ࡱ> `Oh+'0 (4 P \ ht| CHAPTER: HAPChristopher Yoong Koon ShenhrihriNormalpChristopher Yoong Koon Shen12iMicrosoft Word 8.0o@ @/@aŒI ՜.+,D՜.+,< hp   Bears Inc.'    CHAPTER: Title 6> _PID_GUIDAN{875C3EC0-68B0-11D5-B102-D4AF02C10E27} CHAPTER: SINE AND COSINE RULES Contents The Sine Rule The Ambiguous Case 2 The Cosine Rule Introduction The sine and cosine rules are used to solve non-right-angled triangles, i.e. finding the unknown sides and angles. ( 1 The Sine Rule Consider a triangle with angles A, B and C, and sides a, b and c as shown. C The sine rule for the triangle is given by, b a  eq \f(a,sin A) =  eq \f(b,sin B) =  eq \f(c, sin C)  A c B Proof In triangle ABC, CN is perpendicular to AB C and AM is perpendicular to BC. So, sin B =  eq \f(CN,CB) ( CN = CB sin B = a sin B And sin A =  eq \f(CN,AC) ( CN = AC sin A = b sin A M Thus, CN = a sin B = b sin A (  eq \f(a,sin A) =  eq \f(b,sin B)  Similarly, AM = b sin C = c sin B A N B (  eq \f(b,sin B) =  eq \f(c, sin C)  Hence, combining the two equations, we get  eq \f(a,sin A) =  eq \f(b,sin B) =  eq \f(c, sin C)  Note: The sine rule is used to solve triangles when it is given: two sides and a non-included angle, or, two angles and one side. Example 1.1 Solve the triangle ABC in which ( A = 400 , ( B = 650 and side AB = 10. [AC = 9, BC = 7] Solution Example 1.2 (Additional Maths Vol 1) ABC is a triangle in which ( A = 420 and AB = 25 cm. ( C varies such that 300 ( ( C ( 1100 . Calculate the maximum and minimum lengths of AC. [Ans: max = 47.6 cm, min = 12.5 cm] Solution ( 1.1 The Sine Rule : Ambiguous case The ambiguous case occurs when 2 sides and 1 angle of a triangle are given with the angle opposite the shorter side. There is no ambiguity if the given angle is obtuse or the side opposite the given angle is greater than the other side. In the ambiguous case, there will be two distinct triangles satisfying the given data with the result that we get two sets of answers. Note : When there are two possible values for an angle obtained by using sine rule, we can find out which value is correct by: a) using a scale drawing of the triangle b) verifying a known side or angle using the two possible values c) using the rule that the largest angle is opposite the longest side and the smallest angle is opposite the shortest side. Since there is sometimes an ambiguous case when using the sine rule, always try to use cosine rule whenever possible. Example 1.3 (Additional Maths Vol 1) In triangle ABC, ( A = 300 , a = 3 cm and b = 5 cm. Find the possible values of AB, ( B and ( C. [Ans: 6.0 cm, 56.40, 93.60 or 2.7 cm, 123.60 , 26.40 ] Solution Example 1.4 (Additional Maths Vol 1) ABC is a triangle in which ( C = 720 , AC = 7 cm , AB = 12 cm. Find ( A, ( B and BC. [Ans: 740 , 340, 12 cm] Solution ( 2 The Cosine Rule The cosine rule of a triangle ABC is given by a2 = b2 + c2 2bc cos A b2 = a2 + c2 2ac cos B c2 = a2 + b2 2ab cos C Proof y The diagram shows triangle ABC on a coordinate plane C (b cos A, b sin A) with AB on the positive x-axis and A at the origin. Then point B ( (c, 0) and point C ( (b cos A, b sin A). Hence to find length of BC, i.e. a, we have a2 = (b cos A c)2 + (b sin A 0)2 b a = b2 cos2 A 2bc cos A + c2 + b2 sin2 A = b2 (cos2 A + sin2 A) + c2 2bc cos A = b2 + c2 2bc cos A x A c B (c, 0) By placing B at the origin, we would get b2 = a2 + c2 2ac cos B By placing C at the origin, we would get c2 = a2 + b2 2ab cos C Note: The cosine rule is used to