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Mixed examples – which rule to use? Study each of these diagrams and determine which rule to use – Sine Rule or Cosine Rule? If Cosine Rule, which version? Answers & working on next slides. A 35 71 x m 16 m E 80 9 cm 6 cm B 14 cm 10 cm 12 cm D 67 x m 11 m 13 m F 33 9 cm x cm 12 cm

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A 35 71 x m 16 m Example A SSAA – SINE RULE - side version We have a given angle and opposite side (35 and 16m), and the unknown x and the other given (71 ) also form a matching angle and opposite pair. (2 dp) Ans: the length of side x is 26.38 m approximately. Remember to check appropriateness of your answer!

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Example B SSSA - COSINE RULE – the angle version Ans: the size of angle is 44.4 Remember to check appropriateness of your answer! B 14 cm 10 cm 12 cm Let…. C = c = 10 a = 12 b = 14

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Example C SSAA – SINE RULE – side version We have a given angle and opposite side (29 and 12cm), but the unknown x and the other given (119 ) are NOT a matching angle and opposite pair. BUT…the third angle is 180 – 119 – 29 = 32 to two dec pl. Ans: the length of side x is 13.12 cm approximately. Remember to check appropriateness of your answer! 32 Let…. a = x A = 32 b = 12 B = 29

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Example D SSSA – COSINE RULE – side version Ans: the size of side x is 13.35 m (to 2 dec places) Remember to check appropriateness of your answer! Let…. C = 67 c = x a = 11 b = 13 D 67 x m 11 m 13 m c 2 = a 2 + b 2 – 2ab cos C x 2 = 11 2 + 13 2 – 2 × 11 × 13 × cos 67 x 2 = 178.251 x = 13.35

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Example E SSAA – SINE RULE – angle version We have a given angle and opposite side (80 and 9 cm), but the unknown and the other given (6 cm) are NOT a matching angle and opposite side. HOWEVER…we can use the SINE RULE to find the third angle (which forms a matching pair with the 6cm) then use the 180 rule to find Ans: the size of angle is approx. 58.96 Remember to check appropriateness of your answer! Let…. a = 6 A = b = 9 B = 80 E 80 9 cm 6 cm

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Example F F 33 9 cm x cm 12 cm We have a given angle and opposite side (33 and 9 cm), but the unknown x and the other given (12 cm) are insufficient data for Sine Rule. The Cosine Rule won’t work either as the triangle’s data does not match either of the two configurations for the Cosine Rule. HOWEVER…if we let be the angle opposite the 12cm we then have a second matching pair and can begin with using the SINE RULE to find angle . (This is PART 1 ) NOW FOR PART 2 …..Once we know we can then find the third angle (which is opposite to x) and then apply the Sine Rule a second time to find x. Part 1 (finding ) Finding = 180 – 33 – 46.57 = 100.43 Part 2 (finding x) Note!! Here the diagram is quite out of scale. This becomes apparent on checking the reasonableness of your answer

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