35:the sine rule © christine crisp “teach a level maths” vol. 1: as core modules
TRANSCRIPT
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35:The Sine Rule35:The Sine Rule
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
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The Sine Rule
Triangles that aren’t Right Angled
To find unknown sides and angles in non-right angled triangles we can use one or both of 2 rules: • the sine
rule• the cosine rule
The next few slides prove the sine rule. The cosine rule is on the next presentation.
You do not need to learn the proof.
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The Sine Rule
a, b and c are the sides opposite angles A, B and C
A B
C
b a
c
ABC is a scalene triangle
The Sine Rule
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The Sine Rule
A B
C Draw the perpendicular, h, from C to BA.
N
h b a
c
In ,ACNΔ
ABC is a scalene triangle
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The Sine Rule
A
h b a
c
Asin
b a
c
C
B
In ,ACNΔ
ABC is a scalene triangle
N
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The Sine Rule
A B
h b a
c
C In ,ACNΔ
hAb sin
ABC is a scalene triangle
In ,BCNΔ
N
Asinb
h
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The Sine Rule
Bsin
A B N
h b a
c A B N
h b a
c
C In ,ACNΔ
In ,BCNΔ
ABC is a scalene triangle
hAb sin
Asinb
h
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The Sine Rule
hBa sin
In ,ACNΔ
In ,BCNΔ
ABC is a scalene triangle
hAb sin
A B N
h b a
c A B
h b a
c
C
a
hBsin
Asinb
h
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The Sine Rule
BaAb sinsin
so, Ab sin Ba sinandh h
A B
C
hb a
c
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The Sine Rule
BaAb sinsin
A B
C
b a
c B
b
A
a
sinsin
so, Ab sin Ba sinandh h
angle) oppositesin(
side
angle) oppositesin(
side
b
B
a
A sinsin
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The Sine Rule
. . . can be turned so that BC is the base. CbBch sinsin
C
A B
b a
c
A
B C
b
a
c
The triangle ABC . . .
We would then get
hC
c
B
b
sinsin
c
C
b
B sinsin
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The Sine Rule
c
C
b
B sinsinSo,
b
B
a
A sinsin
c
C
b
B sinsin
c
C
b
B
a
A sinsinsin
We now have
So,
and
C
c
B
b
A
a
sinsinsin
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The Sine Rule
The sine rule can be used in a triangle when we know• One side and its opposite angle,
plus• One more side or angle
sinsin
P Q
q p
Tip: We need one complete “pair” to use the sine rule.
e.g. Suppose we know p, q and angle Q in triangle PQR
The angle or side that we can find is the one that completes another pair.
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The Sine Rule
sinsin
A B
b a Solution:
Usesin
sin AB a
b 12
62sin10sin A
We don’t need the 3rd part of the rule
7358.0sin A
e.g. 1 In the triangle ABC, find the size of angles A and C.
A B
C
12 10
62
67044762180 C
447A (3 s.f.)
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The Sine Rule
Solution: As the unknown is a side, we “flip” the sine rule over. The unknown side is then
at the “top”. z
sin
sin Z y
Y
29sin
55sin13 z 022 z ( 3 s.f.)
sinsin
z y
Z Y
e.g. 2 In the triangle XYZ, find the length XY.
Y Z
X
13
5529
z
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The Sine RuleSUMMARY The sine rule can be used in a triangle
when we know• One side and its opposite angle, plus• One more side or angle
If 2 sides and 1 angle are known we use:
If 1 side and 2 angles are known we use:
We write the sine rule so that the unknown angle or side is on the left of the equation
B
b
A sinsina
b
B
a
sinsinA
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The Sine RuleExercises
1. In triangle ABC, b = cm, c = cm and angle C = . Find the size of angles A and B.
1102463
Solution:
sinsin
B C
c b
24
110sin63sin
B
753 B 316 A
2. In triangle PQR, PQ = 23 cm, angle R = and angle P = . Find the size of side QR.
4217
BA
C
a 63
24
110
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The Sine Rule
42sin
17sin23 p
010 p ( 3 s.f.)
2. In triangle PQR, PQ = 23 cm, angle R = and angle P = . Find the size of QR.
4217
sinsin
p r
P R
Solution:
23P Q
R
p
42
17
Exercises
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The Sine Rule
The following may be left out if time is an issue
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The Sine Rule
e.g. In a triangle PQR, p = 5 cm, r = 7.2 cm and angle P = .
