Chap31 - 498-520.qxd 31/5/06 11:59 am Page 498 31 PythagorasвЂ™ theorem and trigonometry (2) CHAPTER 10 Linear equations CHAPTER In Chapter 19, PythagorasвЂ™ theorem and trigonometry were used to find the lengths of sides and the sizes of angles in right-angled triangles. These methods will now be used with three-dimensional shapes. 31.1 Problems in three dimensions In a cuboid all the edges are perpendicular to each other. Problems with cuboids and other 3-D shapes involve identifying suitable right-angled triangles and applying PythagorasвЂ™ theorem and trigonometry to them. Example 1 H ABCDEFGH is a cuboid with length 8 cm, breadth 6 cm and height 9 cm. a i Calculate the length of AC. ii Calculate the length of AG. Give your answer correct to 3 significant figures. b Calculate the size of angle GAC. Give your answer correct to the nearest degree. C 6 cm A E 8 cm D C 6 cm 8 cm B Look for a right-angled triangle where AC is one side and the lengths of the other two sides are known. ABC is a suitable triangle. So draw triangle ABC marking the known lengths. B AC2 П AB2 П© BC2 П 82 П© 62 П 64 П© 36 П 100 AC П 10 cm ii F 9 cm A Solution 1 a i G Use PythagorasвЂ™ theorem for this triangle. G Look for a right-angled triangle where AG is one side and the lengths of the other two sides are known. ACG is a suitable triangle. G So draw triangle ACG marking the known lengths. 9 cm C A 10 cm C AG2 П AC2 П© CG2 AG2 П 102 П© 92 П 100 П© 81 П 181 AG П Н™181 а·† П 13.4536 вЂ¦ AG П 13.5 cm (to 3 s.f.) 498 A Use PythagorasвЂ™ theorem for this triangle. Chap31 - 498-520.qxd 31/5/06 11:59 am Page 499 CHAPTER 31 31.1 Problems in three dimensions b G 9 cm A вњ“ 10 cm tan (angle GAC)П бЋЏ19бЋЏ0 П 0.9 angle GAC П 41.987 вЂ¦В° Angle GAC П 42В° (to the nearest degree) For angle GAC. 9 cm is the opposite side. 10 cm is the adjacent side. opp tan П бЋЏбЋЏ adj C Exercise 31A Where necessary give lengths correct to 3 significant figures and angles correct to one decimal place. 1 ABCDEFGH is a cuboid of length 8 cm, breadth 4 cm and height 13 cm. a Calculate the length of ii GB iii AC iii FA iv GA. b Calculate the size of ii angle GBC iii angle GAC. i angle FAB H G E F 13 cm D C 4 cm A B 8 cm F 2 ABCDEF is a triangular prism. In triangle ABC angle CAB П 90В°, AB П 5 cm and AC П 12 cm. In rectangle ABED the length of BE П 15 cm. a Calculate the length of CB. b Calculate the length of ii AF. i CE c Calculate the size of ii angle FAD. i angle FED C D E 12 cm 15 cm A 5 cm B O 3 The diagram shows a square-based pyramid. The lengths of sides of the square base, ABCD, are 10 cm and the base is on a horizontal plane. The centre of the base is the point M and the vertex of the pyramid is O, so that OM is vertical. The point E is the midpoint of the side AB. OA П OB П OC П OD П 15 cm. D a Calculate the length of i AC ii AM. 10 cm b Calculate the length of OM. c Calculate the size of angle OAM. d Hence find the size of angle AOC. e Calculate the length of OE. f Calculate the size of angle OAB. 15 cm C M B E A B Angle between a line and a plane Imagine a light shining directly above AB onto the plane. AN is the shadow of AB on the plane. A line drawn from point B perpendicular to the plane will meet the line AN and form a right angle with this line. Angle BAN is the angle between the line AB and the plane. N A 499 Chap31 - 498-520.qxd 31/5/06 11:59 am Page 500 CHAPTER 31 PythagorasвЂ™ theorem and trigonometry (2) O Example 2 The diagram shows a pyramid. The base, ABCD, is a horizontal rectangle in which AB П 12 cm and AD П 9 cm. The vertex, O, is vertically above the midpoint of the base and OB П 18 cm. Calculate the size of the angle that OB makes with the horizontal plane. Give your answer correct to one decimal place. Solution 2 18 cm C D O B 9 cm A 18 cm C D M 9 cm B 12 cm A O The base, ABCD, of the pyramid is horizontal so the angle that OB makes with the horizontal plane is the angle that OB makes with the base ABCD. Let M be the midpoint of the base which is directly below O. Join O to M and M to B. As OM is perpendicular to the base of the pyramid the angle OBM is the angle between OB and the base and so is the required angle. Draw triangle OBM marking OB П 18 cm. To find the size of angle OBM find the length of either MB or OM. Calculate the length of MB which is 1бЋЏ2бЋЏ DB. 18 cm D Draw the right-angled triangle ABD marking the known lengths. 