solve a triangle when it is given three sides, or two sides and the included angle. Example 2.1 In triangle ABC, a = 6.5cm, b = 5 cm and c = 7 cm. Calculate all the angles of the triangle. [Ans: 630 ,430, 740 ] Solution Example 2.2 Given that ABC is a triangle in which a = 15 cm, b = 10 cm and c = 7 cm, find angle A. [Ans: 1230 ] Solution Example 2.3 ABC is a triangle in which a = 13 cm, b = 8 cm and c = 12 cm. Calculate the largest and smallest angles. [Ans: largest ( A = 780 , smallest ( B = 370 ] Solution PAGE 5 )]j  @AOPQ~OPRScdfgxyht H*mH  jmH  jmH  jUmH  jUmH>*mH  jx>*mH 5mH mH R )7J\]j@~ *jOz ! & F 8 & F  )7J\]j@~ *jOz&NghtZ ľzupkfa\!x^jk    X"h .l     $&NghtZ  & F !h !   1 2 H I J K L M P Q U V G j ! O [   /:pqxy./3489HIMNRSabg jUmH5mH  jx>*mH >*mH  jmH H*mH mH  jmH U m n v  Z Ŀ~ytoje`[x\de     ! m n v  Z N O u   ! ! N O u  !"#$%&'()*+,-./UĿ~ytoje`[ %}]! !"#$%&'()*+,-./U,F`a !,F`aq+W&Ohi"#$0'()*+,-./012345Ŀ|zzzzzzzzzzzzzzzzzzzzzzz    &'@ij@{aqr0aq+W&Ohi"#$0  & F !h  ! !@CBCDCFCHCJCLCNCPCpCrCDDDHHH2J$ !h&` !'()*+,-./0123456789:;<=>?@L !56789:;<=>?@L 0&P P. A!"#$%x hx x x x 8x 8x x x........ Summary (Sine and Cosine Rule) ==============6 In some cases, both answers may be correct, depending on the question. 2 rCDDDDHHHHJJ KK$KdKfKzK|KKKL LMM0M2MBMDMhMzMMMMMPP QQ$QdQfQzQ|QQQR RSS0S2SBSDShSzSSSSSSVV WW$WdWfWzW|WWWX XYY0Y2YBYDYhYzYYYYYYY\ ;>*mH H*mH  jmH >*mH  jx>*mH 0JmHmH W2JJKBKLhMzM|M~MMMMMMMM2PPQBQRhSzS|S~SSSS$ ! !3 Miscellaneous Examples In some cases, both answers may be correct, depending on the question. .33.7.1 ( 3 Miscellaneous Examples Example 3.1 (SAJC 01/1/7) In triangle ABC, ( ABC is 400, AB is 10 and AC is 8. Find the two possible angles for ( BAC. Assuming that ( ABC and AB remains unchanged, what is the range of values that AC can take such that there will be two possible values for ( BAC? [86.50 or 13.50, 6.43 < AC < 10] Solution S6 |Y~YYYYYYYYYYYYYYYYYYYYY2\\]B]^h_$ ! !h_z_|_~______________________2dde$ ! !3 Miscellaneous Examples In some cases, both answers may be correct, depending on the question. a .33.7.1x ( 3 Miscellaneous Examples Example 3.1 (SAJC 01/1/7) In triangle ABC, ( ABC is 400, AB is 10 and AC is 8. Find the two possible angles for ( BAC. Assuming that ( ABC and AB remains unchanged, what is the range of values that AC can take such that there will be two possible values for ( BAC? [86.50 or 13.50, 6.43 < AC < 10] Solution S ==========================5 eJe fpggggggggggggggggggggggggg$ ! !\\ ]]$]d]f]z]|]]]^ ^__0_2_B_D_h_z_______ddddddee,eleneeeee&f(f g"g8g:gJgLgpgggggggg6mH 0JmH ;>*mH H*mH  jmH mH >*mH  jx>*mH 9 [$@$NormalmH 4@4 Heading 1$ 5>*mH 2@2 Heading 2$>*mH <A@<Default Paragraph Font,@,Header  !&)@& Page NumberB\uv YC)h?gs ,n2 3 4 f o p q r s t u v w x y z { | } ~    & < j  K  E m ~ HXz{|mvwxyz{|}~/8Wq)*lCDL0$%Sm vvvv      @@@  rC\g#0 a2JS|Yh_eg '*-14 5UnknownChristopher Yoong Koon ShenYhk|'111111111111111 !8 @& (    |B NC NEF5%?' N N N'@`@  N N   |B NC NEF5%?'G NG'@` N NHB B # ?f$y:iGHB B # ?&&H(@ 7 N@ L 7 N@ 7 N@HB   # ?> N>HB  B # ?_ 7_ @BB  B  ?_ 79 )>BB    ? )&9?A>HB   # ?R),9R)>B S  ? # t t r t!! t  ~   r u   ? B Z ] gj$'ET%R]gjkq ,9nw $ ' U ]  ) / k m    ! L N X[fl ;LDJ09TVnpChristopher Yoong Koon ShenC:\AUNTY\CJC\Notes\n-trg-0.docChristopher Yoong Koon Shen0C:\windows\TEMP\AutoRecovery save of n-trg-0.asdx\ܐf1;e0,t Wܐf@xhh)hho()@x0@xh8.@x0..@x 0...@ x 0 ....@ x8 .....@ x8 ......@ x`.......@ xX `........@xhh),,DV; x\ W1;eV; @x0V;ab@xh8.W;"@x0..