By the end of this set of exercises, you should be able to. calculate the area of a triangle using trigonometry

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1 TIGOOMETY y the end of this set of exercises, you should be able to (a) (b) calculate the area of a triangle using trigonometry solve problems using Sine and osine rules. Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 3

2 TIGOOMETY Introduction: Sine, osine and Tangent Graphs Exercise 1 1. The Sine Graph (a) Make a copy of this table and use your calculator to help fill it in, giving each answer correct to 2 decimal places. x sin x 0á00 0á34 0á64 0á87 0á98 1á x sin x (b) Use a piece of 2 mm graph paper to draw a set of axes as illustrated below Ð1 (c) (d) lot as accurately as possible the 21 points from your table. Join them up smoothly to create the graph of the function y = sin x. 2. epeat question 1 (a) to (d) for the function y = cos x 3. epeat for the graph of y = tan x (a different scale will be required for the vertical axis). (These graphs will be studied later). Sine, osine and Tangents of angles other than acute angles Exercise 1 1. Use your calculator to find the following trigonometric ratios. Give each answer correct to 3 decimal places. (a) sin 25 (b) cos 95 (c) tan 107 (d) sin 200 (e) cos 315 (f) tan 181 (g) cos 240 (h) sin 330 (i) tan 225 (j) sin 300 (k) tan 315 (l) cos 500 (m) tan (Ð75 ) (n) cos (Ð200 ) (o) sin 360 (p) cos 360 Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 4

3 . rea of a Triangle using Trigonometry. Exercise 2 1. In this question you are being asked to calculate the area of triangle, using two methods. Method 1 (a) Use basic right angled trigonometry on triangle to calculate the height (= h cm). h cm Method 2 (b) ow use the formula rea = 1 / 2 (base x height) to calculate the area of D. Use the formula: rea = 1 / 2 b c sin with b = 12 cm, c = and angle = 72 to calculate the area of triangle. Did you obtain the same answer? Which method was the faster? cm 2. Use the formula rea = 1 / 2 a b sin to calculate the areas of the following six triangles: (Give all answers correct to 1 decimal place). (a) (b) 6 cm 15 cm cm 39 (c) M 9á2 cm (d) 34 D 52 L 16 cm 16 cm 8á5 cm (e) X (f) E 2á1 cm T 128 F S 13 cm 3á2 cm Y cm Z Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 5

4 3. alculate the areas of the following two triangles: (a) (b) What do you notice? 4. alculate the areas of the following two triangles: (a) (b) á5 cm 6á5 cm What do you notice? an you explain your answers to questions 3 and 4? 5. Shown is a sketch of Farmer GilesÕ triangular field. alculate its area in square metres. 52 m 6. alculate the area of this pentagon: m 6á cm 7. alculate the areas of the following two parallelograms: (a) (b) 14 cm 4 cm Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 6

5 . Sine ule. Exercise 3 In this exercise, give all answers correct to 1 decimal place. 1. opy and complete the following: a Sin a Sin 61 b c = = ( ) Sin Sin = 7á5 Sin 39 => a = 7á5 x Sin 61 = cm Sin 39 7á5 cm 61 a cm Use the Sine ule in each of the following to calculate the size of the side marked. (a) (b) 25 cm (c) 50 8á6 cm I (d) J K 60 (e) E 40 (f) T 120 S 45 cm 35 D 70 F Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 7

6 3. (a) Write down the size of Ð. (b) Use the Sine rule to calculate the length of the line In each of the following, calculate the size of the third angle first before attempting to calculate the length of the side marked. (a) (b) M (c) J á5 cm 72 10á4 cm cm 109 F K 5. opy and complete: a Sin 10 Sin b c = = ( ) Sin Sin = 8 Sin 42 => 8 Sin = 10 Sin 42 => Sin x = 10 Sin 42 = 0á... 8 => x = Use the Sine ule in each of the following to calculate the size of the angle marked. (a) (b) (c) X Y 12 cm 12 cm Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 8 Z (d) L cm 8á4 cm 30 cm 75 M 3á2 cm

