GCSE: Non-Right Angled Triangles
Dr J Frost ([email protected])www.drfrostmaths.com
Last modified: 31st August 2015
RECAP: Right-Angled TrianglesWeโve previously been able to deal with right-angled triangles, to find the area, or missing sides and angles.
54
3
6
5
Area = 15
5
3
30.96ยฐ
?
? ?
Using Pythagoras: Using : Using trigonometry:
Labelling Sides of Non-Right Angle Triangles
Right-Angled Triangles:
h๐
๐
Non-Right-Angled Triangles:
๐๐
๐
๐ถ
๐ด๐ต?
?
?
We label the sides and their corresponding OPPOSITE angles
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 Cosine rule
#4 Two sides known and a missing side not opposite known angle
Remaining side Sine rule twice
The Sine Rule
65ยฐ
85ยฐ
30ยฐ
105.02
9.10
For this triangle, try calculating each side divided by the sin of its opposite angle. What do you notice in all three cases?
! Sine Rule:
c
C
b
B
a
A
?
You have You want Use#1: Two angle-side opposite pairs
Missing angle or side in one pair
Sine rule
Examples
45ยฐ
8
11.27
85ยฐ
?
Q1
You have You want Use#1: Two angle-side opposite pairs
Missing angle or side in one pair
Sine rule
100ยฐ
8
15.7630ยฐ ?
Q2
50ยฐ
sin๐5 =
sin 856
Examples
85ยฐ
6
5
56.11ยฐ?
Q3 8
When you have a missing angle, itโs better to โflipโ your formula to get
i.e. in general put the missing value in the numerator.
40.33ยฐ
10
126ยฐ?
Q4
sin๐8 =
sin 126 ยฐ10
Test Your Understanding
๐๐
๐
85 ยฐ20 ยฐ
5๐๐
Determine the length .
82 ยฐ๐ 10๐
12๐
Determine the angle .
? ?
Exercise 1Find the missing angle or side. Please copy the diagram first! Give answers to 3sf.
Q1
85 ยฐ
๐ฅ
15
40 ยฐ
๐ฅ=23.2?
Q2
๐ฅ 30 ยฐ
๐ฅ=53.1 ยฐ?
1610Q3
30 ยฐ
๐ฆ=56.4 ยฐ?
๐ฆ12
40 ยฐ
๐ฅ
10
๐ฅ=6.84?
Q4
20
Q5
๐ผ=16.7 ยฐ?
๐ผ20
1035 ยฐQ6
๐ฅ=5.32?
70 ยฐ๐ฅ
5
Cosine RuleThe 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.
15
12
115ยฐ๐ฅ
Cosine Rule:
๐๐ด
๐
๐
The only angle in formula is , so label angle in diagram , label opposite side , and so on ( and can go either way).
How are sides labelled ?
Calculation?
Sin or Cosine Rule?If you were given these exam questions, which would you use?
Sine Cosine Sine Cosine
Sine Cosine
10
15
๐ฅ70 ยฐ
10
15
๐ฅ
70 ยฐ
10
15
7๐ผ
Sine Cosine
10
70 ยฐ12
๐ผ
Test Your Understanding
๐ฅ
7 8
e.g. 1
๐ฅ=6.05
47 ยฐ
e.g. 2
7
4106.4 ยฐ
๐ฅ
๐ฅ=8.99? ?
You have You want UseTwo sides known and a missing side opposite a known angle
Remaining side Cosine rule
Exercise 2
Use the cosine rule to determine the missing angle/side. Quickly copy out the diagram first.
5 ๐ฅ
7
60 ยฐ
๐ฅ=6.24?
100 ยฐ5 8
๐ฆ๐ฆ=10.14?
135 ยฐ58
70๐ฅ
๐ฅ=50.22?
6๐ฅ
643 ยฐ
๐ฅ=4.398?
Q1 Q2 Q3
Q4 Q5
10
3
8
๐ฅ
65 ยฐ
๐ฅ=9.513? ๐ฅ 3
54 75 ยฐ
๐ฅ=6.2966?
Q6
Dealing with Missing Angles
7
49
๐ผ
๐ผ=25.2 ยฐ? ๐๐=๐๐+๐๐โ (๐ร๐ร๐ร๐๐จ๐ฌ๐ถ )
Label sides then substitute into formula.
