Activities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions

Transcription

1 MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd Solutions ( pge)

2 MEP: Demonstrtion Projet UNIT 4: Trigonometry TIVITY 4. Pythgors' Theorem For eh of the tringles elow, mesure the ngles nd the sides of eh tringle. Mesure the sides in m, to one deiml ple. Reord your nswers in the tle. I II III IV V VI VII Tringle Right ngled? I II III IV V VI VII Wht n you infer from the finl olumn of results?

3 MEP: Demonstrtion Projet UNIT 4: Trigonometry TIVITY 4.2 Spirls Spirls of vrious kinds re found throughout nture in oth plnts nd nimls. Exmples inlude Snil's shell Sheep's horns Seshells Fossils. We will show here how to onstrut spirls nd how they n e used. Geometri onstrution Step onstrut n isoseles tringle with two sides, O nd, of unit length. O Step 2 Step 3 dd nother line,, of unit length, perpendiulr to the longest side, O, of the tringle nd omplete the next tringle y drwing O. ontinue in this wy, eh time dding unit length. D E. Drw the spirl in this wy with 5 linked sides. Wht is the length of () O () O () OD? (Use Pythgors' Theorem.) 2. From your onstrution, estimte the vlue of 4. hek your ury using lultor. 3. Repet the onstrution, strting with right ngled tringle s shown, dding on sides of length 2 units. How mny sides do you need to onstrut in order 2 to estimte 33? 2 Extension s it onsists of stright lines, the onstrution used ove produes only n pproximtion to true spirl. We n, though, otin true spirl using the eqution r= θ 360 y Here r is the distne of the point P from the origin, where OP mkes n ngle θ with the x xis. Plot the P points whih orrespond to θ = 0, 45, 90,......, 360, nd join up the points with smooth urve. ontinue the urve y plotting points orresponding to O θ x θ = 405, 450,......, et.

4 MEP: Demonstrtion Projet UNIT 4: Trigonometry TIVITY 4.3 linometers linometer is n instrument for mesuring the ngle of n inline (slope). To mke your linometer, you need drinking strw, protrtor, some dhesive, smll mss nd piee of thred. Use dhesive to tth the strw to the protrtor. The thred with mss t its end will serve s plum line. dhesive Protrtor Thred Strw Mss You n now use your linometer to find the height of n ojet suh s tree or uilding. Follow this method.. hoose your ojet. 2. Stnding some distne wy from the ojet, view the top of it through the drinking strw. Red nd reord the vlue of ngle x. x Ojet 3. Mesure nd reord your distne, l, from the ojet. 4. Use the redings you hve tken to lulte the height of the ojet, ove your eyeline, from the formul l l tn x = h = h tn x h d l Ojet 5. The height, H, of the ojet is now given y l H = + d tn x when d is the height of your eyes from the ground.

5 MEP: Demonstrtion Projet UNIT 4: Trigonometry TIVITY 4.4 Rdr ship is nhored t se. The following digrm represents the irulr sreen of the ship's rdr disply. It is divided into squres. Eh side of squre represents 0 km. The entre of the sreen, O, represents the ship's own position. nother ship, S, is nhored t ( 2, 3). Its position is indited y lip on the sreen. (In the digrm, we use dot to represent the lip.) North O S

6 MEP: Demonstrtion Projet UNIT 4: Trigonometry TIVITY 4.4 (ontinued) Rdr. Wht is the distne etween the two ships, orret to the nerest km? 2. t 0600, nother lip ppers t (5, 6) with respet to O, trvelling on ering of 200 t n estimted speed of 40 km/h. () () () (d) Use dot to indite the position of this unidentified vessel on the sreen. Wht is its distne from the ship t O, orret to the nerest km? Drw line lerly on the sreen to show the ourse of nvigtion tken y this unidentified vessel. If this unidentified vessel ontinues with the sme ourse, wht will e the shortest distne etween it nd the ship t O? Give your nswer orret to the nerest km. How long will it e from the time the lip first ppers to the time the unidentified vessel moves out of the rdr disply? How fr will it then e from O?

7 MEP: Demonstrtion Projet UNIT 4: Trigonometry TIVITY 4.5 Posting Prels The Royl Mil Dt Post Interntionl prel servie epts prels up to mximum size s given in the rules elow. l Rule Rule 2 Length + height + width must not exeed 900 mm. None of the length, height, width must exeed 600 mm. h w. Whih of the following prels would e epted for this servie? () l = 620 mm, h = 20 mm, w = 50 mm () l = 500 mm, h = 350 mm, w = 50 mm () l = 550 mm, h = 00 mm, w = 50 mm mm piture with frme, 60 mm y 80 mm, is to e pled digonlly in retngulr ox s shown. Find suitle dimensions for the ox so tht it would e epted for the Dt Post servie. 80 mm 3. long, thin tue is to e sent y Dt Post. Wht is the lrgest possile length tht n e sent? Extension Wht re the dimensions of the retngulr ox of mximum volume tht n e sent through the Dt Post servie?

8 MEP: Demonstrtion Projet UNIT 4: Trigonometry TIVITY 4.6 Interloking Pipes Pipes whih re used to drin wy surfe rinwter from rods re designed with lip in suh wy tht 0 metre pipes n hve hnge in ngle of t most 5. The Highwys Deprtment wnts to ly these pipes long rod whih hs urvture of rdius 50 metres. Is this possile? You n nswer this prolem y finding the minimum rdius tht n e otined y using the mximum ngle hnge of 5. The sitution is illustrted elow.. Wht is the vlue of the ngle α s shown in the digrm? 0 m Show tht r = sin 2 α. Hene find the minimum rdius possile. r 3. n urvture of 50 metres e hieved? If so, wht should e the ngle of hnge etween pipes? α lthough you hve solved prtiulr prolem, it would e very useful for the highwy engineer to hve wy of finding generl expression for the minimum rdius possile, sy r metres, for pipes of given length, sy x metres, whih hs mximum ngle hnge of β, s shown elow. 4. Find the reltionship etween r, x nd β. x β r x r Use this reltionship to omplete the redy rekoner tle s shown ove for the required ngle β, given r nd x. 6. If pipes re ville only in sizes of 2, 5, 0 nd 20 metres, find the mximum size to produe rdius of urvture of () 25 metres () 75 metres. (ssume tht 5 is the mximum hnge in ngle possile.)

9 MEP: Demonstrtion Projet UNIT 4: Trigonometry TIVITY 4.7 Sine Rule Mesure the sizes of ll three ngles nd the lengths of ll three sides of eh of the following tringles. omplete the tle tht follows. Give your nswers orret to one deiml ple if pplile. I II III IV Tringle sin sin sin I II III IV Wht n you onjeture from these results? Extension Drw irle of ertin dimeter nd drw ny tringle with its verties on the irumferene of the irle. Mesure the sides nd ngles of the tringle, nd ompre the vlues of, sin sin, sin to the dimeter. Repet with irle of different dimeter. Wht n you onjeture?