SHW6-0 Total: 8 arks Review Exercise 3. Area of C ()(8)sin 9 9.9 BC CA. Let s. 4 9 0 s. Area of C 3. s( s )( s BC)( s CA).(. 4)(. 9)(. 0) 8.0 BAC 80 By the sine forula, BAC 4 80 ( su of ) BC sinc sin A 0 a sin sin80 a 4.0 4. By the cosine forula, PQ QR PR cos PQR ( PQ)( QR) 7 6 cos (7)(6) 44.4 6. By the cosine forula, b c a cos A bc 8 0 (8)() A.77.8 a c b cos B ac 0 8 (0)() B 4.4096 4.4 A B C 80 C 80.77 4.4096 8.8 ( su of ) 7. (a) With the notations in the figure, BAQ P (alt. s, QA // BP) 3 The copass bearing of B fro A S(90 3)W S8W (b) With the notations in the figure, +. By the cosine forula, b a b c 7 4 (7)(4) cos 4.0400.04 0 ac cos B A 00.86 b c a cos A bc.0400 4 7 (.0400)(4) A B C 80 C 80 00.86 4 34. ( su of ) + RCB PBC (alt. s, RC // BP) 8 ACB 80 70 (3 8) ( su of ) 60 The true bearing of A fro C 70 RCB ACB 70 8 60 348
8. (a) With the notations in the figure, 9. With the notations in the figure, (b) PAR P 80 (adj. s on st. line) 6 P 80 P In P, by the sine forula, sin P sin P AP BP sin P sin 0 k 9 k 0sin sin P 9 P 38.68 or 80 38.68 38.7 (cor. to 3 sig.fig.) or 4 (rejected) The copass bearing of P fro B is N38.7 E. P BPT 38. 68 (alt. s, // PT) The true bearing of B fro P is (80 + 38.68 ), i.e. 9. HPR PHS 6 (alt. s, PR // SH) In HPR, HR sin HPR HP 80 sin 6 HP 80 HP sin 6 90.606 HPQ HPR 80 (adj. s on st. line) HPQ 6 80 HPQ 8 In HPQ, by the cosine forula, HQ HQ PQ HP 0 90.606 (0)(90.606) cos8 9.708 ( PQ)( HP)cos HPQ By the cosine forula, HQ PQ HP cos HQP ( HQ)( PQ) 9.708 0 90.606 (9.708)(0) HQP 6.7 The angle of elevation of H fro the an at Q is 6.7.
SHW 6-A Total: arks. (a) Consider ADH. AD DH (Pyth. theore) 4 4 3.66 Consider AGH. AG GH (Pyth. theore) AG ( 3) 48 6.93 4 + (b) The angle between AG and the plane ADHE is G. Consider AGH. GH tan G 4 3 G 3.3 The angle between AG and the plane ADHE is 3.3.. (a) Consider BEC. BE cos CBE BC 80 cos3 BC BC 97.660 97.7 Consider BCD. BC sin BDC BD 97.660 sin 40 BD BD.93 (cor to 3 sig. fig.) Consider AFD. DF tan DAF AF DF tan 3 80 DF 6.066 6.0 ++ (b) The angle between BD and BF is DBF. DF sin DBF BD 6.066.93 DBF.6 The angle between BD and BF is.6. (c) F is the projection of D on the plane EF. BF is the projection of BD on the plane EF. The angle between BD and BF is the sae as the angle between BD and the plane EF. 3. (a) The angle between EG and AG is AGE. Consider EFG. EG EF FG (Pyth. theore) EG 6 0 Consider EAG. EA tan AGE EG 0 0 AGE 6.6 The angle between EG and AG is 6.6. (b) The angle between AG and CDHG is AGD. Consider CDG. DG CD CG (Pyth. theore) DG 6 0 36 Consider ADG. AD tan AGD DG 36 AGD 3. The angle between AG and CDHG is 3.. (c) The angle between HG and CD is D. Consider ADH. DH tan D AD 0 D 39.8 The angle between HG and CD is 39.8.. (a) The angle between OH and OE is EOH. Consider C. AC BC (Pyth. theore) AC 0 0 OC = Consider OCH. OH OC CH (Pyth. theore) OH 7 Siilarly, OE = 0 7 3
