4 Trigonometr MEP Practice ook ES4 4.4 Sine, osine and Tangent 1. For each of the following triangles, all dimensions are in cm. Find the tangent ratio of the shaded angle. c b 4 f 4 1 k 5. Find each of the following, giving our answer correct to 3 decimal places. tan 36 tan 4 tan 55 tan17 (e) tan 68 (f) tan 73 (g) tan 67. 4 (h) tan 75. 5 (i) tan 81. (j) tan 89. 3 (k) tan 16. 9 (l) tan 6. 3. Find the size of angle in each of the following. Give our answer correct to 1 decimal place. tan = 03. tan = 04. tan = 08. tan = 13. (e) tan = 15. (f) tan = (g) tan = 5. (h) tan = 33. (i) tan = 45. (j) tan = 58. (k) tan = 100. 4 (l) tan = 33. 5 4. For each of the following triangles, all dimensions are in cm. Find the sine ratio of the shaded angle. Give our answer correct to decimal places. 9 3 3 10 14 1 11 55
MEP Practice ook ES4 5. Find the value of each of the following. Give our answer correct to 3 decimal places. sin sin 76 sin 19. 6 sin 39. (e) sin 61. 3 (f) sin 85. 7 (g) sin 44. 9 (h) sin 50. 4 (i) sin 67. 1 (j) sin 79. 3 (k) sin 81. (l) sin 9. 6 6. Find the size of angle in each of the following. Give our answer correct to 1 decimal place. sin = 031. sin = 07. sin = 046. sin = 064. (e) sin = 0. 189 (f) sin = 0. 986 (g) sin = 0. 497 (h) sin = 0. 71 (i) sin = 0. 584 (j) sin = 0. 84 (k) sin = 0. 99 (l) sin = 0. 999 7. For each of the following triangles, all dimensions are in cm. Find the cosine ratio of the shaded angle. Give our answer correct to decimal places. 9 6 4 14 1 3 9 8 8. Find the value of each of the following. Give our answer correct to 3 decimal places. cos 9 cos 48 cos30 cos 69 (e) cos 80. (f) cos 54. 7 (g) cos 79. 3 (h) cos 35. 5 (i) cos 43. 8 (j) cos 56. (k) cos 61. (l) cos 83. 8 9. Find the size of angle in each of the following. Give our answer correct to 1 decimal place. cos = 033. cos = 06. cos = 051. cos = 037. (e) cos = 0. 016 (f) cos = 0. 998 (g) cos = 0. 305 (h) cos = 0. 816 (i) cos = 0. 538 (j) cos = 0. 76 (k) cos = 0. 171 (l) cos = 0. 66 56
MEP Practice ook ES4 10. Write epressions for and sin α, cos α, tanα sin β, cos β, tanβ in terms of a, b and c. What do ou notice about the results? c α a b β 4.5 Finding Lengths in Right ngled Triangles 1. In each of the following find the length of, giving our answer correct to decimal places. 9 cm 43 70 16.6 cm 4 cm 57 64.4 55 cm (e) (f) 36. 8 1 cm 300 cm. One end of a pole, 8 metres long, reaches a corner of the ceiling of a room. If the angle made b the pole with the horizontal is 35, what is the height of the ceiling? Give our answer correct to significant figures. Pole 35 8 m 57
MEP Practice ook ES4 3. The length of the shadow of a vertical pole is 3.4 metres long when the ras of the sun are inclined at an angle of 40. 5 to the horizontal. What is the height of the pole? Give our answer correct to decimal places. Pole Sun's ras 3.4 m shadow 40.5 4. The diagram shows two banks of a river which are at different levels. Points P and Q are on opposite sides of the river such that a rope attached from P to Q makes an angle of to the horizontal. If PQ = 70 m, calculate the width of the river, the difference in heights of the two banks. Give our answers correct to the nearest metre. Q ank 70 m River P ank 5. path, 750 metres long, runs straight up the slope of a hill. If the angle made b the path with the horizontal is 16, find the height of the point at the top end of the path. Give our answer correct to 3 significant figures. 16 750 m Path 6. ladder is placed on horizontal ground with its foot metres from a vertical wall. If the ladder makes an angle of 50 with the ground, find Wall the length of the ladder, how far up the wall it reaches. Give our answers correct to 1 decimal place. 50 m 7. One end of a rope of length 45 metres is tied to a point on the ground and the other end to the top of an antenna. When the rope is taut, its inclination to the horizontal is 48. Find, correct to 3 significant figures, the distance of the top of the antenna from the ground. 45 m 48 58