7 7. The diagram shows a roof truss. alculate the size of the angle marked between the wooden supports. 3á2 m 34 2á1 m 8. H.M.S. autilus lies East of H.M.S. Unicorn. The diagram shows where an enemy submarine is in relation to the two ships. 35 km alculate how far the submarine is from H.M.S. autilus Unicorn autilus 9. This is the metal frame used to support and hold a childõs swing. It is in the shape of an isosceles triangle. (a) alculate the size of Ð. (b) Use the Sine rule to calculate how far apart points and are. (nswers to 2 decimal places) (c) Draw a vertical line through, creating two right angled triangles and use right angled trigonometry to check your answer to part (b). 2á8 m 30 2á8 m 10. alculate the size of the angles marked, y and z. (careful!) 108 6á1 cm 65 7á y V z 10á6 cm 9á U 42 Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 9 W

8 . osine ule Exercise 4 1. opy and complete the following: a 2 = b 2 + c 2 Ð (2bc cos ) => x 2 = Ð (2 x 7 x 8 x cos 25 ) => x 2 = Ð (...) => x 2 =... => x = Use the osine rule to calculate the size of each side marked here. (a) (b) cm cm (c) J (d) M cm 9á2 cm I K L 8á 47 (e) G (f) 34 Y 7á5 cm 7á5 cm 20 cm E F V 15 1 W Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 10

9 3. opy and complete the following: a 2 = b 2 + c 2 Ð (2bc cos ) => x 2 = Ð (2 x 8 x 6 x cos 110 ) => x 2 = Ð (96 x (Ð0á342..)) => x 2 =... Ð (Ð32á83..) => x 2 = á83.. => x 2 =... => x = (note) cm 4. alculate the lengths of the sides marked. (a) (b) (c) 1 15 cm U 4á V 5á2 cm W 5. farmer owns a piece of fenced land which is triangular in shape. alculate the length of the third side and then write down the perimeter of the field. 58 m F D m E 6. Two ships leave eterborough harbour at The ightingale sails at 20 miles per hour on a bearing 042. The Mayflower II sails at 25 miles per hour on a bearing 087. (a) alculate the size of ÐM. (b) How far apart will the 2 ships be after 1 hour? (c) How far apart will they be at 1600? orth ightingale M Mayflower II Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 11

10 Exercise 4 1. opy and complete the following to find Ð: a 2 = b 2 + c 2 Ð (2bc cos ) => cos = => cos = b 2 + c 2 Ð a 2 2bc Ð x 6 x 7 6 cm => cos = 0á... => = 2. Use this ÔreverseÕ form of the osine rule to calculate the size of each angle marked x here. cos = b 2 + c 2 Ð a 2 2bc (a) (b) 6 cm 15 cm (c) J (d) 13 cm M 1 1 7á 7á1 cm (e) I 14 cm G K (f) L 8á2 cm Y 6á5 cm 6á5 cm 40 cm 1 E 5á F V 36 cm W Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 12

11 3. opy and complete the following to find Ð: a 2 = b 2 + c 2 Ð (2bc cos ) => cos = => cos = b 2 + c 2 Ð a 2 2bc Ð x 7 x 6 => cos = Ð0á178.. => =????? 6 cm Hint :- try finding SHIFT (or IV) cos (Ð0á178..) if you obtain the correct answer of 100á3, your calculator can handle negatives. if you obtain the wrong answer of Ð79á7, ask your teacher/lecturer for help. 4. alculate the size of each of the obtuse angles in the following three triangles: (a) (b) (c) 3á5 cm 2á5 cm 66 mm 5 cm 5 cm 6 cm U 38 mm V 41 mm W 5. Two guy ropes are used to restrain a balloon. The ropes are 85 metres and 65 metres long, and are tethered at points 100 metres apart. alculate the sizes of the two angles marked and y. 85 m y 100 m 65 m 6. This triangular metal plate has its 3 sides as shown. (a) alculate the size of the angle marked. (b) alculate the area of the triangular plate cm 2 Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 13

12 HEKU FO TIGOOMETY 1. Write down the values of the following to 3 decimal places: (a) sin 200 (b) tan 320 (c) cos (Ð265 ) 2. alculate the area of this triangle: cm 3. alculate the area of this parallelogram: 6á5 cm 57 9á3 cm 4. Use the Sine ule or the osine rule (2 formats) to calculate the value of x each time here: (a) (b) (c) 8á5 cm 22 cm 9á5 cm (d) (e) (f) cm 9á2 cm 9á2 cm 61 (g) (h) (i) cm 13 cm 8á 7á5 cm 9á6 cm 11 cm 11 cm 16á5 cm Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 14

