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GCSE: Non-right angled triangles

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

2 RECAP: Right-Angled Triangles
We’ve previously been able to deal with right-angled triangles, to find the area, or missing sides and angles. 5 6 3 4 Area = 15 ? 30.96Β° ? 5 5 3 ? Using Pythagoras: π‘₯= βˆ’ =3 Using 𝑨= 𝟏 𝟐 𝒃𝒉: π΄π‘Ÿπ‘’π‘Ž= 1 2 Γ—6Γ— =15 Using trigonometry: tan πœƒ = 3 5 πœƒ= tan βˆ’ =30.96Β°

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

4 Labelling Sides of Non-Right Angle Triangles
Right-Angled Triangles: Non-Right-Angled Triangles: π‘Ž β„Ž 𝐢 ? π‘œ 𝑏 𝐡 ? ? 𝐴 π‘Ž 𝑐 We label the sides π‘Ž, 𝑏, 𝑐 and their corresponding OPPOSITE angles 𝐴, 𝐡, 𝐢

5 OVERVIEW: Finding missing sides and angles
You have You want Use #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 side Cosine rule #3 All three sides An angle #4 Two sides known and a missing side not opposite known angle Sine rule twice

6 The Sine Rule 5.02 10 9.10 ! Sine Rule: π‘Ž sin 𝐴 = 𝑏 sin 𝐡 = 𝑐 sin 𝐢 ?
c C b B a A For this triangle, try calculating each side divided by the sin of its opposite angle. What do you notice in all three cases? 65Β° 5.02 10 85Β° ! Sine Rule: π‘Ž sin 𝐴 = 𝑏 sin 𝐡 = 𝑐 sin 𝐢 ? 30Β° 9.10 You have You want Use #1: Two angle-side opposite pairs Missing angle or side in one pair Sine rule

7 Examples 8 Q2 Q1 8 50Β° 85Β° 100Β° 15.76 ? 45Β° 30Β° 11.27 ? 𝒙 𝐬𝐒𝐧 πŸ–πŸ“ = πŸ– 𝐬𝐒𝐧 πŸ’πŸ“ 𝒙= πŸ– 𝐬𝐒𝐧 πŸ–πŸ“ 𝐬𝐒𝐧 πŸ’πŸ“ =𝟏𝟏.πŸπŸ• 𝒙 𝐬𝐒𝐧 𝟏𝟎𝟎 = πŸ– 𝐬𝐒𝐧 πŸ‘πŸŽ 𝒙= πŸ– 𝐬𝐒𝐧 𝟏𝟎𝟎 𝐬𝐒𝐧 πŸ‘πŸŽ =πŸπŸ“.πŸ•πŸ” You have You want Use #1: Two angle-side opposite pairs Missing angle or side in one pair Sine rule

8 Examples When you have a missing angle, it’s better to β€˜flip’ your formula to get 𝐬𝐒𝐧 𝑨 𝒂 = 𝐬𝐒𝐧 𝑩 𝒃 i.e. in general put the missing value in the numerator. 5 Q3 Q4 8 126Β° 85Β° 40.33Β° ? 10 ? 56.11Β° 6 sin πœƒ 5 = sin sin πœƒ = 5 sin πœƒ= sin βˆ’ sin =56.11Β° sin πœƒ 8 = sin 126Β° sin πœƒ = 8 sin πœƒ= sin βˆ’ sin =40.33Β°

9 Test Your Understanding
𝑄 82Β° 𝑃 20Β° 10π‘š πœƒ 85Β° 5π‘π‘š 12π‘š 𝑅 Determine the angle πœƒ. sin 𝜽 𝟏𝟎 = sin πŸ–πŸ 𝟏𝟐 𝜽=π’”π’Š 𝒏 βˆ’πŸ 𝟏𝟎 sin πŸ–πŸ 𝟏𝟐 =πŸ“πŸ“.πŸ”Β° Determine the length 𝑃𝑅. 𝑷𝑹 𝐬𝐒𝐧 𝟐𝟎 = πŸ“ 𝐬𝐒𝐧 πŸ–πŸ“ 𝑷𝑹= πŸ“ 𝐬𝐒𝐧 𝟐𝟎 𝐬𝐒𝐧 πŸ–πŸ“ =𝟏.πŸ•πŸπ’„π’Ž ? ?

