Grade 8 / Cosinus 28 exercises corrected

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ChingQuizz : 6 exercises available for Quizz assessment : 1. Reminders on angles E.6436 Determine the mea-sure of the angle ∠ BCA : E.6437 Consider a triangle ABC such that : ∠ BAC = 54 o ; ∠ ACB = 38 o The points M and N belong to the segments [ AC ] and [ BC ] respectively and are such that ( AB ) == ( MN ) . Determine the measure of the angle ∠ MNC . E.9015 Consider the rectangle ABCD such that ∠ DAC =65 o . The point I of the diagonal [ AC ] is placed such that the triangle ABI is right-angled at I : 1 Determine the measure of the angle ∠ DCA . 2 Determine the measures of all the angles of triangles ADC , AIB and BIC . 2. Introduction E.9017 Definition: in a right triangle: the side opposite the right angle is called the hy-pothènuse relative to an angle carried by the hypotenuse, the other side of the angle is called the side adjacent to the an-gle and the other side is called the side opposite the angle . Consider the two right-angled triangles ABC and DEF re-spectively right-angled at B and F shown below : 1 a Name the triangle mortgage ABC . b Name the side adjacent to the angle ∠ BAC . 2 Name the side opposite the angle ∠ FDE https://chingmath.fr quizzChap/1255 chapExoCorrec/6436 sacados/6436 ABC35o52o chapExoCorrec/6437 sacados/6437 54o38oABCMN(AB==(MN chapExoCorrec/9015 sacados/9015 ABCDI chapExoCorrec/9017 sacados/9017 ABChypothénusecôtéadjacentà¸côtéopposéุABChypothénusecôtéopposéà˛côtéadjacentà˛˛ ABCDEF
E.4917 We consider the four triangles shown below are right-angled and have an angle of 25 o . 1 Using the measurements in the figure above, complete the table below, rounding the lengths to the nearest mil-limeter: Triangle 1 2 3 4 Hypoténuse Side adjacent à the corner of 25 o 2 Using the calculator, give the values of the quotients re-quested rounded to the nearest hundredth : Triangle 1 2 3 4 Longueur from the side adjacent to the corner of 25 o Longueur de l’hypoténuse E.9029 Definition-proposition: In a right-angled triangle, for an acute angle ¸ of this triangle, we define the cosine of the angle α by the value of the quotient : Longueur of the side adjacent to the angle ¸ Longueur of the hypohtenuse The value of this quotient does not depend on the dimen-sions of the right-angled triangle, but only on the measure ¸ of the angle: it is noted cos( α ) Example 1: Consider the triangle ABC shown below où the point H is the foot of the height arising from the point C . 1 In the triangle ABC , give as a function of the lengths of the segments, an expression of the trigonometric ratios : a cos ∠ BAC b cos ∠ ABC 2 a In the triangle ACH , give an expression for cos ∠ CAH b In triangle BCH , give an expression of cos ∠ CBH https://chingmath.fr chapExoCorrec/4917 sacados/4917 A1B1C125oA2B2C225oA3B3C325oA4B4C425o chapExoCorrec/9029 sacados/9029 ABChypothénusecos¸ACAB¸ABChypothénusecos˛BCAB˛ ABCH
E.6439 Example: (angles have been rounded to the nearest degree) Consider the two triangles ABC and DEF below : Give the following values rounded to the nearest hundredth : a cos 28 b cos 64 Hint: cosine values for all angles from 1 o to 89 o are all known and available on a table or using the calculator: E.3579 Consider the triangle AMN right-angled M ; the points B and C belong to the segments [ BC ] and [ MN ] respectively; the line ( BC ) is perpendicular to the line ( AM ) : 1 Using a graduated ruler, give an approximate value for the side measures of the two triangles ABC and AMN . 