MEP Practice ook ES4 8. wire 18 metres long runs from the top of a pole to the ground as shown in the diagram. The wire makes an angle of 35 with the ground. alculate the height of the pole. Give our answer to a reasonable degree of accurac. 9. is a right-angled triangle. = 60 m ngle = 3 Find the length of. 60 m 3 10. and DE are similar triangles. = 1.5 m, DE = 9 cm, = cm 18 m 35 Not drawn accuratel E (NE) (Q) Not drawn accuratel alculate the length of D. 9 cm 1.5 cm cm D (Q) 4.6 Finding ngles in Right ngled Triangles 1. In each of the following find angle, giving our answer correct to 1 decimal place. 1 cm 34 cm 5 cm 30 cm 3.4 cm 15 cm 10 cm 5. cm (e) (f) 40 cm 5 cm 18.6 cm 7.8 59
MEP Practice ook ES4. The diagram shows a roofing frame D. = 7 m, = 5 m, D = 3 m, angle D = angle D = 90. D 3 m 7 m 5 m alculate the length of D. alculate the size of angle D. 3. From the top of a building a man sights a pedestrian on the street below at a distance of 48 metres awa. The pedestrian is 34.5 metres awa from the foot of the building. Find the angle of depression of the pedestrian from the man, correct to the nearest degree. Man 48 m (MEG) 34.5 m Pedestrian 4. Find all unknown angles and lengths for each triangle. Give our answers correct to the nearest cm or degree. D 13 cm 8 cm F 4 cm 6 cm E G 33 cm L I.8 cm H 4 cm J 7.5 cm K 60
MEP Practice ook ES4 4.7 Mied Problems with Trigonometr 1. The angle of elevation of a radio-controlled model aeroplane from the transmitter on the ground is 3. If the aeroplane is 100 metres from the transmitter, find the height of the aeroplane from the ground, horizontal distance of the aeroplane rom the transmitter. Transmitter 100 m 3. weather balloon is at a height of 900 metres. The angle of elevation of the balloon from an observer on the field is 4. What is the distance of the balloon from the observer, correct to the nearest metre? Observer 4 900 m 3. The angle of elevation of the top of a building, 13 metres high, from an observer at point on the ground is 54. How far is he from the base of the building, correct to the nearest metre? 13 m building If he walks further awa from the building to a point such that the angle of elevation is halved, find the distance correct to the nearest metre. 54 7 4. In each of the following cases, find the labelled side or angle. Give each answer correct to the nearest cm or degree. 10 cm 0 cm q 6 cm 7 cm 10 cm p 14 cm 5 1 30 cm 17 cm 1 cm 61
MEP Practice ook ES4 5. The diagram shows the positions of three airports: E (East Midlands) M (Manchester) and L (Leeds). The distance from M to L is 65 km on a bearing of 060. North M 60 65 km ngle LME = 90 and ME = 100 km. alculate, correct to three significant 100 km figures, the distance LE. alculate, to the nearest degree, the E bearing of E from L. n aircraft leaves M at 10.45 a.m. and flies direct to E, arriving at 11.03 a.m. alculate the average speed of the aircraft in kilometres per hour. Give our answer correct to the appropriate number of significant figures. (MEG) L 