9 cm A M 12 cm 12 cm B B DB2 П 92 П© 122 П 81 П© 144 DB2 П 225 DB П Н™225 а·† П 15 Use PythagorasвЂ™ theorem to calculate the length of DB. MB П 1бЋЏ2бЋЏ DB П 7.5 O For angle OBM, 18 cm is the hypotenuse, 7.5 cm is the adjacent side. 18 cm M 7.5 cm B 7.5 cos (angle OBM) П бЋЏбЋЏ 18 angle OBM П 65.37 вЂ¦В° The angle between OB and the horizontal plane is 65.4В° (to one d.p.) 500 adj cos П бЋЏбЋЏ hyp Chap31 - 498-520.qxd 31/5/06 11:59 am Page 501 CHAPTER 31 31.1 Problems in three dimensions вњ“ Exercise 31B Where necessary give lengths correct to 3 significant figures and angles correct to one decimal place. O 1 The diagram shows a pyramid. The base, ABCD, is a horizontal rectangle in which AB П 15 cm and AD П 8 cm. The vertex, O, is vertically above the centre of the base and OA П 24 cm. Calculate the size of the angle that OA makes with the horizontal plane. 24 cm C D B 8 cm 15 cm A 2 ABCDEFGH is a cuboid with a rectangular base in which AB П 12 cm and BC П 5 cm. The height, AE, of the cuboid is 15 cm. Calculate the size of the angle a between FA and ABCD b between GA and ABCD c between BE and ADHE d Write down the size of the angle between HE and ABFE. H G E F 15 cm D C 5 cm A 3 ABCDEF is a triangular prism. In triangle ABC, angle CAB П 90В°, AB П 8 cm and AC П 10 cm. In rectangle ABED, the length of BE П 5 cm. C Calculate the size of the angle between a CB and ABED b CD and ABED c CE and ABED 10 cm d BC and ADFC. B 12 cm F D E 5 cm A 4 The diagram shows a square-based pyramid. The lengths of sides of the square base, ABCD, are 8 cm and the base is on a horizontal plane. The centre of the base is the point M and the vertex of the pyramid is O so that OM is vertical. The point E is the midpoint of the side AB. OA П OB П OC П OD П 20 cm Calculate the size of the angle between OE and the base ABCD. B 8 cm O 20 cm C D M 8 cm A B E 501 Chap31 - 498-520.qxd 31/5/06 11:59 am Page 502 CHAPTER 31 PythagorasвЂ™ theorem and trigonometry (2) 5 ABCD is a horizontal rectangular lawn in a garden and TC is a vertical pole. Ropes run from the top of the pole, T, to the corners, A, B and D, of the lawn. a Calculate the length of the rope TA. b Calculate the size of the angle made with the lawn by ii the rope TD iii the rope TA. i the rope TB T 6m D C 12 m A B 8m 6 The diagram shows a learnerвЂ™s ski slope, ABCD, of length, AB, 500 m. Triangles BAF and CDE are congruent right-angled triangles and ABCD, AFED and BCEF are rectangles. The rectangle BCEF is horizontal and the rectangle AFED is vertical. The angle between AB and BCEF is 20В° and the angle between AC and BCEF is 10В°. D A E C 500 m F Calculate a the length of FB c the distance AC B b the height of A above F d the width, BC, of the ski slope. 7 Diagram 1 shows a square-based pyramid OABCD. Each side of the square is of length 60 cm and OA П OB П OC П OD П 50 cm. O C 50 cm D B 60 cm 60 cm A Diagram 2 shows a cube, ABCDEFGH, in which each edge is of length 60 cm. A solid is made by placing the pyramid on top of the cube so that the base, ABCD, of the pyramid is on the top, ABCD, of the cube. The solid is placed on a horizontal table with the face, EFGH, on the table. a Calculate the height of the vertex O above the table. b Calculate the size of the angle between OE and the horizontal. 502 Diagram 1 C 60 cm 60 cm D B A G 60 cm H F E Diagram 2 Chap31 - 498-520.qxd 31/5/06 11:59 am Page 503 CHAPTER 31 31.2 Trigonometric ratios for any angle 31.2 Trigonometric ratios for any angle The diagram shows a circle, centre the origin O and radius 1 unit. Imagine a line, OP, of length 1 unit fixed at O, rotating in an anticlockwise direction about O, starting from the x-axis. The diagram shows OP when it has rotated through 40В°. y 1 0.8 P 0.6 1 0.4 0.2 40В° ПЄ1.2 ПЄ1 ПЄ0.8 ПЄ0.6 ПЄ0.4 ПЄ0.2 O 0.2 Q 0.4 0.6 0.8 1 x ПЄ0.2 ПЄ0.4 ПЄ0.6 ПЄ0.8 ПЄ1 ПЄ1.2 The right-angled triangle OPQ has hypotenuse OP П 1 Relative to angle POQ, side PQ is the opposite side and side OQ is the adjacent side. This means that OQ П cos 40В° and PQ П sin 40В° For P, x П cos 40В° and y П sin 40В° so the coordinates of P are (cos 40В°, sin 40В°). In general when OP rotates through any angle вђЄ В°, the position of P on the circle, radius П 1 is given by x П cos вђЄ В°, y П sin вђЄ В°. The coordinates of P are (cos вђЄ В°, sin вђЄ В°). So when OP rotates through 400В° the coordinates of P are (cos 400В°, sin 400В°). A rotation of 400В° is 1 complete revolution of 360В° plus a further rotation of 40В°. The position of P is the same as in the previous diagram so (cos 400В°, sin 400В°) is the same point as (cos 40В°, sin 40В°), therefore cos 400В° П cos 40В° and sin 400В° П sin 40В°. If OP rotates through ПЄ40В° this means OP rotates through 40В° in a clockwise direction. 