10 Exercise 1 Find the missing angle or side. Please copy the diagram first! Give answers to 3sf. Q1 Q2 Q3 15 10 16 85Β° 12 𝑦 30Β° π‘₯ 30Β° 40Β° π‘₯ 20 π‘₯=53.1Β° ? 𝑦=56.4Β° ? π‘₯=23.2 ? π‘₯ Q4 Q6 Q5 70Β° 35Β° 10 40Β° 5 10 𝛼 20 𝛼=16.7Β° ? π‘₯=5.32 ? π‘₯ π‘₯=6.84 ?

11 Cosine Rule How are sides labelled ? Calculation? Cosine Rule: 15
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. π‘Ž 𝐴 𝑏 𝑐 Cosine Rule: π‘Ž 2 = 𝑏 2 + 𝑐 2 βˆ’2𝑏𝑐 cos 𝐴 15 The only angle in formula is 𝐴, so label angle in diagram 𝐴, label opposite side π‘Ž, and so on (𝑏 and 𝑐 can go either way). π‘₯ 2 = βˆ’ 2Γ—15Γ—12Γ— cos π‘₯ 2 = … π‘₯=22.83 How are sides labelled ? 115Β° π‘₯ Calculation? 12

12 Sin or Cosine Rule? οƒΌ Sine  Cosine  Sine Cosine οƒΌ  Sine Cosine οƒΌ
If you were given these exam questions, which would you use? π‘₯ 10 π‘₯ 10 70Β° 70Β° 15 15 οƒΌ Sine  Cosine  Sine Cosine οƒΌ 10 10 𝛼 7 𝛼 70Β° 15 12  Sine Cosine οƒΌ Sine οƒΌ Cosine 

13 Test Your Understanding
e.g. 1 e.g. 2 π‘₯ π‘₯ 4 7 8 47Β° 106.4Β° 7 π‘₯=6.05 ? π‘₯=8.99 ? You have You want Use Two sides known and a missing side opposite a known angle Remaining side Cosine rule

14 Exercise 2 Use the cosine rule to determine the missing angle/side. Quickly copy out the diagram first. Q1 Q2 Q3 135Β° 58 5 5 8 π‘₯ 100Β° 60Β° 70 π‘₯ 7 𝑦 π‘₯=6.24 ? 𝑦=10.14 ? π‘₯=50.22 ? Q6 Q5 π‘₯ Q4 6 4 5 75Β° π‘₯ 10 43Β° 65Β° 8 6 3 π‘₯=4.398 ? π‘₯ 3 π‘₯=9.513 ? π‘₯=6.2966 ?

15 Dealing with Missing Angles
You have You want Use All three sides An angle Cosine rule 𝒂 𝟐 = 𝒃 𝟐 + 𝒄 𝟐 βˆ’πŸπ’ƒπ’„ 𝐜𝐨𝐬 𝑨 7 𝛼 Label sides then substitute into formula. 4 πŸ’ 𝟐 = πŸ• 𝟐 + πŸ— 𝟐 βˆ’ πŸΓ—πŸ•Γ—πŸ—Γ— 𝐜𝐨𝐬 𝜢 πŸπŸ”=πŸπŸ‘πŸŽβˆ’πŸπŸπŸ” 𝐜𝐨𝐬 𝜢 πŸπŸπŸ” 𝐜𝐨𝐬 𝜢 =πŸπŸ‘πŸŽβˆ’πŸπŸ” 𝐜𝐨𝐬 𝜢 = πŸπŸπŸ’ πŸπŸπŸ” 𝜢= 𝐜𝐨𝐬 βˆ’πŸ πŸπŸπŸ’ πŸπŸπŸ” =πŸπŸ“.𝟐° ? 9 ? Simplify each bit of formula. ? Rearrange (I use β€˜swapsie’ trick to swap thing you’re subtracting and result) ? 𝛼=25.2Β° ?

