using the Lw of osines, would still be 27.7 but: sin c = sin sin = sin sin = sin = 68.2 The correct result for is = To void this problem, lwys find the lrgest ngle first in n SSS solution. opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

21 Foundtions of Mth 11 Section 3.3 Lw of osines 145 Using the Lw of osines for SS Exmple 2 Solve Δ, given = 50, b = 12 nd c = = b 2 + c 2 2bccos 2 = cos50 2 = = The Lw of Sines my be used to find nother ngle of the tringle. To void obtining two solutions for the ngle, it is best to find the ngle opposite the shortest side, since tht ngle is lwys cute. sin = sin c sin = sin 5 sin = 5sin Therefore, =180 = 6.4 = 23.6 Note: If ws solved for first, it would led to wrong ngle. sin = sin b sin = sin 12 sin = 12sin = 73.6 The correct result for is = 6.4. To void this problem, lwys find the smllest ngle first in n SS problem. Summry of Lw of Sines nd Lw of osines Given S or S SS SS SSS Method of Solving 1. Find the remining ngle using + + = Find the remining sides using the Lw of Sines. e wre of the mbiguous cse. There my be two solutions. 1. Find n ngle using the Lw of Sines. 2. Find the remining ngle using + + = Find the remining side using the Lw of Sines. 1. Find the remining side using the Lw of osines. 2. Find the smller of the two remining ngles using the Lw of Sines. 3. Find the remining ngle using + + = Find the lrgest ngle using the Lw of osines. 2. Find one remining ngle by using the Lw of Sines. 3. Find the remining ngle using + + = 180. opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

22 146 hpter 3 Non-Right ngle Tringles Foundtions of Mth Exercise Set 1. Fill in the blnk. ) Use the Lw of osines when the informtion given for the tringle is or. b) If the Lw of osines 2 = b 2 + c 2 2bccos is pplied to right tringle, the result is theorem, since cos90 =. c) Write version of the Lw of osines tht is needed to solve ΔXYZ, with YXZ = 23, z = 12 nd y = Determine whether the Lw of Sines or the Lw of osines would be used to begin the solution process for ech tringle. ) b) c) d) e) f) opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

23 Foundtions of Mth 11 Section 3.3 Lw of osines Solve ech Lw of osines for the unknown prt. nswer to 2 deciml plces. ) 2 = cos 43 b) b 2 = cos115 c) c 2 = cos90 d) 7 2 = cos e) = (2.7)(4.6)cos f) = (6.2)(4.5)cos 4. In Δ, if = 48, b = 12 nd c = 6, which of the two ngles or cn be sid for certin is cute, nd why? 5. In Δ, if = 95, b = 5ndc = 9, which of the two ngles or cn be sid for certin is cute, nd why? opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

24 148 hpter 3 Non-Right ngle Tringles Foundtions of Mth Given the indicted prts of Δ, wht ngle or side should be found first, nd which formul should be used to find it? ) b) c b c) c d) b b e) f) g) h) b opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

25 Foundtions of Mth 11 Section 3.3 Lw of osines Solve Δ. Round nswers to one deciml plce. ) = 50, b =, c = 15 b) = 36, = 4, c = c) = 60, b = 4, = 8 d) = 2, b = 3, c = 4 e) = 7, b = 24, c = 25 f) = 9, b = 14, c = 11 g) b = 4, c = 1, = 120 h) = 6, b = 7, c = 12 opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

26 150 hpter 3 Non-Right ngle Tringles Foundtions of Mth Solve Δ, using either the Lw of Sines or the Lw of osines to begin the solution. ) = 126, b = 9, c = 12.2 b) = 28, = 42, c = 18.2 c) = 63, b = 8, c = d) = 41, = 11, c = 6 e) = 12.3, b = 9.6, c = 8.9 f) = 38, b = 9, c = 7 g) = 0, =, c = h) = 60, = 2 3, c = 4 opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

27 162 hpter 3 Non-Right ngle Tringles Foundtions of Mth Determine if the following leds to 0, 1 or 2 tringles. ) Δ, = 19, = 25, = 30 b) Δ, = 28, = 50, b = 20 c) ΔXYZ, X = 58, x = 9.3, z = 6.8 d) ΔXYZ, X = 1, x = 90, z = 0 8. Solve Δ using the Lw of Sines. ) = 65, = 93, c = b) = 54, b = 9, c = opyright by rescent ech Publishing ll rights reserved. ncopy hs ruled tht this book is not covered by