Simplify each bit of formula.
Rearrange (I use โswapsieโ trick to swap thing youโre subtracting and result)
? ? ?
?
๐๐=๐๐+๐๐โ๐๐๐๐๐จ๐ฌ ๐จ
You have You want UseAll three sides An angle Cosine rule
Test Your Understanding
๐5
82=72+52โ (2ร7ร5ร cos๐ )?
8
7๐7๐๐
4๐๐
9๐๐
42=72+92โ (2ร7ร9ร cos๐ )?
๐
76
6
๐=71.4 ยฐ?
12
513.2
๐ฝ
๐ฝ=92.5 ยฐ?
5.211
8
๐
๐=111.1 ยฐ?
Exercise 3
1 2 3
Using sine rule twiceYou have You want Use#4 Two sides known and a missing side not opposite known angle
Remaining side Sine rule twice
4
๐ฅ
3 32 ยฐ
Given there is just one angle involved, you might attempt to use the cosine rule:
This is a quadratic equation!Itโs possible to solve this using the quadratic formula (using ). However, this is a bit fiddly and not the primary method expected in the examโฆ
?
Using sine rule twiceYou have You want Use#4 Two sides known and a missing side not opposite known angle
Remaining side Sine rule twice
4
๐ฅ
3 32 ยฐ
๐๐๐โ๐๐โ๐๐ .๐๐๐๐=๐๐๐ .๐๐๐๐
1: We could use the sine rule to find this angle.
2: Which means we would then know this angle.
3: Using the sine rule a second time allows us to find
๐ฅsin 103.0444=
3sin 32
!
?
?
?
Test Your Understanding
61 ยฐ
53 ยฐ
10
9
๐ฆ
34
๐ฆ
๐ฆ=6.97
๐ฆ=5.01
?
?
Area of Non Right-Angled Triangles
Area = Where C is the angle wedged between two sides a and b.
59ยฐ
3cm
7cm
Area = 0.5 x 3 x 7 x sin(59) = 9.00cm2
!
?
Test Your Understanding
61 ยฐ 10
9
6.97
5 5
5
๐ด=12ร6.97ร10ร๐ ๐๐61
๐ด=12ร5ร5รsin 60
?
?
Harder Examples
Q1 (Edexcel June 2014)
Finding angle :
Area of
67
8Using cosine rule to find angle opposite 8:
? ?
Q2
Exercise 4
64 ยฐ 49 ยฐ
8 .7๐๐Q5
5100ยฐ
Q1
Area = 7.39? 8
3Q2
๐ด๐๐๐=โ34
=0.433?
1 1
1 5.2
3.63.8
Q3
75ยฐ
Area = 9.04?
Q4
Area = 8.03
5
70ยฐ
? Q6
๐ด๐๐๐=29.25๐๐2 is the midpoint of and the midpoint of . is a sector of a circle. Find the shaded area.
( 12ร62รsin 60)โ 16 ๐ (32 )=10.9๐๐2
?
?
Q7
3cm
2cm
110ยฐ
Area = ? Q8
3m
4.2m
5.3m
Area = ?
Segment Area
๐
๐ด
๐ต
70 ยฐ
10๐๐ is a sector of a circle, centred at .Determine the area of the shaded segment.
? ?
?
๐ด=3๐ โ9?
Test Your Understanding
๐ด=119๐2?
Exercise 5 - Mixed ExercisesQ1
8 0ยฐ
๐ฅ
27
40 ยฐ
b) ? ?
8 ๐ฆ
10
70 ยฐ
๐ฆ=10.45?
Q2
?
Q3
๐ผ=17.79 ยฐ?
๐ผ18
1130 ยฐ๐ง
? ?
๐๐ =12.6๐๐?
130 ยฐ90๐
60๐
๐๐๐๐๐๐๐ก๐๐?
Q4
Q5
Q6
4.615
12
๐
๐=122.8 ยฐ?
6๐๐52 ยฐ
๐ด๐๐๐=2.15๐๐2?
Q7
61 ยฐ57
๐ฅ
๐ฅ=7.89
Q8
? ?
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