3 7. 7.2 Trigonometry in three dimensions Questions re trgeted t the grdes indicted The digrm shows the ck of truck used to crry scffold poles. L K G m J F C 0.8 m H E 3 m D 6.5 m Use Pythgors Theorem to find the length of the scffold pole i C...... ii F...... iii G...... Use trigonometry to find the ngle i C...... ii F...... iii GF...... c The scffold pole EL is 8 m long. How fr does it etend eyond the line JK?.. d Wht ngle does it mke with the floor of the truck?.. 343
3 7. 7.2 Trigonometry in three dimensions 2 rocket ws lunched from this plnt pot. Wht ngle does it mke with the ground?... 5 cm...... 20 cm 8 cm 3 cue of stone of side 6 mm is mde into ed y drilling hole through two opposite corners. G H F E Hint Clculte the length of digonl C first. C D 6 mm Clculte the length of the hole... Find the ngle the hole mkes with the fce CD... 4 The digrm shows cmer tripod with four legs of equl length. Clculte the lengths E * i C... ii EF....5 m Use your nswers to prt to find the ngle the leg E mkes with the floor. F 0.9 m 0.9 m C......... The legs re djusted to length of.2 m. c Clculte the new ngle ech leg mkes with the floor. 343
3 7. 7.2 Trigonometry in three dimensions 5 Clculte the ngle F of this rmp. * D E C 2 m 7 m F 5 m skteorder trvels from to C. How fr does she trvel? c Clculte the distnce FC. d Find the ngle to the horizontl t which the skteorder trvels. Hint Find ngle CF. 343C
3 7.3 Trigonometric rtios for ny ngle Questions re trgeted t the grdes indicted Give your nswers correct to deciml plce, where necessry. * The sketch elow helps you to solve the eqution sin θ = 0.8 for vlues of θ in the rnge 360 to 360. Sin θ 0.8 360 80 0 80 360 θ Use your clcultor to find θ. Mrk the vlue on the sketch. Hint press the [sin ] key. Use the symmetry of the grph to find the other vlues of θ etween 360 nd 360. 2 Use the sketch elow to solve the eqution cos θ = 0.2 for vlues of θ in the rnge 0 to 720. Cos θ 0 80 360 540 720 0.2 θ...... 3 Use the sketches to solve the equtions in the given rnge. sin = 0.3 for vlues of etween 0 nd 360. Sin......... 0 80 360... 345
3 7.3 Trigonometric rtios for ny ngle cos = 0.5 for vlues of etween 80 nd 80.... Cos *......... 80 0 80 c sin = 0.7 for vlues of etween 360 nd 360. Sin 360 80 0 80 360 d cos = 0.5 for vlues of etween 540 nd 80. Cos 540 360 80 0 80 345
3 7.3 Trigonometric rtios for ny ngle 4 Solve the equtions in the given rnge. Mke sketch to show your solutions. * sin = 0.62 for vlues of etween 0 nd 360. cos = 0.44 for vlues of etween 360 nd 0. c sin = 0.05 for vlues of etween 80 nd 360. d cos = 0.707 for vlues of etween 0 nd 270. () Show tht one solution of 8 sin = is 7.2. () Hence solve the eqution for vlues of in the rnge 360 to 360. () 8 sin = sin = 8 (divide oth sides y 8) sin = 0.25 = 7.2 (to d.p.) () Solutions etween 360 nd 360 re = 80 7.2 = 72.8 = 360 + 7.2 = 352.8 = 80 7.2 = 87.2 Cos 0.25 360 80 0 7.2 80 360 345C
3 7.3 Trigonometric rtios for ny ngle 5 Show tht one solution of 8 cos = 5 is 5.3. *... Hence solve the eqution for vlues of in the rnge 0 to 720.... 6 Show tht one solution of 50 sin = 49 is 78.5.... Hence solve the eqution for vlues of in the rnge 360 to 360.... 7 Show tht one solution of cos = 3 2 is 30.... Hence solve the eqution for vlues of in the rnge 720 to 0.... 345D
3 7.4 Finding the re of tringle using 2 sin C Questions re trgeted t the grdes indicted Rememer: re of tringle C = sin C 2 C c Clculte lengths nd res correct to 3 significnt figures nd ngles correct to deciml plce. Clculte the re of ech tringle. c d 0 cm 70 8 cm 50 mm 0 200 mm.6 m 5 cm 25 5 cm 30.4 m........................ 2 Clculte the re of ech shpe. * Hint Split the shpe into two tringles. c 20 cm 4.2 m 22 2.7 m 7 cm Prllelogrm Drt 70 30 Kite. m.5 m 347
3 7.4 Finding the re of tringle using 2 sin C Hint Find the ngle t the centre first. 3 Clculte the re of ech regulr polygon. * 7 cm 20 mm.................. c Regulr polygon with 9 sides nd rdius.2 m... d Regulr polygon with 0 sides nd rdius 200 mm... 4 The re of ech tringle is given. Clculte the mrked ngle. re = sin C 2 2 = 0.5 6. 7 sin 2 = 2.35 sin 2 2.35 = sin (divide oth sides y 2.35) sin = 0.9836 (press [sin ] on your clcultor) = 79.6 ( d.p.) 6. mm 7 mm re = 2 mm 2 c 2 mm 9 mm 39 cm re = 27 mm 2 30 cm re = 20 cm 2.7 m 2.4 m re =.2 m 2 347
