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GCSE: Non-right angled triangles Dr J Frost Last modified: 16 th November 2014

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RECAP: Right-Angled Triangles We’ve previously been able to deal with right-angled triangles, to find the area, or missing sides and angles Area = ° ? ? ?

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Learning Objectives There’s 2 things you’ll need to be able to do with non-right angle triangles: 59° 3cm 7cm 1.How would I find the missing side and angle? 2.How do I find the area of this non-right angled triangle? ? ? We’ll revisit this later...

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Labelling Sides of Non-Right Angle Triangles Right-Angled Triangles:Non-Right-Angled Triangles: ? ? ?

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OVERVIEW: Finding missing sides and angles You haveYou wantUse #1: Two angle-side opposite pairs Missing angle or side in one pair Sine rule #2 Two sides known and a missing side opposite a known angle Remaining sideCosine rule #3 All three sides An angleCosine rule #4 Two sides known and a missing side not opposite known angle Remaining sideSine rule twice

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The Sine Rule 65° 85° 30° For this triangle, try calculating each side divided by the sin of its opposite angle. What do you notice in all three cases? c C b B a A ? You haveYou wantUse #1: Two angle-side opposite pairs Missing angle or side in one pair Sine rule

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Examples 45° ° ? Q1 You haveYou wantUse #1: Two angle-side opposite pairs Missing angle or side in one pair Sine rule 100° ° ? Q2 50°

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Examples 85° ° ? Q ° ° ? Q4

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Test Your Understanding ? ?

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Exercise 1 Find the missing angle or side. Please copy the diagram first! Give answers to 3sf. Q1 ? Q2 ? Q3 ? ? Q4 Q5 ? Q6 ?

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Cosine Rule The sine rule could be used whenever we had two pairs of sides and opposite angles involved. However, sometimes there may only be one angle involved. We then use something called the cosine rule. How are sides labelled ? Calculation?

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Sin or Cosine Rule? If you were given these exam questions, which would you use? Sine Cosine SineCosine Sine Cosine SineCosine

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Test Your Understanding e.g. 1 e.g. 2 ? ? You haveYou wantUse Two sides known and a missing side opposite a known angle Remaining sideCosine rule

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Exercise 2 Use the cosine rule to determine the missing angle/side. Quickly copy out the diagram first. ? ? ? ? Q1Q2 Q3 Q4 Q5 ? ? Q6

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Dealing with Missing Angles ? Label sides then substitute into formula. Simplify each bit of formula. Rearrange (I use ‘swapsie’ trick to swap thing you’re subtracting and result) ? ? ? ? You haveYou wantUse All three sidesAn angleCosine rule

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Test Your Understanding ??

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? ? ? Exercise 3 123

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Using sine rule twice You haveYou wantUse #4 Two sides known and a missing side not opposite known angle Remaining sideSine rule twice ?

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Using sine rule twice You haveYou wantUse #4 Two sides known and a missing side not opposite known angle Remaining sideSine rule twice 1: We could use the sine rule to find this angle. 2: Which means we would then know this angle. ? ? ?

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Test Your Understanding ? ?

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Area of Non Right-Angled Triangles 59° 3cm 7cm Area = 0.5 x 3 x 7 x sin(59) = 9.00cm 2 ?

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Test Your Understanding ? ?

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Harder Examples Q1 (Edexcel June 2014) ? ? Q2

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Exercise 4 Q5 100° Q1 Area = 7.39 ? Q2 ? Q3 75° Area = 9.04 ? Q4 Area = ° ? Q6 ? ? Q7 3cm 2cm ? Q8 3m 4.2m 5.3m ?

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Segment Area ? ? ?

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? Test Your Understanding ?

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Exercise 5 - Mixed Exercises Q1 ? ? ? Q2 ? Q3 ? ? ? ? ? Q4 Q5 Q6 ? ? Q7 Q8

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