6. The diagram shows a smmetrical framework for a bridge. = 100 m, = = 70 m. (i) alculate the angle D. E F (ii) alculate the length ED. D similar framework is made with the length corresponding to = 180 m. (i) alculate the length corresponding to. (ii) What is the size of the angle corresponding to angle D? 7. acht is moored to the side of a qua b a rope 3 metres long. The rope is tied to the acht at R and the qua at T. t low tide, when the acht is as far awa from the qua as possible, the rope makes an angle of 64 with the horizontal, as shown. 3 m alculate the vertical distance of R R below T when the acht is in this 64 position. Give our answer correct to one decimal place. t high tide, the water level at the qua is 1.6 metres higher than at low tide. t high tide, when the acht is as far awa from the qua as possible, calculate the distance from R to the side of the qua. T (SEG) 6
MEP Practice ook ES4 8. When an aeroplane takes off, its ascent is in two stages. These two stages are shown in the diagram below as and. 7 35 000 feet 15 1 miles Ground Distance 1 mile = 580 feet In the first stage the aeroplane climbs at an angle of 15 to the horizontal. alculate the height it has reached when it has covered a ground distance of 1 miles. Give our answer correct to the nearest thousand feet. In the second stage the aeroplane climbs at an angle of 7 to the horizontal. t the end of its ascent it has reached a height of 35 000 feet above the ground. alculate the total distance it has covered. Give our answer to a reasonable degree of accurac. (NE) 9. Paul's ladder is 4 metres long. Paul leans his ladder against a vertical wall, with the end,, on horizontal ground. The angle between the ladder and the ground is 70. 4 m Wall alculate the distance of from the wall. 70 Pamela moves the ladder and uses it to reach a windowsill which is 3.8 metres above the ground. For safel, the angle between the ladder and the ground should be within of 70. 4 m 3.8 m Is the ladder safel placed? You must show some calculation to eplain our answer. 63
MEP Practice ook ES4 10. D = 4 cm, = 6 cm, angle D = 35. D is perpendicular to. 6 cm alculate D. alculate angle. 4 cm D 35 Triangle is similar to triangle. The area of triangle is nine times the area of triangle. (i) What is the size of angle? (ii) Work out the length of. 11. tall vertical fence GY is supported b a post which is 3.5 m long as shown. The foot of the post is 1.1 m from the fence on horizontal gound XG. (i) alculate the length of G. Y Not to scale 3.5 m (ii) To be safe, the post must make an angle of at least 70 with the ground. G 1.1 m X Is the post safe? Show the calculations ou make. nother post makes an angle of 78 with the ground. Its foot is also 1.1 m from the fence. What is the length of this angled post? (OR) 1. is a right-angled triangle. = 19 cm and = 9 cm. Not to scale 19 cm 9 cm c) and D are right-angled triangles. D is parallel to. = 1 cm, = 5 cm and D = 33.8 cm. D alculate the length of. Not to scale PQR is a right-angled triangle. PQ = 11 cm and QR = 4 cm. 33.8 cm P Not to scale 11 cm 5 cm 1 cm Q 4 cm R alculate the size of angle PRQ. alculate the size of angle D. (Q) 64
MEP Practice ook ES4 4.8 Sine and osine Rules 1. Find the side marked with a letter in each triangle below. s 16 cm 69 37 58 40 cm q 34.3 w 78 cm 6 cm 75 110 30 n 55. Find the shaded angles in the triangles shown. F E 43 19 cm cm D 3 cm 54 cm 15 E 18 cm X 108 G F 46 cm U Y 5 cm Z S 39 cm 93 T 3. Find in triangle, given that angle = 34, angle = 75 and = 10 cm. 10 34 75 4. Find angle ZXY in triangle XYZ, given that XY = 17 cm, XZ = 30 cm and angle XYZ = 40. Z 30 17 40 Y X 65