\W;#@x 0...W;$@ x 0 ....W;en@ x8 .....LX;fn@ x8 ......X;'@ x`.......Y;h @ xX `........@HP DeskJet 660CLPT1:HPFDJC02HP DeskJet 660CHP DeskJet 660C4 d,,HP DeskJet 660CLPT1 ,,HP DeskJet 660C4 d,,HP DeskJet 660CLPT1 ,,MNtMMC$Eƀb'GmH !C$Eƀ3 Miscellaneous Examples In some cases, both answers may be correct, depending on the question. .33.7.1 ( 3 Miscellaneous Examples Example 3.1 (SAJC 01/1/7) In triangle ABC, ( ABC is 400, AB is 10 and AC is 8. Find the two possible angles for ( BAC. Assuming that ( ABC and AB remains unchanged, what is the range of values that AC can take such that there will be two possible values for ( BAC? [86.50 or 13.50, 6.43 < AC < 10] Solution S ==========================6 In some cases, both answers may be correct, depending on the question. .33.7.13 3 Miscellaneous Examples In some cases, both answers may be correct, depending on the question. .33.7.1 ( 3 Miscellaneous Examples Example 3.1 (SAJC 01/1/7) In triangle ABC, ( ABC is 400, AB is 10 and AC is 8. Find the two possible angles for ( BAC. Assuming that ( ABC and AB remains unchanged, what is the range of values that AC can take such that there will be two possible values for ( BAC? [86.50 or 13.50, 6.43 < AC < 10] Solution 6 XYijz{PQUVZ["0'@LCClCnCpCrC0JmH0J j0JU jmH >*mH H*mH mH  jmH V\\ ]]$]d]f]z]|]]]^ ^__0_2_B_D_h_z_______ddddddee,eleneeeee&f(f g"g8g:gJgLgpggggggggjjjjjjkk,klknkkkkk&l(l m"m8m:mJmLmpmmmmm6mH 0JmH ;>*mH H*mH  jmH mH >*mH  jx>*mH UeJe fpggggggggggggggggggggggggg$ ! !SSSSSSSSSSSSSSSSSSSS2VVWBWXhYzY|Y$ ! ! 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N N N'@`@  N N   |B NC NEF5%?'G NG'@` N NHB B # ?f$y:iGHB B # ?&&H(@ 7 N@ L 7 N@ 7 N@HB   # ?> N>HB  B # ?_ 7_ @BB  B  ?_ 79 )>BB    ? )&9?A>HB   # ?R),9R)>B S  ? # t t r t!! t  }   q t  > A Y \ fi$'ET%R]gjkq ,9nw # & T \  ( . j l    K M WZek :KCI/8SUmoChristopher Yoong Koon ShenC:\AUNTY\CJC\Notes\n-trg-0.docChristopher Yoong Koon Shen0C:\windows\TEMP\AutoRecovery save of n-trg-0.asdx\ܐf1;e0,t Wܐf@xhh)hho()@x0@xh8.@x0..@x 0...@ x 0 ....@ x8 .....@ x8 ......@ x`.......@ xX `........@xhh),,V; x\ W1;eTV; @x0V;ab@xh8.V;"@x3 Miscellaneous Examples In some cases, both answers may be correct, depending on the question. .33.7.1 ( 3 Miscellaneous Examples Example 3.1 (SAJC 01/1/7) In triangle ABC, ( ABC is 400, AB is 10 and AC is 8. Find the two possible angles for ( BAC. Assuming that ( ABC and AB remains unchanged, what is the range of values that AC can take such that there will be two possible values for ( BAC? [86.50 or 13.50, 6.43 < AC < 10] Solution S ==========================6 b'GmHmH GmHFC$Eƀb'GmHmH GmHC$Eƀb'3 Miscellaneous Examples In some cases, both answers may be correct, depending on the question. a .33.7.1x ( 3 Miscellaneous Examples Example 3.1 (SAJC 01/1/7) In triangle ABC, ( ABC is 400, AB is 10 and AC is 8. Find the two possible angles for ( BA0.. W;#@x 0...lW;$@ x 0 ....W;en@ x8 .....X;fn@ x8 ......hX;'@ x`.......X;h @ xX `........@HP DeskJet 660CLPT1:HPFDJC02HP DeskJet 660CHP  C. Assuming that ( ABC and AB remains unchanged, what is the range of values that AC can take such that there will be two possible values for ( BAC? [86.50 or 13.50, 6.43 < AC < 10] Solution S ==========================6 g2jjkJk lpmmmmmmmmmmmmmmmmmmmmmmm !mmmmmH 0JmHDeskJet 660C4 d,,HP DeskJet 660CLPT1 ,,HP DeskJet 660C4 d,,HP DeskJet 660CLPT1 ,,tC$Eƀb'GmH !C$Eƀb'GmHmH GmHFC$Eƀb'GmHmH GmHC$Eƀb' & FC$Eƀb'C$Eƀ6>*b'$C$Eƀmmm$ ! 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