503 Chap31 - 498-520.qxd 31/5/06 11:59 am Page 504 CHAPTER 31 PythagorasвЂ™ theorem and trigonometry (2) For вђЄ П 136, вђЄ П 225, вђЄ П 304 and вђЄ П ПЄ40 the position of P is shown on the diagram. y 1.2 1 0.8 P 0.6 0.4 136В° 0.2 ПЄ1.2 ПЄ1 ПЄ0.8 ПЄ0.6 ПЄ0.4 ПЄ0.2 O 0.2 ПЄ0.2 225В° 0.4 0.6 ПЄ40В° 0.8 x 1.2 ПЄ0.4 ПЄ0.6 P 304В° P 1 ПЄ0.8 P ПЄ1 ПЄ1.2 For P when вђЄ П 136, x П cos 136В° and y П sin 136В°. From the diagram, cos 136В° ПЅ 0 and sin 136В° Пѕ 0 For P when вђЄ П 225, x П cos 225В° and y П sin 225В°. From the diagram, cos 225В° ПЅ 0 and sin 225В° ПЅ 0 For P when вђЄ П 304, x П cos 304В° and y П sin 304В°. From the diagram, cos 304В° Пѕ 0 and sin 304В° ПЅ 0 For P when вђЄ П ПЄ40, x П cos ПЄ40В° and y П sin ПЄ40В°. From the diagram, cos ПЄ40В° Пѕ 0 and sin ПЄ40В° ПЅ 0 The diagram shows for each quadrant whether the sine and cosine of angles in that quadrant are positive or negative. sin П© cos П© sin П© cos ПЄ 2nd 1st 3rd 4th sin ПЄ cos П© sin ПЄ cos ПЄ The sine and cosine of any angle can be found using your calculator. The following table shows some of these values corrected where necessary to 3 decimal places. вђЄ 0 30 40 45 60 90 136 180 sin вђЄ В° 0 0.5 0.643 0.707 0.866 1 0.695 0 cos вђЄ В° 1 0.866 0.766 0.707 0.5 0 225 270 304 ПЄ0.707 ПЄ1 ПЄ0.829 ПЄ0.719 ПЄ1 ПЄ0.707 0 0.559 Using these values and others from a calculator the graphs of y П sin вђЄ В° and y П cos вђЄ В° can be drawn. A graphical calculator would be useful here. 504 360 0 1 Chap31 - 498-520.qxd 31/5/06 11:59 am Page 505 CHAPTER 31 31.2 Trigonometric ratios for any angle Graph of y вЂ« ШЌвЂ¬sin вђЄ В° y 1 0.5 ПЄ180 O 180 360 540 Оё ПЄ0.5 Notice that the graph: в—Џ cuts the вђЄ-axis at вЂ¦ , ПЄ180, 0, 180, 360, 540, вЂ¦ в—Џ repeats itself every 360В°, that is, it has a period of 360В° в—Џ has a maximum value of 1 at вђЄ П вЂ¦ , 90, 450, вЂ¦ в—Џ has a minimum value of ПЄ1 at вђЄ П вЂ¦ , ПЄ90, 270, вЂ¦ ПЄ1 Graph of y вЂ« ШЌвЂ¬cos вђЄ В° y 1 0.5 ПЄ180 O 180 360 540 Оё ПЄ0.5 Notice that the graph: в—Џ cuts the вђЄ-axis at вЂ¦ ПЄ90, 90, 270, 450, вЂ¦ в—Џ repeats itself every 360В°, that is it has a period of 360В° в—Џ has a maximum value of 1 at вђЄ П вЂ¦ , 0, 360, вЂ¦ в—Џ has a minimum value of ПЄ1 at вђЄ П вЂ¦ , ПЄ180, 180, 540, вЂ¦ ПЄ1 Notice also that the graph of y П sin вђЄ В° and the graph of y П cos вђЄ В° are horizontal translations of each other. sin вђЄ В° To find the value of the tangent of any angle, use tan вђЄ В° П бЋЏбЋЏ cos вђЄ В° From the graph of y П cos вђЄ В° it can be seen that cos вђЄ В° П 0 at вђЄ П 90, 270, 450, вЂ¦ for example. As it is not possible to divide by 0 there are no values of tan вђЄ В° at вђЄ П 90, 270, 450, вЂ¦ that is, the graph is discontinuous at these values of вђЄ. Graph of y вЂ« ШЌвЂ¬tan вђЄ В° y 8 6 4 2 ПЄ180 O ПЄ2 ПЄ4 ПЄ6 ПЄ8 180 360 540 Оё Notice that the graph: в—Џ cuts the вђЄ-axis where tan вђЄ В° П 0, that is, at вЂ¦ ПЄ180, 0, 180, 360, 540 вЂ¦ в—Џ repeats itself every 180В°, that is it has a period of 180В° в—Џ does not have values at вђЄ П П®90, П®270, П®450, вЂ¦ в—Џ does not have any maximum or minimum points. Notice also that tan вђЄ В° can take any value. 