16 Test Your Understanding
4π‘π‘š 8 7π‘π‘š 5 πœƒ πœƒ 9π‘π‘š 7 ? 4 2 = βˆ’ 2Γ—7Γ—9Γ— cos πœƒ 16=130βˆ’126 cos πœƒ 114=126 cos πœƒ cos πœƒ = πœƒ= cos βˆ’ =πŸπŸ“.𝟐𝟏° ? 8 2 = βˆ’ 2Γ—7Γ—5Γ— cos πœƒ 64=74βˆ’70 cos πœƒ 10=70 cos πœƒ cos πœƒ = πœƒ= cos βˆ’ =πŸ–πŸ.πŸ•πŸ—Β°

17 Exercise 3 1 2 3 12 5.2 7 6 𝛽 πœƒ 11 5 πœƒ 13.2 6 8 𝛽=92.5Β° ? πœƒ=71.4Β° ? πœƒ=111.1Β° ?

18 Using sine rule twice ? 4 32Β° 3 π‘₯
You have You want Use #4 Two sides known and a missing side not opposite known angle Remaining side Sine rule twice Given there is just one angle involved, you might attempt to use the cosine rule: πŸ‘ 𝟐 = 𝒙 𝟐 + πŸ’ 𝟐 βˆ’ πŸΓ—π’™Γ—πŸ’Γ— 𝐜𝐨𝐬 πŸ‘πŸ πŸ—= 𝒙 𝟐 +πŸπŸ”βˆ’πŸ–π’™ 𝐜𝐨𝐬 πŸ‘πŸ 4 ? 32Β° 3 π‘₯ This is a quadratic equation! It’s possible to solve this using the quadratic formula (using 𝒂=𝟏, 𝒃=βˆ’πŸ– 𝐜𝐨𝐬 πŸ‘πŸ , 𝒄=πŸ•). However, this is a bit fiddly and not the primary method expected in the exam…

19 ! Using sine rule twice ? ? ? 4 32Β° 3 π‘₯
You have You want Use #4 Two sides known and a missing side not opposite known angle Remaining side Sine rule twice ! 2: Which means we would then know this angle. 4 πŸπŸ–πŸŽβˆ’πŸ‘πŸβˆ’πŸ’πŸ’.πŸ—πŸ“πŸ“πŸ”=πŸπŸŽπŸ‘.πŸŽπŸ’πŸ’πŸ’ ? 1: We could use the sine rule to find this angle. 32Β° 3 3: Using the sine rule a second time allows us to find π‘₯ ? sin 𝐴 4 = sin 32 3 𝐴= Β° π‘₯ π‘₯ sin = 3 sin 32 π‘₯=5.52 π‘‘π‘œ 3𝑠𝑓 ?

20 Test Your Understanding
9 ? 𝑦=6.97 𝑦 61Β° 10 4 3 53Β° 𝑦=5.01 ? 𝑦

21 Where C is the angle wedged between two sides a and b.
Area of Non Right-Angled Triangles 3cm Area = 0.5 x 3 x 7 x sin(59) = 9.00cm2 ? 59Β° 7cm ! Area = 1 2 π‘Ž 𝑏 sin 𝐢 Where C is the angle wedged between two sides a and b.

