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Finding the length of sides and size of angles in non right-angled triangles when the Sine rule won’t do the job! The non right-angled triangles we have worked with so far, have all had one thing in common. 110° 35° x m 12 m Find x. Q. What can you say about the information (sides and angles) given in the triangles for all Sine rule questions???? 25° 28 m 15 m Find . A. Each two pieces of information (sides and angles) have ALWAYS been opposite each other. So, what happens here? B Let’s try to use the Sine rule! 120° a b? c = = Sin A Sin B Sin C 18 m 25 m C Find x. x m A We know these parts of the triangle And we’re asked to find this. You can’t use the Sine rule! Does it work finding an angle? B Sin A Sin B Sin C ? = a = b c 35 m Or here? It won’t work here either! We need another rule! 50 m Find . A 22 m C Here’s how to derive the next rule: C b A In BCD, by Pythagoras, a h x D a2 = h2 + (c - x)2 = h2 + c2 - 2cx + x2 = (h2 + x2) + c2 - 2cx c - x c If we say this part of c is x, then B then this part must be.. 1 But in ACD, by Pythagoras b2 = h2 + x2 2 And in ACD, by normal Trig Cos A = x b i.e. x = b Cos A So, we have 3 equations to use……. 3 a2 = (h2 + x2) + c2 - 2cx C a b A xD b2 = h2 + x2 c - x c B x = b Cos A 1 2 3 Use # 2 to replace h2 + x2 with…. b2 So, a2 = (h2 + x2) + c2 - 2cx becomes a2 = b2 + c2 - 2cx And use # 3 to replace x with…. b Cos A So, the finished rulea2 = b2 + c2 - 2cb Cos A This is known as the Cosine rule :a2 = b2 + c2 - 2cb Cos A a2 = b2 + c2 - 2bc Cos A More simply written as... a2 = b2 + c2 - 2bcCos A This rule allows you to find a. (But only after you have found the square root of a2). What happens if you are asked to find b or c? There is a pattern in the way the rule is written………... What is it? The three sides are written first, all squared, minus twice the second two sides from the rule, times the So, Cos of the side you’re finding……. and b2 = a2 + c2 - 2acCos B c2 = a2 + b2 - 2abCos A Now, to use it…………... C Step 1. Work out what you are trying to find within the rule, by labelling the triangle. 16m Step 2. Write the correct rule A In this instance we need the rule to find c. c2 = a2 + b2 - 2abCos A 45° 20 m x B Find x, to the nearest metre Step 3. Substitute then evaluate Solution: c2 = 20 2 + 162 - 2 20 16 Cos 45 c2 400 + 256 - 640 0.7071…. c2 656 - 452.54834…. c2 203.45166…. c 203.45166..... Don’t forget to do this - many do! c 14.26m c 14 m Now, to find an angle……. B 55m A Step 1. Work out what you are trying to find within the 27 m rule, by labelling the triangle. C 75m Find , to the nearest metre Here, we’re after B, but our rules only allow us to find sides! Write the rule to find side b then change the subject. b2 = a2 + c2 - 2acCos B Here the CosB is with the -2ac, so its negative. It will become positive if we move everything after the minus sign to the right 2acCos B + b2 = a2 + c2 2acCos B = a2 + c2 - b2 a2 + c2 - b2 Cos B = 2ac Move the b2 to the other side by…. subtracting it, so Now, move the 2ac to the other side by…. dividing it, so In future you either do all these steps at the start of the question or you remember the Cos rule in its angle form as well (I suggest the latter) So what’s the pattern to be able to write any Cos rule for an angle? a2 + c2 - b2 Cos B = 2ac First, write the Cos of the angle you’re finding,then equal to the other 2 sides each squared, minus the side related to the angle squared, all divided by twice the other 2 sides So…. b2 + c2 - a2 Cos A = 2bc And…. a2 + b2 - c2 Cos C = 2ab Now to use it…. B 55m A 27 m We were trying to find B C 75m Find , to the nearest minute Cos B = 27 2 + 552 - 752 Cos B 2 27 55 -1871 2970 -0.6299…. B B 129.04….° 129° 3’ Cos B = Now use your calculator with the INV key to find the size of angle A. Your turn…… Set 6I Page 273 of Coroneus