2 Using a calculator, compare the approximate values of the following two quotients : BC AC ; MN AM 3 a Justify that the lines ( BC ) and ( MN ) are parallel. b Justify the equality of the following two quotients : BC MN = AC AN c Derive the following equality: BC AC = MN AN d Compare this equality with your results from question 1 . Was there an error in your calculations? Where d the error come from? 4 Using measurements and approximate values, compare the pairs of quotients below : a AB AC ; AM AN b BC AC ; MN AM E.1154 Let ABC 4 be a right-angled trian-gle at B . The angle C 4 AB measures 40 o , it is divided into four angles of the same measure using the points C 1 , C 2 , C 3 belonging to the segment BC 4 1 Using your graduated ruler, we’ll give the measurements of the lengths rounded to the nearest millimetre. The quotients in the last column will be rounded to the near-est hundredth : i BAC i AC i AB AC i 1 2 3 4 2 Complete the following table using the cos key on your calculator. The values are rounded to the nearest thou-sandth : ¸ 10 o 20 o 30 o 40 o cos( ¸ ) 3. Use of the cosine: search for the adjacent side E.4919 Consider the four triangles shown below : https://chingmath.fr chapExoCorrec/6439 sacados/6439 ABC4;9cm6;7cmDEF14km43km43o71o4;96;70;73114430;326 6km3,2km6,8kmABC28o8mm3,9mm8,9mmDEF26o chapExoCorrec/3579 sacados/3579 ABCMN chapExoCorrec/1154 sacados/1154 C110oC2C3C4AB chapExoCorrec/4919 sacados/4919 ABC5cm25oGIH3cm35o
Determine the lengths of the following segments. Round your answers to the nearest millimeter: a [ AC ] b [ GH ] Hints: Below is part of the trigonometric table for cosine. ¸ o cos ¸ o ¸ o cos ¸ o ¸ o cos ¸ ¸ o cos ¸ o ¸ o cos ¸ o 0 o 1 20 o 0 ; 94 40 o 0 ; 766 60 o 0 ; 5 80 o 0 ; 174 5 o 0 ; 996 25 o 0 ; 906 45 o 0 ; 707 65 o 0 ; 423 85 o 0 ; 087 10 o 0 ; 985 30 o 0 ; 866 50 o 0 ; 643 70 o 0 ; 342 90 o 0 15 o 0 ; 966 35 o 0 ; 819 55 o 0 ; 574 75 o 0 ; 259 E.9025 Consider the four triangles shown below : Determine the lengths of the following segments. Round your answers to the nearest millimeter: a [ EF ] b [ KJ ] Hints: Below is part of the trigonometric table for cosine. ¸ o cos ¸ o ¸ o cos ¸ o ¸ o cos ¸ ¸ o cos ¸ o ¸ o cos ¸ o 0 o 1 20 o 0 ; 94 40 o 0 ; 766 60 o 0 ; 5 80 o 0 ; 174 5 o 0 ; 996 25 o 0 ; 906 45 o 0 ; 707 65 o 0 ; 423 85 o 0 ; 087 10 o 0 ; 985 30 o 0 ; 866 50 o 0 ; 643 70 o 0 ; 342 90 o 0 15 o 0 ; 966 35 o 0 ; 819 55 o 0 ; 574 75 o 0 ; 259 E.6440 We consider the four triangles rep-resented below : Determine the measure of the lengths of the following seg-ments rounded to the nearest millimeter: a [ AB ] b [ GI ] 4. Use of the cosine: search for the hypothenuse E.9027 We consider the four triangles rep-resented below : Determine the measure of the lengths of the following seg-ments rounded to the nearest millimeter: a [ EF ] b [ KJ ] E.9020 At an acrobatic park, management wants to install a new zip line. The cable will be installed at a height of 8 m in height and the slope of the cable must be 10 o . Determine the length of cable, rounded to the nearest meter, needed to make this tirolienne. 5. Use of the cosine E.4918 Consider the four triangles shown below : Determine the lengths of the following segments, rounded to the nearest millimeter: a [ AB ] b [ JL ] Hints: Below is part of the trigonometric table for cosine. ¸ o cos ¸ o ¸ o cos ¸ o ¸ o cos ¸ ¸ o cos ¸ o ¸ o cos ¸ o 0 o 1 20 o 0 ; 94 40 o 0 ; 766 60 o 0 ; 5 80 o 0 ; 174 5 o 0 ; 996 25 o 0 ; 906 45 o 0 ; 707 65 o 0 ; 423 85 o 0 ; 087 10 o 0 ; 985 30 o 0 ; 866 50 o 0 ; 643 70 o 0 ; 342 90 o 0 15 o 0 ; 966 35 o 0 ; 819 55 o 0 ; 574 75 o 0 ; 259 https://chingmath.fr chapExoCorrec/9025 sacados/9025 EDF3;5cm75oJLK6;5cm20o chapExoCorrec/6440 sacados/6440 ABC6cm33oGIH3cm65o chapExoCorrec/9027 sacados/9027 JLK4;5cm40oEDF1;5cm48o chapExoCorrec/9020 sacados/9020 10o chapExoCorrec/4918 sacados/4918 ABC4;5cm20oJKL1;5cm75o