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED

MATH SPEAK - TO BE UNDERSTOOD AND MEMORIZED FOM 11 T20 RIGHT TRINGLE TRIGONOMETRY 1 MTH SPEK - TO E UNDERSTOOD ND MEMIZED 1) TRINGLE = 2-dimentionl she hving 3 sides nd 3 ngles. HRTERISTI OF TRINGLES I) Every tringle is n enclosed she tht hs these

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

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

Unit 6 Solving Oblique Triangles - Classwork

Unit 6 Solving Oblique Triangles - Classwork Unit 6 Solving Oblique Tringles - Clsswork A. The Lw of Sines ASA nd AAS In geometry, we lerned to prove congruence of tringles tht is when two tringles re exctly the sme. We used severl rules to prove

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

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

THE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE NO CALCULATORS 90 MINUTES

THE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE NO CALCULATORS 90 MINUTES THE 08 09 KENNESW STTE UNIVERSITY HIGH SHOOL MTHEMTIS OMPETITION PRT I MULTIPLE HOIE For ech of the following questions, crefully blcken the pproprite box on the nswer sheet with # pencil. o not fold,

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

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

Trigonometry. VCEcoverage. Area of study. Units 3 & 4 Geometry and trigonometry

Trigonometry. VCEcoverage. Area of study. Units 3 & 4 Geometry and trigonometry Trigonometry 9 VEcoverge re of study Units & Geometry nd trigonometry In this ch chpter 9 Pythgors theorem 9 Pythgoren trids 9 Three-dimensionl Pythgors theorem 9D Trigonometric rtios 9E The sine rule

More information

An Introduction to Trigonometry

An Introduction to Trigonometry n Introduction to Trigonoetry First of ll, let s check out the right ngled tringle below. The LETTERS, B & C indicte the ngles nd the letters, b & c indicte the sides. c b It is iportnt to note tht side

More information

Topics Covered: Pythagoras Theorem Definition of sin, cos and tan Solving right-angle triangles Sine and cosine rule

Topics Covered: Pythagoras Theorem Definition of sin, cos and tan Solving right-angle triangles Sine and cosine rule Trigonometry Topis overed: Pythgors Theorem Definition of sin, os nd tn Solving right-ngle tringles Sine nd osine rule Lelling right-ngle tringle Opposite (Side opposite the ngle θ) Hypotenuse (Side opposite

More information

6.2 The Pythagorean Theorems

6.2 The Pythagorean Theorems PythgorenTheorems20052006.nb 1 6.2 The Pythgoren Theorems One of the best known theorems in geometry (nd ll of mthemtics for tht mtter) is the Pythgoren Theorem. You hve probbly lredy worked with this

More information

Consolidation Worksheet

Consolidation Worksheet Cmbridge Essentils Mthemtics Core 8 NConsolidtion Worksheet N Consolidtion Worksheet Work these out. 8 b 7 + 0 c 6 + 7 5 Use the number line to help. 2 Remember + 2 2 +2 2 2 + 2 Adding negtive number is

More information

Trigonometry and Constructive Geometry

Trigonometry and Constructive Geometry Trigonometry nd Construtive Geometry Trining prolems for M2 2018 term 1 Ted Szylowie tedszy@gmil.om 1 Leling geometril figures 1. Prtie writing Greek letters. αβγδɛθλµπψ 2. Lel the sides, ngles nd verties

More information

I1.1 Pythagoras' Theorem. I1.2 Further Work With Pythagoras' Theorem. I1.3 Sine, Cosine and Tangent. I1.4 Finding Lengths in Right Angled Triangles

I1.1 Pythagoras' Theorem. I1.2 Further Work With Pythagoras' Theorem. I1.3 Sine, Cosine and Tangent. I1.4 Finding Lengths in Right Angled Triangles UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet STRAND I: Geometry nd Trigonometry I1 Pythgors' Theorem nd Trigonometric Rtios Tet Contents Section I1.1 Pythgors' Theorem I1. Further Work With Pythgors'

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

More information

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.) MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

More information

2) Three noncollinear points in Plane M. [A] A, D, E [B] A, B, E [C] A, B, D [D] A, E, H [E] A, H, M [F] H, A, B

2) Three noncollinear points in Plane M. [A] A, D, E [B] A, B, E [C] A, B, D [D] A, E, H [E] A, H, M [F] H, A, B Review Use the points nd lines in the digrm to identify the following. 1) Three colliner points in Plne M. [],, H [],, [],, [],, [],, M [] H,, M 2) Three noncolliner points in Plne M. [],, [],, [],, [],,

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

ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 1 - Thurs 28th Sept 17 Review of trigonometry and basic calculus

ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 1 - Thurs 28th Sept 17 Review of trigonometry and basic calculus ES 111 Mthemticl Methods in the Erth Sciences Lecture Outline 1 - Thurs 28th Sept 17 Review of trigonometry nd bsic clculus Trigonometry When is it useful? Everywhere! Anything involving coordinte systems

More information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60. Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

More information

USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year

USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year 1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.