505 Chap31 - 498-520.qxd 31/5/06 11:59 am Page 506 CHAPTER 31 PythagorasвЂ™ theorem and trigonometry (2) Example 3 For values of вђЄ in the interval вЂ“ 180 to 360 solve the equation ii sin вђЄ В° П 0.7 ii 5 cos вђЄ В° П 2 Give each answer correct to one decimal place. Solution 3 i sin вђЄ В° П 0.7 Use a calculator to find one value of вђЄ . вђЄ П 44.4 y 1 ПЄ180 y П 0.7 O 180 360 Оё To find the other solutions draw a sketch of y П sin вђЄ В° for вђЄ from ПЄ180 to 360 The sketch shows that there are two values of вђЄ in the interval ПЄ180 to 360 for which sin вђЄ В° П 0.7 One solution is вђЄ П 44.4 and by symmetry the other solution is вђЄ П 180 ПЄ 44.4 ПЄ1 вђЄ П 44.4, 180 ПЄ 44.4 вђЄ П 44.4, 135.6 ii 5 cos вђЄ В° П 2 Divide each side of the equation by 5 cos вђЄ В° П 2бЋЏ5бЋЏ П 0.4 вђЄ П 66.4 Use a calculator to find one value of вђЄ . y To find the other solutions draw a sketch of y П cos вђЄ В° for вђЄ from ПЄ180 to 360 1 y П 0.4 ПЄ180 O 180 The sketch shows that there are three values of вђЄ in the interval ПЄ180 to 360 for which cos вђЄ В° П 0.4 360 Оё ПЄ1 вђЄ П 66.4, ПЄ66.4, 360 П© ПЄ66.4 вђЄ П 66.4, ПЄ66.4, 293.6 One solution is вђЄ П 66.4 and by symmetry another solution is вђЄ П ПЄ66.4 Using the period of the graph the other solution is вђЄ П 360 П© ПЄ66.4 Exercise 31C 1 For ПЄ360 СЂ вђЄ СЂ 360 sketch the graph of a y П sin вђЄ В° b y П cos вђЄ В° 2 Find all values of вђЄ in the interval 0 to 360 for which a sin вђЄ В° П 0.5 b cos вђЄ В° П 0.1 c y П tan вђЄ В°. c tan вђЄ В° П 1 3 a Show that one solution of the equation 3 sin вђЄ В° П 1 is 19.5, correct to 1 decimal place. b Hence solve the equation 3 sin вђЄ В° П 1 for values of вђЄ in the interval 0 to 720 506 Chap31 - 498-520.qxd 31/5/06 11:59 am Page 507 CHAPTER 31 31.3 Area of a triangle 4 a Show that one solution of the equation 10 cos вђЄ В° П ПЄ3 is 107.5 correct to 1 decimal place. b Hence find all values of вђЄ in the interval ПЄ360 to 360 for which 10 cos вђЄ В° П ПЄ3 5 Solve 4 tan вђЄ В° П 3 for values of вђЄ in the interval ПЄ180 to 360 31.3 Area of a triangle Labelling sides and angles The vertices of a triangle are labelled with capital letters. The triangle shown is triangle ABC. B c a C A b The sides opposite the angles are labelled so that a is the length of the side opposite angle A, b is the length of the side opposite angle B and c is the length of the side opposite angle C. Area of a triangle П бЋЏ12бЋЏ base П« height B Area of triangle ABC П 1бЋЏ2бЋЏ bh In the right-angled triangle BCN a h П a sin C So area of triangle ABC П 1бЋЏ2бЋЏ b П« a sin C that is C c h A N b area of triangle ABC вЂ« ШЌвЂ¬ab sin C 1 бЋЏбЋЏ 2 The angle C is the angle between the sides of length a and b and is called the included angle. The formula for the area of a triangle means that Area of a triangle П 1бЋЏ2бЋЏ product of two sides П« sine of the included angle. For triangle ABC there are other formulae for the area. Area of triangle ABC П 1бЋЏ2бЋЏ ab sin C П 1бЋЏ2бЋЏ bc sin A П 1бЋЏ2бЋЏ ac sin B. These formulae give the area of a triangle whether the included angle is acute or obtuse. Example 4 Find the area of each of the triangles correct to 3 significant figures. a b B 7.3 cm C 16.2 m 37В° 5.8 cm A 118В° 7.4 m 507 Chap31 - 498-520.qxd 31/5/06 11:59 am Page 508 CHAPTER 31 PythagorasвЂ™ theorem and trigonometry (2) Solution 4 a Area П 1бЋЏ2бЋЏ П« 7.3 П« 5.8 П« sin 37В° Substitute a П 7.3 cm, b П 5.8 cm, C П 37В° into area П 1бЋЏ2бЋЏ ab sin C Area П 12.74 вЂ¦ Area П 12.7 cm2 Give the area correct to 3 significant figures and state the units. b Area П 1бЋЏ2бЋЏ П« 7.4 П« 16.2 П« sin 118В° Area П 52.92 вЂ¦ Substitute into area of a triangle П бЋЏ12бЋЏ product of two sides П« sine of the included angle. Area П 52.9 m2 Example 5 xВ° 2 The area of this triangle is 20 cm . Find the size of the acute angle xВ°. Give your angle correct to one decimal place. Solution 5 1 бЋЏбЋЏ П« 8.1 П« 6.4 П« sin xВ° П 20 2 8.1 cm 6.4 cm Use area of a triangle П 1бЋЏ2бЋЏ product of two sides П« sine of the included angle. 2 П« 20 sin xВ° П бЋЏбЋЏ П 0.7716 8.1 П« 6.4 Find the value of sin xВ°. xВ° П 50.49 вЂ¦В° xВ° П 50.5В° Give the angle correct to one decimal place. Exercise 31D Give lengths and areas correct to 3 significant figures and angles correct to one decimal place. 