E.9026 Consider the four triangles shown below : Determine the lengths of the following segments, rounded to the nearest millimeter: a [ ED ] c [ GI ] Hint: Below is part of the trigonometric table for cosine. ¸ o cos ¸ o ¸ o cos ¸ o ¸ o cos ¸ ¸ o cos ¸ o ¸ o cos ¸ o 0 o 1 20 o 0 ; 94 40 o 0 ; 766 60 o 0 ; 5 80 o 0 ; 174 5 o 0 ; 996 25 o 0 ; 906 45 o 0 ; 707 65 o 0 ; 423 85 o 0 ; 087 10 o 0 ; 985 30 o 0 ; 866 50 o 0 ; 643 70 o 0 ; 342 90 o 0 15 o 0 ; 966 35 o 0 ; 819 55 o 0 ; 574 75 o 0 ; 259 E.1156 Consider the two triangles ABC and MNP shown below : Determine the lengths of the following segments, rounded to the nearest millimetre: a [ AC ] b [ MN ] 6. Use of the reciprocal cosine E.2114 Note: when we look at the cosine trigonometric table, we see the cosine values decreasing from 1 to 0 : it never takes the same value twice. Proposition-definition: in a right-angled triangle, the value : r = longueur of the adjacent longueur side of the hypoténuse of an acute angle is uniquely associated with the measure ¸ of that angle. Note cos − 1 ( r ) the measure of this angle. Consider the three right-angled triangles below : 1 a Carry out, on the figure above, the measurements below, transferring them to the table : AB AC BC DE DF EF GH GI HI b Give the value of the following quotients rounded to the nearest hundredth : AC AB BC AB DF DE EF DE HI HG GI HG Indication : we recall the extract from the trigonometric table below : α cos ¸ α cos ¸ α cos ¸ α cos ¸ α cos ¸ 0 1.000 2 0.999 4 0.998 6 0.995 8 0.990 10 0.985 12 0.978 14 0.970 16 0.961 18 0.951 20 0.940 22 0.927 24 0.914 26 0.899 28 0.883 30 0.866 32 0.848 34 0.829 36 0.809 38 0.788 40 0.766 42 0.743 44 0.719 46 0.695 48 0.669 50 0.643 52 0.616 54 0.588 56 0.559 58 0.530 60 0.500 62 0.469 64 0.438 66 0.407 68 0.375 70 0.342 72 0.309 74 0.276 76 0.242 78 0.208 80 0.174 82 0.139 84 0.105 86 0.070 88 0.035 3 Using the trigonometric table, give an approximate value for each of the following angles : BAC ABC FDE FED GHI HGI https://chingmath.fr chapExoCorrec/9026 sacados/9026 GHI5cm25oDEF3;5cm35o chapExoCorrec/1156 sacados/1156 4cm6cm40o20oABCMNP chapExoCorrec/2114 sacados/2114 ABCDEFGHI
E.4931 Below is an excerpt from the trigonometric table of cosines with a step size of 2 o : α cos ¸ α cos ¸ α cos ¸ α cos ¸ α cos ¸ 0 1 ; 000 2 0 ; 999 4 0 ; 998 6 0 ; 995 8 0 ; 990 10 0 ; 985 12 0 ; 978 14 0 ; 970 16 0 ; 961 18 0 ; 951 20 0 ; 940 22 0 ; 927 24 0 ; 914 26 0 ; 899 28 0 ; 883 30 0 ; 866 32 0 ; 848 34 0 ; 829 36 0 ; 809 38 0 ; 788 40 0 ; 766 42 0 ; 743 44 0 ; 719 46 0 ; 695 48 0 ; 669 50 0 ; 643 52 0 ; 616 54 0 ; 588 56 0 ; 559 58 0 ; 530 60 0 ; 500 62 0 ; 469 64 0 ; 438 66 0 ; 407 68 0 ; 375 70 0 ; 342 72 0 ; 309 74 0 ; 276 76 0 ; 242 78 0 ; 208 80 0 ; 174 82 0 ; 139 84 0 ; 105 86 0 ; 070 88 0 ; 035 The figure below shows four triangles with measurements marked on them : Give the approximate value of each angle to the nearest two degrees : a BAC c HGI E.9028 The figure below shows four trian-gles ; measurements are plotted on it: Determine the measure of the following angles rounded to the nearest tenth of a degree : a ∠ DEF b ∠ LJK 7. Use of cosine and inverse cosine E.9019 Consider the figure below where : points A , B , D and points A , C , E are aligned ; triangles ABC and AED are rectangles at B and E re-spectively. 1 Determine, to the nearest tenth of a degree, the measure of the angle ∠ BAC . 2 Use the previous question, to determine the measure of segment [ CE ] rounded to the nearest millimeter. E.4932 Consider a triangle ABC right-angled C such that : AC = 4 cm ; AB = 7 cm 1 Determine the measure of the angle ∠ CAB rounded to the tenth degree. In the remainder of the exercise, the rounded value of the angle ∠ CAB obtained in the previous question will be used : 2 In this question, we will not use the Pythagorean theo-rem : a Determine the measure of the angle ∠ CBA rounded to the nearest tenth of a degree. b Derive the length of side [ BC ] rounded to the nearest millimeter. https://chingmath.fr chapExoCorrec/4931 sacados/4931 ABC3.5cm4.5cmGIH3cm4.5cm chapExoCorrec/9028 sacados/9028 EDF2.5cm3.5cmJLK3cm5.7cm chapExoCorrec/9019 sacados/9019 ABCDE5cm4;5cm2cm chapExoCorrec/4932 sacados/4932 ABC7cm4cm