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

Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions

Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,

More information

Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivatives/Indefinite Integrals of Basic Functions Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second

More information

Section 1.3 Triangles

Section 1.3 Triangles Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior

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

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

10. AREAS BETWEEN CURVES

10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

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

Trigonometry Revision Sheet Q5 of Paper 2

Trigonometry Revision Sheet Q5 of Paper 2 Trigonometry Revision Sheet Q of Pper The Bsis - The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.

More information

Similar Right Triangles

Similar Right Triangles Geometry V1.noteook Ferury 09, 2012 Similr Right Tringles Cn I identify similr tringles in right tringle with the ltitude? Cn I identify the proportions in right tringles? Cn I use the geometri mens theorems

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

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

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

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

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

More information

Math 113 Exam 2 Practice

Math 113 Exam 2 Practice Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

More information

Math Lesson 4-5 The Law of Cosines

Math Lesson 4-5 The Law of Cosines Mth-1060 Lesson 4-5 The Lw of osines Solve using Lw of Sines. 1 17 11 5 15 13 SS SSS Every pir of loops will hve unknowns. Every pir of loops will hve unknowns. We need nother eqution. h Drop nd ltitude

More information

Lesson 1.6 Exercises, pages 68 73

Lesson 1.6 Exercises, pages 68 73 Lesson.6 Exercises, pges 68 7 A. Determine whether ech infinite geometric series hs finite sum. How do you know? ) + +.5 + 6.75 +... r is:.5, so the sum is not finite. b) 0.5 0.05 0.005 0.0005... r is:

More information

Duality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.

Duality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below. Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we

More information

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40 Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

More information

AP Calculus Multiple Choice: BC Edition Solutions

AP Calculus Multiple Choice: BC Edition Solutions AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this

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

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph. nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $

More information

Non Right Angled Triangles

Non Right Angled Triangles Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit

More information

8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.

8Similarity UNCORRECTED PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8. 8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8. re nd volume scle fctors 8. Review U N O R R E TE D P G E PR O O FS 8.1 Kick off with S Plese refer to the Resources t in the Prelims

More information

青藜苑教育 The digrm shows the position of ferry siling between Folkestone nd lis. The ferry is t X. X 4km The pos

青藜苑教育 The digrm shows the position of ferry siling between Folkestone nd lis. The ferry is t X. X 4km The pos 青藜苑教育 www.thetopedu.com 010-6895997 1301951457 Revision Topic 9: Pythgors Theorem Pythgors Theorem Pythgors Theorem llows you to work out the length of sides in right-ngled tringle. c The side opposite

More information

A P P E N D I X POWERS OF TEN AND SCIENTIFIC NOTATION A P P E N D I X SIGNIFICANT FIGURES

A P P E N D I X POWERS OF TEN AND SCIENTIFIC NOTATION A P P E N D I X SIGNIFICANT FIGURES A POWERS OF TEN AND SCIENTIFIC NOTATION In science, very lrge nd very smll deciml numbers re conveniently expressed in terms of powers of ten, some of wic re listed below: 0 3 0 0 0 000 0 3 0 0 0 0.00

More information

Is there an easy way to find examples of such triples? Why yes! Just look at an ordinary multiplication table to find them!

Is there an easy way to find examples of such triples? Why yes! Just look at an ordinary multiplication table to find them! PUSHING PYTHAGORAS 009 Jmes Tnton A triple of integers ( bc,, ) is clled Pythgoren triple if exmple, some clssic triples re ( 3,4,5 ), ( 5,1,13 ), ( ) fond of ( 0,1,9 ) nd ( 119,10,169 ). + b = c. For

More information

A sequence is a list of numbers in a specific order. A series is a sum of the terms of a sequence.

A sequence is a list of numbers in a specific order. A series is a sum of the terms of a sequence. Core Module Revision Sheet The C exm is hour 30 minutes long nd is in two sections. Section A (36 mrks) 8 0 short questions worth no more thn 5 mrks ech. Section B (36 mrks) 3 questions worth mrks ech.