1 Work out the area of each of these triangles. i ii iii 28В° 9.3 cm 10.6 cm 13.5 cm 9.2 cm 34.7В° 43В° 6.9 cm 9.1 cm iv v vi 8.6 cm 148.6В° 13.4 cm 76.3В° 4.6 cm 4.6 cm 9.6 cm 137В° 4.7 cm 2 ABCD is a quadrilateral. Work out the area of the quadrilateral. D 57В° 9.4 cm C 12.6 cm 8.6 cm 80В° A 508 B Chap31 - 498-520.qxd 31/5/06 11:59 am Page 509 CHAPTER 31 31.3 Area of a triangle 3 The area of triangle ABC is 15 cm2 Angle A is acute. Work out the size of angle A. C 6.5 cm A B 8.4 cm C 4 The area of triangle ABC is 60.7 m2 Work out the length of BC. A 35В° B 12.6 m 5 a Triangle ABC is such that a П 6 cm, b П 9 cm and angle C П 25В°. Work out the area of triangle ABC. b Triangle PQR is such that p П 6 cm, q П 9 cm and angle R П 155В°. Work out the area of triangle PQR. c What do you notice about your answers? Why do you think this is true? 6 The diagram shows a regular octagon with centre O. a Work out the size of angle AOB. OA П OB П 6 cm. b Work out the area of triangle AOB. c Hence work out the area of the octagon. A B O 7 Work out the area of the parallelogram. 5.7 cm 63В° 12.8 cm 8 a An equilateral triangle has sides of length 12 cm. Calculate the area of the equilateral triangle. b A regular hexagon has sides of length 12 cm. Calculate the area of the regular hexagon. 9 The diagram shows a sector, AOB, of a circle, centre O. The radius of the circle is 8 cm and the size of angle AOB is 50В°. a Work out the area of triangle AOB. b Work out the area of the sector AOB. c Hence work out the area of the segment shown shaded in the diagram. A 8 cm 50В° O B 8 cm 509 Chap31 - 498-520.qxd 31/5/06 11:59 am Page 510 CHAPTER 31 PythagorasвЂ™ theorem and trigonometry (2) 31.4 The sine rule B a c C A b The last section showed that 1 1 1 Area of triangle ABC П 2 ab sin C П 2 bc sin A П 2 ca sin B 1 бЋЏбЋЏ 2 ab sin C вЂ« ШЌвЂ¬1бЋЏ2бЋЏ bc sin A and cancelling 1бЋЏ2бЋЏ and b from both sides a sin C вЂ« ШЌвЂ¬c sin A 1 бЋЏбЋЏ 2 bc sin A вЂ« ШЌвЂ¬1бЋЏ2бЋЏ ca sin B cancelling 1бЋЏ2бЋЏ and c from both sides and or b sin A вЂ« ШЌвЂ¬a sin B or a c бЋЏбЋЏ вЂ« ШЌвЂ¬бЋЏбЋЏ sin A sin C and b a бЋЏбЋЏ вЂ« ШЌвЂ¬бЋЏбЋЏ sin B sin A Combining these results a b c бЋЏбЋЏ вЂ« ШЌвЂ¬бЋЏбЋЏ вЂ« ШЌвЂ¬бЋЏбЋЏ sin A sin B sin C This result is known as the sine rule and can be used in any triangle. Using the sine rule to calculate a length Example 6 Find the length of the side marked a in the triangle. Give your answer correct to 3 significant figures. 74В° a 37В° Solution 6 a 8.4 бЋЏбЋЏ П бЋЏбЋЏ sin 74В° sin 37В° 8.4 П« sin 37В° a П бЋЏбЋЏ sin 74В° a П 5.258 вЂ¦ a П 5.26 cm 510 8.4 cm a b Substitute A П 37В°, b П 8.4, B П 74В° into бЋЏбЋЏ П бЋЏбЋЏ. sin A sin B Multiply both sides by sin 37В°. Chap31 - 498-520.qxd 31/5/06 11:59 am Page 511 CHAPTER 31 31.4 The sine rule Example 7 Find the length of the side marked x in the triangle. Give your answer correct to 3 significant figures. 18В° x Solution 7 Missing angle П 180 ПЄ (18 П© 124) П 38В° 9.7 cm 124В° 18В° The angle opposite 9.7 cm must be found before the sine rule can be used. Use the angle sum of a triangle. x 9.7 cm 124В° 38В° x 9.7 бЋЏбЋЏ П бЋЏбЋЏ sin 124В° sin 38В° 9.7 П« sin 124В° x П бЋЏбЋЏ sin 38В° Write down the sine rule with x opposite 124В° and 9.7 opposite 38В°. Multiply both sides by sin 124В°. x П 13.06 вЂ¦ x П 13.1 cm Using the sine rule to calculate an angle When the sine rule is used to calculate an angle it is a good idea to turn each fraction upside down (the reciprocal). This gives sin A sin B sin C бЋЏбЋЏ П бЋЏбЋЏ П бЋЏбЋЏ a b c Example 8 Find the size of the acute angle x in the triangle. Give your answer correct to one decimal place. 74В° 7.9 cm x Solution 8 sin x sin 74В° бЋЏбЋЏ П бЋЏбЋЏ 8.4 7.9 8.4 cm Write down the sine rule with x opposite 7.9 and 74В° opposite 8.4 7.9 П« sin 74В° sin x П бЋЏбЋЏ 8.4 Multiply both sides by 7.9 sin x П 0.904 вЂ¦ Find the value of sin x. x П 64.69 вЂ¦В° x П 64.7В° 511 Chap31 - 498-520.qxd 31/5/06 11:59 am Page 512 CHAPTER 31 PythagorasвЂ™ theorem and trigonometry (2) Exercise 31E Give lengths and areas correct to 3 significant figures and angles correct to 1 decimal place. 1 Find the lengths of the sides marked with letters in these triangles. a b c a 79В° b 27В° 46В° 58В° 4.2 cm 51В° 62В° 11 cm 6.1 cm d e 64В° d c f 17В° 13.6 cm 113В° f e g 22В° 76В° 134В° 14.9 cm 6.1 cm 2 Calculate the size of each of the acute angles marked with a letter. a b c 6 cm A 8 cm 32В° 17 cm 21В° 6.8 cm d C 18.4 cm 73В° 9.1 cm B E 104В° 7.6 cm 3 The diagram shows quadrilateral ABCD and its diagonal AC. a In triangle ABC, work out the length of AC. b In triangle ACD, work out the size of angle DAC. c Work out the size of angle BCD. D D 12.7 cm 8.6 cm C 102В° 5.7 cm 46В° 18В° B A 4 In triangle ABC, BC П 8.6 cm, angle BAC П 52В° and angle ABC П 63В°. a