E.4935 Consider the figure below : The straight lines ( AC ) and ( BD ) intersect at the point O . 1 a Determine the measure of the angle ∠ BAO rounded to the nearest tenth of a degree. b Derive the measure of the angle ∠ COD . 2 Determine the perimeter of the triangle ODC rounded to the nearest millimeter. E.1157 An explorer arrives in front of the Cheops pyramid. He places his measuring instruments (theodolite) at point H . Studying the pyramid, he observes that it is a regular pyramid: the foot C of the height from the vertex S is also the center of the base. It also estimates the distance HC to 550 m . From point H to vertex S , his measuring instru-ments reveal an angle of 15 o . Determine the measure of height SC of the Pyramid of Cheops rounded to the nearest metre. 8. Cosine, Pythagorean theorem and Thales theorem E.9018 Consider the triangle ABC right-angled C shown below : 1 Determine the measure of segment [ AC ] . 2 Deduce the measure, to the nearest degree, of the angle ∠ CAB . E.5988 Opposite is shown the right-angled triangle BPQ at B such that : PB =6 cm ; PQ =8 cm The circle C is the circum-scribed circle of the trian-gle BPQ . We construct the circle C with center P and passing through point B . We de-note A as the point of in-tersection of circle C and segment [ PQ ] . The aim of the exercise is to determine an approximate value for the area of the shaded part. 1 Determine the area of the disk C rounded to the nearest tenth of cm 2 . Hint: we will use : ı ≈ 3.1416 2 a Determine the measure, rounded to the nearest tenth of a degree, of angle APB . b Deduce the area, rounded to the nearest tenth of cm 2 , of the angular sector of the circle C defined by the arc AB . 3 Deduce a measurement, rounded to the nearest tenth of cm 2 , of the shaded part. https://chingmath.fr chapExoCorrec/4935 sacados/4935 3;5cm2;5cm4cmABCDO chapExoCorrec/1157 sacados/1157 HCS15o chapExoCorrec/9018 sacados/9018 ABC13km5km chapExoCorrec/5988 sacados/5988 CCOPABQ6cm
E.9016 Pour mountain running, some athletes measure their per-formance by the ascension velocity , noted V a . V a is the quotient of the difference in elevation of the race, expressed in meters, by the duration, expressed in hours. Example: for a gradient of 4 500 m and a run time of 3 h : V a = 1 500 m = h Reminder : the altitude difference of the race is the differ-ence between the altitude at the finish and the altitude at the start. A top-level runner wants to achieve an ascent speed of at least 1 400 m = h during his next race The figure below is not a full-scale representation. The route breaks down into two stages (see figure 2) : First step of 3 800 m for a horizontal displacement of 3 790 m . Second stage of 4.1 km with a slope angle of approxi-mately 12 o . 1 Check that the gradient of the first stage is approxi-mately 275.5 m . 2 How steep is the second stage? (we’ll round to the near-est decimeter) 3 From the start, the runner takes 48 minutes to reach the summit. Does the runner reach his goal? 9. Unclassified exercises E.1155 Let OAB be a right triangle at A . Con-sider points M and N , which lie on line segments [ OA ] and [ OB ] , respectively, such that triangle OMN is a right triangle at M . 1 Justify the following equality: ON × OA = OB × OM . 2 a Using your protractor, find the measure of angle AOB . b Take measurements, to the nearest millimeter, on the figure to complete the table below : Segment [ OA ] [ OM ] [ OB ] [ ON ] Length c Give the values of the ratios, rounded to the nearest thousandth : OM ON ; OA OB 3 a What conjecture can be made about the quotients OM ON and OA OB ? b Using question 1 , , justify that the two quotients OM ON and OA OB are equal. https://chingmath.fr chapExoCorrec/9016 sacados/9016 altitude d’arrivéealtitude départdéniveléangledepenteABCFigure 1 SommetdénivelésecondeétapedénivelépremièreétapeDépartDEFGH3800m4100m3790m12oFigure 2 chapExoCorrec/1155 sacados/1155 OABMN