3.1 Review of Sine, Cosine and Tangent for Right Angles

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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