Calculate the length of AC. b Calculate the length of AB. c Calculate the area of triangle ABC. 5 In triangle PQR all the angles are acute. PR П 7.8 cm and PQ П 8.4 cm. Angle PQR П 58В°. a Work out the size of angle PRQ. b Work out the length of QR. 6 The diagram shows the position of a port (P), a lighthouse (L) and a buoy (B). The lighthouse is due east of the buoy. B The lighthouse is on a bearing of 035В° from the port and the buoy is on a bearing of 312В° from the port. ii angle PLB. a Work out the size of i angle PBL The lighthouse is 8 km from the port. b Work out the distance PB. c Work out the distance BL. d Work out the shortest distance from the port (P) to the line BL. 512 N L P Chap31 - 498-520.qxd 31/5/06 11:59 am Page 513 CHAPTER 31 31.5 The cosine rule 31.5 The cosine rule The diagram shows triangle ABC. The line BN is perpendicular to AC and meets the line AC at N so that AN П x and NC П (b ПЄ x). The length of BN is h. In triangle ANB PythagorasвЂ™ theorem gives c2 П x2 П© h2 1 In triangle BNC PythagorasвЂ™ theorem gives a2 П (b ПЄ x)2 П© h2 a2 П b2 ПЄ 2bx П© x2 П© h2 Using 1 substitute c2 for x2 П© h2 a2 П b2 ПЄ 2bx П© c2 2 B c a h x (b ПЄ x) N A b C In the right-angled triangle ANB, x П c cos A Substituting this into 2 a2 вЂ« ШЌвЂ¬b2 Ш‰ c2 ШЉ 2bc cos A This result is known as the cosine rule and can be used in any triangle. b2 вЂ« ШЌвЂ¬a2 Ш‰ c2 ШЉ 2ac cos B c2 вЂ« ШЌвЂ¬a2 Ш‰ b2 ШЉ 2ab cos C Similarly and Using the cosine rule to calculate a length Example 9 Find the length of the side marked with a letter in each triangle. Give your answers correct to 3 significant figures. a b B a x 8 cm 7.3 cm 117В° 24В° C 12 cm Solution 9 a a2 П 122 П© 82 ПЄ 2 П« 12 П« 8 П« cos 24В° a П 144 П© 64 ПЄ 175.4007 вЂ¦ 2 5.8 cm A Substitute b П 12, c П 8, A П 24В° into a2 П b2 П© c2 ПЄ 2bc cos A. Evaluate each term separately. a2 П 32.599 27 вЂ¦ a П Н™32.599 а·†а·† 27 вЂ¦ a П 5.709 577 вЂ¦ a П 5.71 cm b x2 П 7.32 П© 5.82 ПЄ 2 П« 7.3 П« 5.8 П« cos 117В° x2 П 53.29 П© 33.64 ПЄ 84.68 П« (ПЄ0.4539 вЂ¦) x2 П 86.93 П© 38.44 вЂ¦ x2 П 125.37 вЂ¦ x П Н™125.37 а·†а·† вЂ¦ x П 11.19 вЂ¦ x П 11.2 cm Take the square root. Substitute the two given lengths and the included angle into the cosine rule. cos 117В° ПЅ 0 Take the square root. 513 Chap31 - 498-520.qxd 31/5/06 11:59 am Page 514 CHAPTER 31 PythagorasвЂ™ theorem and trigonometry (2) Using the cosine rule to calculate an angle To find an angle using the cosine rule, when the lengths of all three sides of a triangle are known, rearrange a2 П b2 П© c2 ПЄ 2bc cos A. 2bc cos A П b2 П© c2 ПЄ a2 b2 Ш‰ c2 ШЉ a2 cos A вЂ« ШЌвЂ¬бЋЏбЋЏ 2bc Similarly a2 Ш‰ c2 ШЉ b2 cos B вЂ« ШЌвЂ¬бЋЏбЋЏ 2ac and a2 Ш‰ b2 ШЉ c2 cos C вЂ« ШЌвЂ¬бЋЏбЋЏ 2ab Example 10 Find the size of a angle BAC b angle X. Give your answers correct to one decimal place. a b B 12.7 cm 16 cm 13 cm 8.6 cm X A 11 cm C Solution 10 112 П© 162 ПЄ 132 a cos A П бЋЏбЋЏ 2 П« 11 П« 16 208 cos A П бЋЏбЋЏ 352 cos A П 0.590 909 вЂ¦ A П 53.77 вЂ¦В° A П 53.8В° 8.62 П© 6.92 ПЄ 12.72 b cos X П бЋЏбЋЏ 2 П« 8.6 П« 6.9 ПЄ39.72 cos X П бЋЏбЋЏ 118.68 cos X П ПЄ0.334 68 вЂ¦ X П 109.55 вЂ¦В° X П 109.6В° 514 6.9 cm b2 П© c2 ПЄ a2 Substitute b П 11, c П 16, a П 13 into cos A П бЋЏбЋЏ. 2bc Substitute the three lengths into the cosine rule noting that 12.7 cm is opposite the angle to be found. The value of cos X is negative so X is an obtuse angle. Chap31 - 498-520.qxd 31/5/06 11:59 am Page 515 CHAPTER 31 31.5 The cosine rule Exercise 31F Where necessary give lengths and areas correct to 3 significant figures and angles correct to 1 decimal place. 1 Calculate the length of the sides marked with letters in these triangles. a b c a 8 cm 11.3 cm b 18В° 16.2 cm 62В° 15.5 cm 75В° 9 cm 9.2 cm c d e d f 10.2 cm 9.6 cm 8.4 cm e 147В° 134В° 52В° 9.6 cm f 8.4 cm 6.3 cm 2 Calculate the size of each of the angles marked with a letter in these triangles. a b 7 cm 9 cm 9.4 cm 15.3 cm A B 11 cm 13.6 cm c d 8.6 cm C D 8.7 cm 8.7 cm 7.2 cm 14.4 cm 6.8 cm 3 The diagram shows the quadrilateral ABCD. a Work out the length of DB. b Work out the size of angle DAB. c Work out the area of quadrilateral ABCD. C 26.4 cm D 56В° 8.4 cm 9.8 cm A 16.3 cm B 4 Work out the perimeter of triangle PQR. R 8.6 cm Q 27В° P 10.9 cm 5 In triangle ABC, AB П 10.1 cm, AC П 9.4 