More information

8Similarity ONLINE PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8.

8Similarity ONLINE PAGE PROOFS. 8.1 Kick off with CAS 8.2 Similar objects 8.3 Linear scale factors. 8.4 Area and volume scale factors 8. 8.1 Kick off with S 8. Similr ojects 8. Liner scle fctors 8Similrity 8.4 re nd volume scle fctors 8. Review Plese refer to the Resources t in the Prelims section of your eookplus for comprehensive step-y-step

More information

PYTHAGORAS THEOREM,TRIGONOMETRY,BEARINGS AND THREE DIMENSIONAL PROBLEMS

PYTHAGORAS THEOREM,TRIGONOMETRY,BEARINGS AND THREE DIMENSIONAL PROBLEMS PYTHGORS THEOREM,TRIGONOMETRY,ERINGS ND THREE DIMENSIONL PROLEMS 1.1 PYTHGORS THEOREM: 1. The Pythgors Theorem sttes tht the squre of the hypotenuse is equl to the sum of the squres of the other two sides

More information

2008 Mathematical Methods (CAS) GA 3: Examination 2

2008 Mathematical Methods (CAS) GA 3: Examination 2 Mthemticl Methods (CAS) GA : Exmintion GENERAL COMMENTS There were 406 students who st the Mthemticl Methods (CAS) exmintion in. Mrks rnged from to 79 out of possible score of 80. Student responses showed

More information

Right Triangle Trigonometry

Right Triangle Trigonometry 44 CHPTER In Exercises 9 to 96, find the re, to the nerest squre unit, of the sector of circle with the given rdius nd centrl ngle. 9. r = inches, u = p rdins nerest 0 miles, the given city is from the

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +

More information

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS 33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

More information

Math Sequences and Series RETest Worksheet. Short Answer

Math Sequences and Series RETest Worksheet. Short Answer Mth 0- Nme: Sequences nd Series RETest Worksheet Short Answer Use n infinite geometric series to express 353 s frction [ mrk, ll steps must be shown] The popultion of community ws 3 000 t the beginning

More information

Pythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides

Pythagoras Theorem. The area of the square on the hypotenuse is equal to the sum of the squares on the other two sides Pythgors theorem nd trigonometry Pythgors Theorem The hypotenuse of right-ngled tringle is the longest side The hypotenuse is lwys opposite the right-ngle 2 = 2 + 2 or 2 = 2-2 or 2 = 2-2 The re of the

More information

Pythagoras Theorem. Pythagoras

Pythagoras Theorem. Pythagoras 11 Pythgors Theorem Pythgors Theorem. onverse of Pythgors theorem. Pythgoren triplets Proof of Pythgors theorem nd its converse Problems nd riders bsed on Pythgors theorem. This unit fcilittes you in,

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

6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS

6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS 6. CONCEPTS FOR ADVANCED MATHEMATICS, C (475) AS Objectives To introduce students to number of topics which re fundmentl to the dvnced study of mthemtics. Assessment Emintion (7 mrks) 1 hour 30 minutes.

More information

Markscheme May 2016 Mathematics Standard level Paper 1

Markscheme May 2016 Mathematics Standard level Paper 1 M6/5/MATME/SP/ENG/TZ/XX/M Mrkscheme My 06 Mthemtics Stndrd level Pper 7 pges M6/5/MATME/SP/ENG/TZ/XX/M This mrkscheme is the property of the Interntionl Bcclurete nd must not be reproduced or distributed

More information

Trigonometric Functions

Trigonometric Functions Trget Publictions Pvt. Ltd. Chpter 0: Trigonometric Functions 0 Trigonometric Functions. ( ) cos cos cos cos (cos + cos ) Given, cos cos + 0 cos (cos + cos ) + ( ) 0 cos cos cos + 0 + cos + (cos cos +

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

A study of Pythagoras Theorem

A study of Pythagoras Theorem CHAPTER 19 A study of Pythgors Theorem Reson is immortl, ll else mortl. Pythgors, Diogenes Lertius (Lives of Eminent Philosophers) Pythgors Theorem is proly the est-known mthemticl theorem. Even most nonmthemticins

More information

Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...

Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx... Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting

More information

List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.