13 5. The diagram shows the side view of a house with a sloping roof. alculate the size of the angle, x, between the two sloping sides of the roof. 4á5 m 3á5 m orth 6á9 m 6. From a radar station at, signals from 65 km two ships are picked up. Ship is on a bearing 041 from and 53 km 041 is 65 kilometres away. Ship is on a bearing 295 from and is 53 kilometres away. alculate how far apart the two ships are farmer owns a triangular piece of land trapped between 2 main roads and the farm track. alculate the length of the farm track to the nearest whole metre. farm track m alculate the shaded area of this rectangular metal plate with a triangular hole cut out of it cm Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 15

14 SWES TO MTHEMTIS 2 (IT 2) Trigonometry Exercise 1 1. (a) 0á00 0á34 0á64 0á87 0á98 1á00 0á98 0á87 0á64 0á34 0á00 Ð0á34 Ð0á64 Ð0á87 Ð0á98 Ð1á00 Ð0á98 Ð0á87 Ð0á64 Ð0á34 0á00 1 Ð1 y = sin x y = cos x Ð1 90 y = tan x Exercise 1 1. (a) 0á423 (b) Ð0á087 (c) Ð3á271 (d) Ð0á342 (e) 0á707 (f) 0á017 (g) Ð0á5 (h) Ð0á5 (i) 1 (j) Ð0á866 (k) Ð1 (l) Ð0á776 (m) Ð3á732 (n) Ð0á940 (o) 0 (p) 1 Exercise 2 1. Method : (a) h = 9á51 (b) 57á06 cm 2 Method 2: Ñ> 57á06 cm 2 2. (a) 14á1 cm 2 (b) 56á6 cm 2 (c) 30á 2 (d) 71á6 cm 2 (e) 100á5 cm 2 (f) 2á6 cm 2 3. (a) 25á 2 (b) 25á 2 same answer 4. (a) 18á2 cm 2 (b) 18á2 cm 2 same answer because sin 53 = sin cm á 2 7 (a) 215á 2 (b) 26á2 cm 2 Exercise 3 1. a = 10á4 cm 2. (a) 17á6 cm (b) 14á0 cm (c) 7á6 cm (d) 8á2 cm (e) 13á2 cm (f) 29á 3. (a) 40 (b) 13á 4. (a) 67 ; 5á6 cm (b) 49 ; 9á2 cm (c) 29 ; 13á3 cm , 56á8 6. (a) 37á6 (b) 76á1 (c) 50á6 (d) 21á á á9 km 9. (a) 75 (b) 1á45 m (c) 1á45 m 10. (a) x = 60á3 (b) y = 38á8 (c) z = 106á3 Exercise 4 1. x = 3á39 2. (a) 5á6 cm (b) 14á3 cm (c) 33á5 cm (d) 7á2 cm (e) 4á4 cm (f) 5á3 cm 3. x = 11á5 cm 4. (a) 11á (b) 27á (c) 9á1 cm 5. 67á4 m; 196á4 m 6. (a) 45 (b) 17á8 km (c) 53á5 km Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 32

15 Exercise á3 2. (a) 40á8 (b) 83á9 (c) 47 (d) 61á7 (e) 54á0 (f) 26á á3 4. (a) 109á5 (b) 111á8 (c) 113á3 5. x = 40á1, y = 57á4 6. (a) x = 56á9 (b) rea = 214á 2 heckup for Trigonometry 1. (a) Ð0á342 (b) Ð0á839 (c) Ð0á cm á 2 4. (a) 11á4 cm (b) 11á1 cm (c) 12á5 cm (d) 9á3 cm (e) 8á (f) 22á2 cm (g) 80á4 (h) 72á3 (i) 131á á á5 km m á 2 Simultaneous Linear Equations Exercise 1 1. (a) 3 (b) W = 3 (c) 30kg (d)(e) 2. (a) 1/10 2/20 3/30 4/40 5/50 6/60 in table (b) 10 (c) E = 10 (d) 90 (e) (f) 3. (a) 1/20 2/40 3/60 4/80 5/100 6/120 in table (b) T = 20W (c) 200 mins 4. (a) (b) = 8h (a) 1/12 2/16 3/20 4/24 5/28 in table (b) = 4D + 8 (c) (d) (0,8) (e) osts 8 before even paying for any days!! 8 6. (a) 1/15 2/25 3/35 4/45 5/55 6/65 in table (b) T = 10W + 5 (c) 105 mins 7. (a) = 5k + 50 (b) 100 (c) (a) W = (b) 280 (c) 80 Mathematics Support Materials: Mathematics 2 (Int 2) - Student Materials 33