cm and BC П 8.7 cm. Calculate the size of angle BAC. 6 In triangle XYZ, XY П 20.3 cm, XZ П 14.5 cm and angle YXZ П 38В°. Calculate the length of YZ. 515 Chap31 - 498-520.qxd 31/5/06 11:59 am Page 516 CHAPTER 31 PythagorasвЂ™ theorem and trigonometry (2) 7 AB is a chord of a circle with centre O. The radius of the circle is 7 cm and the length of the chord is 11 cm. Calculate the size of angle AOB. O 7 cm B 11 cm A 8 The region ABC is marked on a school field. The point B is 70 m from A on a bearing of 064В°. The point C is 90 m from A on a bearing of 132В°. a Work out the size of angle BAC. b Work out the length of BC. N B 70 m A 90 m C 9 Chris ran 4 km on a bearing of 036В° from P to Q. He then ran in a straight line from Q to R where R is 7 km due East of P. Chris then ran in a straight line from R to P. Calculate the total distance run by Chris. 10 The diagram shows a parallelogram. Work out the length of each diagonal of the parallelogram. 6 cm 65В° 8 cm 31.6 Solving problems using the sine rule, the cosine rule and 1 бЋЏбЋЏ ab sin C 2 Example 11 The area of triangle ABC is 12 cm2 AB П 3.8 cm and angle ABC П 70В°. ii AC. a Find the length of i BC Give your answers correct to 3 significant figures. b Find the size of angle BAC. Give your answer correct to 1 decimal place. A 3.8 cm 70В° B Solution 11 a i 1бЋЏ2бЋЏ П« 3.8 П« BC sin 70В° П 12 C Substitute c П 3.8, B П 70В° into area П бЋЏ12бЋЏ ac sin B. 2 П« 12 BC П бЋЏбЋЏ 3.8 sin 70В° BC П 6.721 вЂ¦ BC П 6.72 cm ii b2 П 6.721 вЂ¦2 П© 3.82 ПЄ 2 П« 6.721вЂ¦ П« 3.8 cos 70В° b2 П 59.613 вЂ¦ ПЄ 17.470 вЂ¦ b2 П 42.142 вЂ¦ b П 6.491 вЂ¦ AC П 6.49 cm 516 Substitute a П 6.721 вЂ¦, c П 3.8 and B П 70В° into b2 П a2 П© c2 ПЄ 2ac cos B. Chap31 - 498-520.qxd 31/5/06 11:59 am Page 517 1 31.6 Solving problems using the sine rule, the cosine rule and бЋЏ2бЋЏab sin C sin A sin 70В° b бЋЏбЋЏ П бЋЏ бЋЏ 6.721 вЂ¦ 6.491 вЂ¦ 6.721 вЂ¦ П« sin 70В° sin A П бЋЏбЋЏ 6.491 вЂ¦ CHAPTER 31 sin A sin B Substitute a П 6.721 вЂ¦, b П 6.491 вЂ¦ and B П 70В° into бЋЏбЋЏ П бЋЏбЋЏ. a b sin A П 0.9728 вЂ¦ A П 76.62 вЂ¦В° Angle BAC П 76.6В° Exercise 31G Where necessary give lengths and areas correct to 3 significant figures and angles correct to 1 decimal place, unless the question states otherwise. 1 A triangle has sides of lengths 9 cm, 10 cm and 11 cm. a Calculate the size of each angle of the triangle. b Calculate the area of the triangle. 2 In the diagram ABC is a straight line. a Calculate the length of BD. b Calculate the size of angle DAB. c Calculate the length of AC. D 12.9 cm A 5.4cm B 12 cm 65В° 47В° C 2 3 The area of triangle ABC is 15 cm . AB П 4.6 cm and angle BAC П 63Лљ. a Work out the length of AC. b Work out the length of BC. c Work out the size of angle ABC. 4 ABCD is a kite with diagonal DB. a Calculate the length of DB. b Calculate the size of angle BDC. c Calculate the value of x. d Calculate the length of AC. D x cm 8 cm A 50В° C 30В° 8 cm x cm B 5 Kultar walked 9 km due South from point A to point B. He then changed direction and walked 5 km to point C. Kultar was then 6 km from his starting point A. a Work out the bearing of point C from point B. Give your answer correct to the nearest degree. b Work out the bearing of point C from point A. Give your answer correct to the nearest degree. 6 The diagram shows a pyramid. The base of the pyramid, ABCD, is a rectangle in which AB П 15 cm and AD П 8 cm. The vertex of the pyramid is O where OA П OB П OC П OD П 20 cm. Work out the size of angle DOB correct to the nearest degree. O 20 cm C D 8 cm A 15 cm B 517 Chap31 - 498-520.qxd 31/5/06 11:59 am Page 518 CHAPTER 31 PythagorasвЂ™ theorem and trigonometry (2) 7 The diagram shows a vertical pole, PQ, standing on a hill. The hill is at an angle of 8В° to the horizontal. The point R is 20 m downhill from Q and the line PR is at 12В° to the hill. a Calculate the size of angle RPQ. b Calculate the length, PQ, of the pole. 12В° R P Q 20 m 8В° 8 A, B and C are points on horizontal ground so that AB П 30 m, BC П 24 m and angle CAB П 50В°. AP and BQ are vertical posts, where AP П BQ П 10 m. a Work out the size of angle ACB. b Work out the length of AC. c Work out the size of angle PCQ. d Work out the size of the angle between QC and the ground. P Q 10 m 10 m 30 m A B 50В° 