3 7.4 Finding the re of tringle using 2 sin C 5 Clculte the re of ech segment. * ngle of sector 50 re of sector = πr 2 = π 8 2 = 27.925 cm 360 360 2 re of tringle = sin C = 0.5 8 8 sin 50 = 24.53 cm 2 2 So re of segment = 27.925 cm 2 24.53 cm 2 = 3.4 cm 2 (3 s.f.) C 50 8 cm c 20 mm 0 O 2 3 mm 85.3 cm O 347C
3 7.5 7.6 The sine rule nd clculting n ngle Questions re trgeted t the grdes indicted Use sin = sin to find side, nd sin = sin to find n ngle. Use this sine rule when you know side nd the opposite ngle. Clculte lengths correct to 3 significnt figures nd ngles correct to deciml plce. C Find the length of the mrked side. = 30 sin 25 sin 00 = 30 sin 25 (multiply oth sides y sin 25 ) sin 00 = 2.9 mm (3 s.f.) 00 30 mm 25 c d 53 23.2 m 35 8 cm y 32 50 mm 52 62 68 45 37 cm t............ 2 Clculte the missing ngle. Then use the sine rule to find the mrked side. c 30 25 9 cm 7 mm 4 72 00 mm 70 c 349
3 7.5 7.6 The sine rule nd clculting n ngle 3 Use the sine rule to find the size of the mrked ngle. Then clculte the third ngle. sin sin 00 = 22 38 sin = 22 sin 00 38 sin = 0.5705 = 34.8 ( d.p.) 22 mm 38 mm 00 Third ngle = 80 34.8 00 = 45.2 c d.5 cm d 40 5 mm 2.4 cm c 40 200 mm 0 cm 7 mm 23 300 mm 54 6.7 cm............ 349
3 7.5 7.6 The sine rule nd clculting n ngle 4 For ech tringle find i the mrked ngle ii the third ngle iii the mrked side iv the re of the tringle (use 2 sin C). c 30 mm 2.5 cm 20 mm 2 35.8 cm 40 m 95 200 m c i... i... i... ii... ii... ii... iii... iii... iii... iv... iv... iv... 5 Use the sine rule to clculte the length of D. D... Find the ngle D nd use it to clculte the length of. 3 C 270 m 200 m... c Use the sine rule to clculte ngle DC. 46 60 349C
3 7.5 7.6 The sine rule nd clculting n ngle d Find the ngle DC nd use it to clculte the length of DC. e Use the formul 2 sin C to clculte the res of the two tringles nd hence the re of the qudrilterl CD. 6 The mp shows some of the ncient monuments of Egypt. N Sphin 5 0.3 km N Khum y 9.6 km 57 Sqqr Use the sine rule to find ngle... ngle y... c the distnce from the Sphin to Sqqr...... Use your nswer to prt to find the ering of d Khum from the Sphin...... e the Sphin from Khum...... 349D
3 7.7 7.8 The cosine rule nd clculting n ngle Questions re trgeted t the grdes indicted Use 2 = 2 + c 2 2c cos to find side, 2 2 2 nd cos = + c 2c to find n ngle. C c Use the cosine rule when you know: ll three sides or two sides nd their included ngle. Clculte lengths correct to 3 significnt figures nd ngles correct to deciml plce. Find the length of the mrked side. 2 = 2.3 2 + 0.8 2 2 2.3 0.8 cos 45 2 = 5.29 + 0.64 3.68 ( 0.89 ) 2 = 5.29 + 0.64 + 3.04.. ( = +) 2 = 8.944 = 2.99 cm (3 s.f.) 2.3 cm 45 0.8 cm c 2.8 m.3 m 85 250 mm 59 300 mm.28 cm c 42 0.3 cm 35
3 7.7 7.8 The cosine rule nd clculting n ngle 2 Use the cosine rule to find side. Then use the sine rule to find ngle y. c y 25 mm 74 y 24 2 cm 9. cm y 50 mm km 4 km 26 3 Use the cosine rule to find ngle. 200 2 + 230 2 20 2 78 500 cos = = = 0.853 2 200 230 92 000 = 3.4 ( d.p.) 200 mm 20 mm 230 mm c 4 cm.5 cm 6.5 cm 7 m 8 cm 3 m 4 cm 5 cm 9 m 35
3 7.7 7.8 The cosine rule nd clculting n ngle 4 Use the cosine rule to find ngle. Then use the sine rule to find ngle y. c y 0.8 m 0.65 m 90 cm 5 mm 72 mm y 0.6 m 0 cm 50 cm 40 mm y 5 PQR is tringle where PQ = 30 mm, QR = 40 mm nd ngle PQR = 65. Sketch the tringle.... Use the cosine rule to find the length of PR.... c Use the sine rule to find ngle QPR.... d Clculte ngle PRQ.... 35C
3 7.9 Using trigonometry to solve prolems Questions re trgeted t the grdes indicted Use the sine rule when you know side nd the ngle opposite. Otherwise, use the cosine rule. Clculte lengths nd res correct to 3 significnt figures nd ngles correct to deciml plce. Which rule cn e used to find missing side or ngle? 5 cm 28 23 mm 5 mm 2.3 cm 2. cm 23 cm 20 20 35 mm.2 cm 0 mm...... c... d... 2 Use the sine nd cosine rules to clculte the unknown sides nd ngles in Question....... c... d........................... 3 Use the sine nd cosine rules to clculte the mrked ngles nd sides. c d 50 mm 5 m y 2.3 m y 26 70 mm 9.2 cm 7 m 8 m 2.2 m 2.7 m 52 80.................................... 353
3 7.9 Using trigonometry to solve prolems 4 C is tringle where = 24 mm, C = 20 mm nd ngle C = 28. Sketch the tringle. Clculte the length of side C. c Clculte ngle i C ii C d Clculte the re of the tringle Hint use the formul sin C. 2 353