Consider OEH. By the cosine forula, OH OE EH coseoh ( OH)( OE) 7 7 7 7 EOH 48. (cor. to 3 sig.fig.) The angle between OH and OE is 48.. M+ (b) Let I be the projection of O on the plane GHEF. The angle between OH and the plane GHEF is OHI. 0 IH = OC = Consider OHI. OI tan OHI IH 0 OHI 4.7 The angle between OH and the plane GHEF is 4.7. 4
SHW 6-B Total: 8 arks. The angle between the planes ACDE and BCDF is ACB. By the sine forula, AC sinacb sinc 7 sin ACB sin40 sin 40 sin ACB 7 ACB 7.0 The angle between the planes ACDE and BCDF is 7.0.. The angle between the planes ACFD and BCFE is ACB. By the cosine forula, BC AC cos ACB ( BC)( AC) 9 7 (9)() ACB 39.4 The angle between the planes ACFD and BCFE is 39.4. 3. (a) V is an isosceles triangle. VM AM = 8 = 4 Consider AMV. AM VM VA (Pyth. theore) VM 8 4 48 6.93 (b) C is an isosceles triangle. CM The angle between the planes V and C is CMV. By the cosine forula, VM CM VC cos CMV ( VM )( VC) ( CMV 70. 48) ( ( 48)( 48) 8 48) The angle between the planes V and C is 70.. ()() cos 4 8.490 8.4 4. (a) Consider CDF. By the cosine forula, CF CD DF ( CD)( DF) cos CDF CF (b) Consider ADF. AF AF AD DF (Pyth. theore) 8 44. (c) The angle between AC and AF is CAF. Consider CAF. AC AF 44 By the cosine forula, AC AF CF coscaf ( AC)( AF) ( 44) 44)( 44) CAF 3.0 The angle between AC and AF is 3.0.. (a) (i) Consider C. AC AC ( ( 44) 8.490 BC (Pyth. theore) 8 6 0 OC AC (prop. of rectangle) 0 (ii) The angle between VC and the plane CD is VCO. Consider VCO. VO tan VCO OC VCO 67.4 The angle between VC and the plane CD is 67.4. (b) VBC and OBC are isosceles triangles. VM BC and OM BC The angle between the planes VBC and CD is VMO. OM 4 VO tan VMO OM 4 VMO 7.6 The angle between the planes VBC and CD is 7.6.
SHW6-C Total: 7 arks. (a) Consider O. OB sin O OB sin 40 0 OB 0sin 40 Consider OBT. OT tan OBT OB OT tan 0sin 40 OT 0sin 40 tan 0 The height of the tower OT is 0. (b) Consider O. OA cos O OA cos 40 0 OA 0 cos 40 Consider OAT. OT tan OAT OA 0sin 40 tan 0 cos 40 OAT 0. The angle of elevation of T fro A is 0... (a) Consider POQ. With the notations in the figure, O OQ = Q + OQP (ext. of ) 60 = + OQP OQP = 4 By the sine forula, OQ sin OQP sin Q OQ sin 4 sin OQ 9.06 9. (b) Consider VOQ. VO tan VQO OQ 0 9.06 VQO 6.4 The angle of elevation of V fro Q is 6.4. 3. (a) Consider B. tan HBA 8 tan 0 8 tan 0 76.994 Consider C. 76.9 By the cosine forula, AC BC ( )( AC) cosc AC (76.994) 3.9 7 3.6 (b) Consider ACH. tan ACH ACH (76.994)(7) cos AC 8 3.9 40.7 + The angle of elevation of H fro C is 40.7.
4. (a) Consider H. OH tan H tan 4 tan 4 Consider ORH. OH tan ORH OR tan 4 OR OR Consider POR. OR tan R tan 4 tan 4 R 4 The true bearing of R fro P is 80 4 = 6. (b) Consider POR. cosr PR cos4 tan 4 3OQ OQ sin 4 Consider OQH. OH tan OQH OQ sin 4 OQH 67.6 The angle of elevation of H fro Q is 67.6.