22 5 5 5 Test Your Understanding ? ? 𝐴= 1 2 Γ—6.97Γ—10×𝑠𝑖𝑛61 =30.48 9 6.97
𝐴= 1 2 Γ—6.97Γ—10×𝑠𝑖𝑛 =30.48 ? 9 6.97 61Β° 10 5 5 𝐴= 1 2 Γ—5Γ—5Γ— sin = ? 5

23 Harder Examples 6 7 8 ? ? Q1 (Edexcel June 2014) Q2
Finding angle ∠𝐴𝐢𝐷: sin 𝐷 9 = sin 𝐷= Β° ∠𝐴𝐢𝐷=180βˆ’100βˆ’ = Β° π΄π‘Ÿπ‘’π‘Ž π‘œπ‘“ Ξ”= 1 2 Γ—9Γ—11Γ— sin =21.945 Area of 𝐴𝐡𝐢𝐷=2Γ—21.945=43.9 π‘‘π‘œ 3𝑠𝑓 ? Using cosine rule to find angle opposite 8: cos πœƒ = πœƒ= Β° π΄π‘Ÿπ‘’π‘Ž= 1 2 Γ—6Γ—7Γ— sin 75.5…=𝟐𝟎.πŸ‘

24 Exercise 4 Q3 Q1 Q2 Q4 5 3.6 3 1 1 5 100Β° 3.8 75Β° 8 1 5.2 Area = 7.39 ? 70Β° π΄π‘Ÿπ‘’π‘Ž= =0.433 ? Area = 9.04 ? Area = 8.03 ? 2cm Q5 Q7 Q6 110Β° 8.7π‘π‘š 3cm Area = 3.11𝑐 π‘š 2 ? 49Β° 64Β° Q8 4.2m 3m π΄π‘Ÿπ‘’π‘Ž=29.25𝑐 π‘š 2 ? 𝑃 is the midpoint of 𝐴𝐡 and 𝑄 the midpoint of 𝐴𝐢. 𝐴𝑃𝑄 is a sector of a circle. Find the shaded area. 5.3m ? 1 2 Γ— 6 2 Γ— sin 60 βˆ’ 1 6 πœ‹ =10.9𝑐 π‘š 2 Area = 6.29 π‘š 2 ?

25 Segment Area 𝐴 𝑂𝐴𝐡 is a sector of a circle, centred at 𝑂. Determine the area of the shaded segment. 10π‘π‘š 𝑂 70Β° π΄π‘Ÿπ‘’π‘Ž π‘œπ‘“ π‘ π‘’π‘π‘‘π‘œπ‘Ÿ= Γ— πœ‹Γ— = 𝑐 π‘š 2 π΄π‘Ÿπ‘’π‘Ž π‘œπ‘“ π‘‘π‘Ÿπ‘–π‘Žπ‘›π‘”π‘™π‘’= 1 2 Γ— 10 2 Γ— sin 70 = 𝑐 π‘š 2 π΄π‘Ÿπ‘’π‘Ž π‘œπ‘“ π‘ π‘’π‘”π‘šπ‘’π‘›π‘‘= βˆ’ =14.1 π‘‘π‘œ 3𝑠𝑓 ? ? ? 𝐡

26 Test Your Understanding
𝐴=119π‘š2 ? 𝐴=3πœ‹βˆ’9 ?

27 Exercise 5 - Mixed Exercises
90π‘š Q4 Q1 Q2 Q3 𝑧 27 30Β° 130Β° 11 80Β° 8 𝑦 𝛼 60π‘š 40Β° 70Β° 18 π‘₯ 𝛼=17.79Β° 𝑧=26.67 π΄π‘Ÿπ‘’π‘Ž=73.33 ? 10 π‘ƒπ‘’π‘Ÿπ‘–π‘šπ‘’π‘‘π‘’π‘Ÿ =286.5π‘š π‘₯=41.37 b) π΄π‘Ÿπ‘’π‘Ž=483.63 ? ? 𝑦=10.45 π΄π‘Ÿπ‘’π‘Ž=37.59 ? ? ? ? ? Q5 4.6 Q7 Q8 15 πœƒ 7 5 6π‘π‘š 61Β° Q6 52Β° 12 π‘₯ 𝑄𝑅=12.6π‘π‘š ? πœƒ=122.8Β° ? π΄π‘Ÿπ‘’π‘Ž=2.15𝑐 π‘š 2 ? π‘₯=7.89 π΄π‘Ÿπ‘’π‘Ž=17.25


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