List all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1. Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show

More information

Math 113 Exam 1-Review

Math 113 Exam 1-Review Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between

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

7.2 The Definite Integral

7.2 The Definite Integral 7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where

More information

Main topics for the First Midterm

Main topics for the First Midterm Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the

More information

Shape and measurement

Shape and measurement C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do

More information

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student) A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision

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

Comparing the Pre-image and Image of a Dilation

Comparing the Pre-image and Image of a Dilation hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity

More information

03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t

03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t A-PDF Wtermrk DEMO: Purchse from www.a-pdf.com to remove the wtermrk Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Functions Asolute Vlue Function Inverse Function If f ( x ), if f ( x ) 0 f ( x) y f

More information

03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t

03 Qudrtic Functions Completing the squre: Generl Form f ( x) x + x + c f ( x) ( x + p) + q where,, nd c re constnts nd 0. (i) (ii) (iii) (iv) *Note t A-PDF Wtermrk DEMO: Purchse from www.a-pdf.com to remove the wtermrk Add Mths Formule List: Form 4 (Updte 8/9/08) 0 Functions Asolute Vlue Function Inverse Function If f ( x ), if f ( x ) 0 f ( x) y f

More information

(e) if x = y + z and a divides any two of the integers x, y, or z, then a divides the remaining integer

(e) if x = y + z and a divides any two of the integers x, y, or z, then a divides the remaining integer Divisibility In this note we introduce the notion of divisibility for two integers nd b then we discuss the division lgorithm. First we give forml definition nd note some properties of the division opertion.

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

The Law of Cosines. Pace: 2.9 ft. Stride: 5.78 ft u Pace angle. Pace: 3.0 ft

The Law of Cosines. Pace: 2.9 ft. Stride: 5.78 ft u Pace angle. Pace: 3.0 ft 694 hpter 6 dditionl Topics in Trigonometry ETIO 6.2 The Lw of osines Ojectives Use the Lw of osines to solve olique tringles. olve pplied prolems using the Lw of osines. Use Heron s formul to fi nd the

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

Triangles The following examples explore aspects of triangles:

Triangles The following examples explore aspects of triangles: Tringles The following exmples explore spects of tringles: xmple 1: ltitude of right ngled tringle + xmple : tringle ltitude of the symmetricl ltitude of n isosceles x x - 4 +x xmple 3: ltitude of the

More information

Chapter 1: Logarithmic functions and indices

Chapter 1: Logarithmic functions and indices Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

More information

SAINT IGNATIUS COLLEGE

SAINT IGNATIUS COLLEGE SAINT IGNATIUS COLLEGE Directions to Students Tril Higher School Certificte 0 MATHEMATICS Reding Time : 5 minutes Totl Mrks 00 Working Time : hours Write using blue or blck pen. (sketches in pencil). This

More information

Math 1B, lecture 4: Error bounds for numerical methods

Math 1B, lecture 4: Error bounds for numerical methods Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the

More information

1 The Riemann Integral

1 The Riemann Integral The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re

More information

Use the diagram to identify each angle pair as a linear pair, vertical angles, or neither.

Use the diagram to identify each angle pair as a linear pair, vertical angles, or neither. inl xm Review hpter 1 6 & hpter 9 Nme Use the points nd lines in the digrm to identify the following. 1) Three colliner points in Plne M. [],, H [],, [],, [],, [],, M [] H,, M 2) Three noncolliner points

More information

NOT TO SCALE. We can make use of the small angle approximations: if θ á 1 (and is expressed in RADIANS), then

NOT TO SCALE. We can make use of the small angle approximations: if θ á 1 (and is expressed in RADIANS), then 3. Stellr Prllx y terrestril stndrds, the strs re extremely distnt: the nerest, Proxim Centuri, is 4.24 light yers (~ 10 13 km) wy. This mens tht their prllx is extremely smll. Prllx is the pprent shifting

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

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

More information

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3 2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

More information

Proportions: A ratio is the quotient of two numbers. For example, 2 3

Proportions: A ratio is the quotient of two numbers. For example, 2 3 Proportions: rtio is the quotient of two numers. For exmple, 2 3 is rtio of 2 n 3. n equlity of two rtios is proportion. For exmple, 3 7 = 15 is proportion. 45 If two sets of numers (none of whih is 0)

More information

Theorems Solutions. Multiple Choice Solutions

Theorems Solutions. Multiple Choice Solutions Solutions We hve intentionlly included more mteril thn cn be covered in most Student Study Sessions to ccount for groups tht re ble to nswer the questions t fster rte. Use your own judgment, bsed on the

More information