36: The Cosine Rule36: The Cosine Rule
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
The Cosine Rule
Module C2
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The Cosine Rule
The cosine rule is used to find sides and angles of a scalene triangle when
• 2 sides and the angle formed by them are known, or
• all 3 sides are known
In both these cases, we don’t know a pair of side and opposite angle so the sine rule cannot be used.
We will now prove the cosine rule but you do not need to learn the proof.
The Cosine Rule
In the triangle ABC, draw the perpendicular, h, from C to AB.
Proof of the Cosine Rule
N
h
Let AN = x. Then, NB = c x.
x c - x Using Pythagoras’ theorem:In triangle
ANC,In triangle BNC,So, )2()( 2222 xcaxb
From triangle ANC,
)1(cos Abxb
xA cos
A B
C
b a
c
h2 22 xb
h2 22 )( xca
The Cosine Rule
)2( 22222 xcxcaxb
222 cos2 aAbccb
Acbcab cos2222
)2()( 2222 xcaxbWe have
Simplifying:22222 2 xcxcaxb
cxcab 2222
Abccba cos2222
)cos( Abx Substituting for x from equation (1),Rearranging:
Proof of the Cosine Rule
The Cosine Rule
• The letters can be switched to find any side provided it is opposite the given angle.
Abccba cos2222
The Cosine Rule for triangle ABC
• We use this arrangement when 2 sides and the angle formed by them are known.
The Cosine Rule
cos222 bccb 2a A
• The letters can be switched to find any side provided it is opposite the given angle.
The Cosine Rule for triangle ABC
• We use this arrangement when 2 sides and the angle formed by them are known.
• If we want to find an angle, we use the sine rule after we have used the cosine rule.
The Cosine Rule
Cbaabc cos2222 19
e.g. Find side c and angle B in the triangle ABC
A
B C
15 c
30
30cos)19)(15(21915 222 c
( 3 s.f.)619c
Solution: Use the Cosine rule
The Sine rule: c
C
b
B sinsin
619
30sin15sin
B
351B ( 3 s.f.)
Tip: Do the whole calculation in one go on
your calculator. It avoids errors!
Tip: Leave the answer on your calculator as it will be needed to find angle B
a
b
The Cosine RuleExercise
Pqrrqp cos2222
7
1. Find p in the triangle PQR
P R
Q
6
p
120
120cos)6)(7(267 222 p
( 3 s.f.)311p
Solution:
The Cosine Rule
. . . belongs to the side opposite the angle we are finding
The 2nd form of the Cosine Rule
Abccba cos2222
222cos2 acbAbc
We know that
Rearranging,
bc
acbA
2cos
222
We use this form to find any angle of a triangle when we know all 3 sides.
The minus sign . . .
The Cosine Rule
The Cosine Rule
Xcos
Solution: Use the Cosine Rule
6
e.g. 1 Find angle X in triangle XYZ
Y
Z 8
4
X
029X
8750cos X)6)(8(2
468 222
yz
xzyX
2cos
222
The Cosine Rule
bc
acbA
2cos
222 Acos
Solution: Let’s find A first
6e.g. 2 Find all the
angles in triangle ABC B
A 9
5
C
938A77780cos A)5)(9(2
659 222
We can now use the Cosine rule again or switch to the Sine rule. If we use the Sine rule, we must avoid the largest angle ( opposite the longest side ) as we don’t know whether it is less than or greater than . 90
The Cosine Rule
OR: Using the Sine rule for C :631C
6
938sin5sin
C
EITHER: Using the Cosine rule for B or C: 5109 B
)6)(5(2
965cos
222 B 3330cos B
a
A
c
C sinsin
e.g.
6
B
A 9
5
C
938A
6319385109180 C Then
5109938631180 B Then
The Cosine Rule
The Cosine Rule
SUMMARY
Abccba cos2222
• In a triangle that isn’t right angled, if we know 2 sides and the angle formed by the 2 sides, we use
• If we know 3 sides, we use
bc
acbA
2cos
222
to find the 3rd side.
to find any angle.
The Cosine RuleExercise
7
1. Find all the angles in the triangle XYZ giving the answers to 1 decimal place.
Y Z
X
4 9
)9)(4(2
794cos
222 X 248X
Solution: Use the Cosine rule for any angle. e.g.
yz
xzyX
2cos
222
)9)(7(2
497cos
222 Y 225Y
(1 d. p.)6106248225180 Zxz
yzxY
2cos
222
The Cosine Rule
The Cosine 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.
The Cosine Rule
The cosine rule is used to find sides and angles of a scalene triangle when
• 2 sides and the angle formed by them are known, or
• all 3 sides are known
In both these cases, we don’t know a side and its opposite angle so the sine rule cannot be used.
The Cosine Rule
• The letters can be switched to find any side provided it is opposite the given angle.
Abccba cos2222
The Cosine Rule for triangle ABC
• We use this arrangement when 2 sides and the angle formed by them are known.
• If we want to find an angle, we use the sine rule after we have used the cosine rule.
The Cosine Rule
Cbaabc cos2222
19
e.g. Find side c and angle B in the triangle ABC
A
B C
15 c
30
30cos)19)(15(21915 222 c
( 3 s.f.)619c
Solution: Use the Cosine rule
The Sine rule: c
C
b
B sinsin
619
30sin15sin
B
351B ( 3 s.f.)
Tip: Do the whole calculation in one go on your calculator and leave your answer so it can be used to find B.
a
b
The Cosine Rule
The 2nd form of the Cosine Rule
Abccba cos2222
222cos2 acbAbc
We know that
Rearranging,
bc
acbA
2cos
222
We use this form to find any angle of a triangle when we know all 3 sides.
The minus sign goes with the side opposite the angle we are finding.
The Cosine Rule
bc
acbA
2cos
222 Acos
Solution: Let’s find A first
6e.g. 2 Find all the
angles in triangle ABC B
A 9
5
C
938A77780cos A)5)(9(2
659 222
We can now use the Cosine rule again or switch to the Sine rule. If we use the Sine rule, we must avoid the largest angle ( opposite the longest side ) as we don’t know whether it is less than or greater than . 90
The Cosine Rule
OR: Using the Sine rule for C :631C
6
938sin5sin
C
EITHER: Using the Cosine rule for B or C: 5109 B
)6)(5(2
965cos
222 B 3330cos B
a
A
c
C sinsin
e.g.
6
B
A 9
5
C
938A
6319385109180 C Then
5109938631180 B Then
The Cosine Rule
The Cosine Rule
SUMMARY
Abccba cos2222
• In a triangle that isn’t right angled, if we know 2 sides and the angle formed by the 2 sides, we use
• If we know 3 sides, we use
bc
acbA
2cos
222
to find the 3rd side.
to find any angle.