MEP Practice ook ES4 5. The figure shows two trees, P and Q, on a bank of a river. R is another tree on the opposite bank. P 70 50 m Q 55 alculate angle PRQ, RQ. R 6. Find the side marked with a letter in each triangle below. s 9 cm p 5 cm 60 13 cm 8 cm 140 w 14 cm k 16 cm 16 cm 8 cm 30 7. Find the shaded angles in the triangles shown. Z 7 cm 8 cm 16 cm 5 cm 9 cm X 13 cm Y R 8 cm Q 11 cm F 10 cm 1 cm G 15 cm 0 cm H P 66
MEP Practice ook ES4 8. parallelogram has sides of lengths 30 cm and 70 cm. One of its angles is 60 as shown. Find the lengths of its diagonals. 30 60 70 9. is a chord of a circle, centre O and radius 7 cm. If = 86. cm, calculate angle O. O 7 8.6 10. In triangle, = 7 cm, = 1 cm and angle = 15. 15 7 cm 1 cm alculate the length of. Give our answer correct to 3 significant figures. (LON) 11. surveor wishes to measure the height of a church. Measuring the angle of elevation, she finds that the angle increases from 30 to 35 after walking 0 metres towards the church. 30 0 m 35 What is the height of the church? (SEG) 67
MEP Practice ook ES4 1. Two ships, and, leave Dover Docks at the same time. Ship travels at 5 km/h on a bearing of 10. Ship travels at 30 km/h on a bearing of 130. alculate how far apart the two ships are after 1 hour. 13. is a triangle. = 19 cm, = 17 cm and angle = 60. alculate the size of angle. 60 19 cm Q 17 cm Not drawn accuratel (SEG) PQR is a triangle. PR = 3 cm, PQ = cm and angle QPR = 48. alculate the length of QR. 4.9 ngles Larger than 90 P cm 48 3 cm Not drawn accuratel R (Q) 1. Using the values 1 cos 45 = sin 45 = tan 45 = 1 1 cos60 = sin30 = tan 60 = 3 3 cos30 = sin 60 = tan30 = find, without using a calculator, the value of sin135 sin180 sin10 cos180 (e) cos135 (f) cos 10 (g) sin 70 (h) sin 40 (i) tan135 (j) tan 40 (k) tan150 (l) sin 480 (m) cos 405 (n) cos315 (o) sin 315. Use a calculator, if needed, and a sketch to find all solutions in the range 360 θ 360 of the following equations. 1 3 cosθ = 1 sinθ = 1 sinθ = 0 sin θ = 03. (e) cos θ = 0. (f) sin θ = 04. (g) tanθ = 1 (h) tanθ = (i) cos θ = 08. (j) sin θ = 06. (k) cos θ = 08. (l) sinθ = 1 68
MEP Practice ook ES4 3. Use a sketch to find how man solutions, in the range 0 θ 70 eist for the equation sin θ = 07.. Evaluate each solution, to the nearest degree. 4. The depth of water in a harbour t hours after midda, d metres, is given b d = 10 + 7cos( 30t) Draw a graph of depth against time for a 4-hour period. When is high tide and low tide? ship needs a minimum depth of 11.5 metres to berth in the harbour. For how long during the 4-hour period can the ship remain in the harbour? 5. In the triangle, = 6 cm, = 5 cm, and angle = 45. There are two possible triangles that can be constructed. 6 cm 5 cm 45 alculate the two possible values of the angle. (SEG) 6. The diagram shows the graph of = cos for 0 360. 1 0 0 90 180 70 360 1 (i) op the diagram and show the location of the two solutions of the equation cos. = 05. (ii) The angle is between 0 and 360. Work out accuratel the two solutions of the equation cos = 05.. On a particular da the height, h metres, of the tide at Wemouth, relative to a certain point, can be modelled b the equation h = 5sin( 30t) where t is the time in hours after midnight. (i) Sketch the graph h against t for 0 t 1. (ii) Estimate the height of the tide, relative to the same point, at pm that da. (SEG) 69