37
Drawing side r and angle P, we have:
P Q7.2
37
R1
5 This is one possible complete triangle.
If an unknown angle is opposite the longest side, 2 triangles may be possible: one will have an angle greater than90
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The Sine Rule
This is the other.
P Q
R1
7.2
5
37
5
R2
e.g. In a triangle PQR, p = 5 cm, r = 7.2 cm and angle P = .
37
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The Sine Rule
This is the other.
P Q
R1
7.2
5
37
5
R2
The 2 possible values of R are connected since
x
xy
xR 1
• Triangle is isosceles
21 RQR
180 yx•
e.g. In a triangle PQR, p = 5 cm, r = 7.2 cm and angle P = .
37
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The Sine Rule
This is the other. The 2 possible values of R are connected since
xR 1
• Triangle is isosceles
21 RQR
180 yx•
e.g. In a triangle PQR, p = 5 cm, r = 7.2 cm and angle P = .
37
P Q
R1
7.2
5
37
5
R2
x
xy180so, 2R1R
The calculator will give the acute angle ( < ).We subtract from to find the other possibility.180
90
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The Sine Rule
e.g. In a triangle PQR, p = 5 cm, r = 7.2 cm andangle P = . Find 2 possible values of angle R and the corresponding values of angle Q. Give the answers correct to the nearest degree.
37
Solution:
sinsin
R P p r
5
37sin27sin
R
p
PrR
sinsin
86660sin R601 R or
120601802 R8360371801 Q23120371802 Q
601 R1202 R
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The Sine Rule
or:
We have either:
60R 83Qand
120R 23Qand
120
23
PQ27.2
37
5
R2
P
83
60
Q1
R1
7.2
5
37
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The Sine RuleSUMMARY
If the sine rule is used to find the angle opposite the longest side of a triangle, 2 values may be possible.
Use each value to find 2 possible values for the 3rd angle.
Use the sine rule and a calculator to find 1 value. This will be an acute angle ( less than ).
90
Subtract from to find the other possibility.
180
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The Sine RuleExercise
Solution:
sinsin
C B b c
91420sin C12
47sin15sin C
96647166180 A
119479113180 A
166C9113 C
Find 2 possible values of angle ACB in triangle ABC if AB = 15 cm, AC = 12 cm and angle B = . Sketch the triangles obtained.
47
9113166 CC or
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The Sine Rule
966A 119A
AB = 15 cm, AC = 12 cm, angle B = 47
A B15
166
C
12
47966
9113
A B15
119
C12
47
166C(i)
9113 C(ii)
Exercise
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The Sine Rule
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The Sine Rule
The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
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The Sine Rule
c
C
b
B
a
A sinsinsin
a, b and c are the sides opposite angles A, B and C
A B
C
b a
c
ABC is a scalene triangle
The Sine Rule
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The Sine Rule
The sine rule can be used in a triangle when we know• One side and its opposite angle,
plus• One more side or angle
If 2 sides and 1 angle are known ( a, b and B ) we use:
If 1 side and 2 angles are known ( A, b and B ) we use:
We write the sine rule so that the unknown angle or side is on the left of the equation
B
b
A sinsina
b
B
a
sinsinA
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The Sine Rule
We don’t need the 3rd part of the rule
sinsin
A B
b a Solution:
Use
sinsin A
B a
b 12
62sin10sin A
7358.0sin A
67044762180 C
447A (3 s.f.)
e.g. 1 In the triangle ABC, find the size of angles A and C.
A B
C
12 10
62
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The Sine Rule
Solution: As the unknown is a side, we “flip” the sine rule over. The unknown side is then
at the “top”.
zsin
sin Z y
Y
29sin
55sin13 z 022 z ( 3 s.f.)
sinsin
z y
Z Y
e.g. 2 In the triangle XYZ, find the length XY.
Y Z
X
13
5529
z
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The Sine RuleIf an unknown angle is opposite the longest side, 2 triangles may be possible: one will have an angle greater than90
e.g. In a triangle PQR, p = 5 cm, r = 7.2 cm andangle P = . Find 2 possible values of angle R and the corresponding values of angle Q. Give the answers correct to the nearest degree.
37
Solution:
sinsin
R P p r
5
37sin27sin
R
p
PrR
sinsin
86660sin R601 R or
120601802 R8360371801 Q23120371802 Q
601 R1202 R