24 m C Chapter summary You should now be able to: в�… use PythagorasвЂ™ theorem to solve problems in 3 dimensions в�… use trigonometry to solve problems in 3 dimensions в�… work out the size of the angle between a line and a plane в�… draw sketches of the graphs of y П sin x В°, y П cos x В°, y П tan x В° and use these graphs to solve simple trigonometric equations в�… use the formula area П бЋЏ12бЋЏ ab sin C to calculate the area of any triangle в�… a b c use the sine rule бЋЏбЋЏ П бЋЏбЋЏ П бЋЏбЋЏ and the cosine rule a2 П b2 П© c2 ПЄ 2bc cos A in sin A sin B sin C triangles and in solving problems. Chapter 31 review questions 1 In the diagram, XY represents a vertical tower on level ground. A and B are points due West of Y. The distance AB is 30 metres. The angle of elevation of X from A is 30Вє. The angle of elevation of X from B is 50Вє. Calculate the height, in metres, of the tower XY. Give your answer correct to 2 decimal places. A X Diagram NOT accurately drawn 50В° 30В° 30 m B Y (1384 June 1996) 2 The diagram shows triangle ABC. AC П 7.2 cm BC П 8.35 cm Angle ACB П 74В°. a Calculate the area of triangle ABC. Give your answer correct to 3 significant figures. b Calculate the length of AB. Give your answer correct to 3 significant figures. 518 C Diagram NOT accurately drawn 74В° 7.2 cm A 8.35 cm B (1385 June 2002) Chap31 - 498-520.qxd 31/5/06 11:59 am Page 519 CHAPTER 31 Chapter 31 review questions 3 In triangle ABC AC П 8 cm CB П 15 cm Angle ACB П 70В°. a Calculate the area of triangle ABC. Give your answer correct to 3 significant figures. X 8 cm 70В° C 4 The diagram shows a cuboid. A, B, C, D and E are five vertices of the cuboid. AB П 5 cm BC П 8 cm CE П 3 cm. B 15 cm X is the point on AB such that angle CXB П 90В°. b Calculate the length of CX. Give your answer correct to 3 significant figures. (1387 June 2003) Diagram NOT accurately drawn E 3 cm D C Calculate the size of the angle the diagonal AE makes with the plane A ABCD. Give your answer correct to 1 decimal place. 5 In triangle ABC AC П 8 cm BC П 15 cm Angle ACB П 70В°. a Calculate the length of AB. Give your answer correct to 3 significant figures. b Calculate the size of angle BAC. Give your answer correct to 1 decimal place. Diagram NOT accurately drawn A 8 cm 5 cm B Diagram NOT accurately drawn A 8 cm 70В° B 6 This is a sketch of the graph of y П cos xВ° for values of x between 0 and 360. Write down the coordinates of the point ii A ii B. C 15 cm (1387 June 2003) y A 360 x O B 7 Angle ACB П 150В° BC П 60 m. The area of triangle ABC is 450 m2 Calculate the perimeter of triangle ABC. Give your answer correct to 3 significant figures. C Diagram NOT accurately drawn 150В° 60 m A B (1385 November 2000) 519 Chap31 - 498-520.qxd 31/5/06 11:59 am Page 520 CHAPTER 31 PythagorasвЂ™ theorem and trigonometry (2) 8 The diagram shows a quadrilateral ABCD. AB П 4.1 cm BC П 7.6 cm AD П 5.4 cm Angle ABC П 117В° Angle ADC П 62В°. a Calculate the length of AC. Give your answer correct to 3 significant figures. b Calculate the area of triangle ABC. Give your answer correct to 3 significant figures. c Calculate the area of the quadrilateral ABCD. Give your answer correct to 3 significant figures. Diagram NOT accurately drawn A 5.4 cm 4.1 cm 62В° B 117В° D 7.6 cm C (1385 June 2000) 9 This is a graph of the curve y П sin xВ° for 0 СЂ x СЂ 180 y 1 0.5 O 45 90 135 x 180 ПЄ0.5 ПЄ1 a Using the graph or otherwise, find estimates of the solutions in the interval 0 СЂ x СЂ 360 of the equation ii sin xВ° П ПЄ0.6. i sin xВ° П 0.2 cos xВ° П sin (x П© 90)В° for all values of x. b Write down two solutions of the equation cos xВ° П 0.2 (1385 November 2002) 10 In the diagram, ABCD, ABFE and EFCD are rectangles. The plane EFCD is horizontal and the plane ABFE is vertical. EA П 10 cm DC П 20 cm ED П 20 cm. B A 10 cm 20 cm Calculate the size of the angle that the line AC makes with the plane EFCD. 11 In triangle ABC AB П 10 cm AC П 14 cm BC П 16 cm. a Calculate the size of the smallest angle in the triangle. Give your answer correct to the nearest 0.1В°. b Calculate the area of triangle ABC. Give your answer correct to 3 significant figures. 520 F E D C Diagram NOT accurately drawn A 14 cm 10 cm B 20 cm 16 cm C

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