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1
CHAPTER:
SINE AND COSINE RULES
Contents
1
The Sine Rule
1.1 The Ambiguous Case
2
The Cosine Rule
3
Miscellaneous Examples
Introduction
The sine and cosine rules are used to solve non-right-angled triangles, i.e. finding the unknown sides and
angles.
 1 The Sine Rule
Consider a triangle with angles A, B and C, and sides a, b and c as shown.
The sine rule for the triangle is given by,
a
b
c
=
=
sin A sin B sin C
C
b
a
A
Proof
In triangle ABC, CN is perpendicular to AB
and AM is perpendicular to BC.
CN
So,
sin B =
 CN = CB sin B = a sin B
CB
CN
And
sin A =
 CN = AC sin A = b sin A
AC
Thus,
CN = a sin B = b sin A
a
b

=
sin A sin B
Similarly, AM = b sin C = c sin B
b
c

=
sin B sin C
a
b
c
Hence, combining the two equations, we get
=
=
sin A sin B sin C
c
B
C
M
A
N
Note: The sine rule is used to solve triangles when it is given:
a) two sides and a non-included angle, or,
b) two angles and one side.
Example 1.1
Solve the triangle ABC in which  A = 400 ,  B = 650 and side AB = 10.
Solution
[AC = 9, BC = 7]
B
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Example 1.2 (Additional Maths Vol 1)
ABC is a triangle in which  A = 420 and AB = 25 cm.  C varies such that 300   C  1100 .
Calculate the maximum and minimum lengths of AC. [max = 47.6 cm, min = 12.5 cm]
Solution
 1.1 The Sine Rule : Ambiguous case
The ambiguous case occurs when 2 sides and 1 angle of a triangle are given with the angle opposite the shorter
side. There is no ambiguity if the given angle is obtuse or the side opposite the given angle is greater than the
other side. In the ambiguous case, there will be two distinct triangles satisfying the given data with the result
that we get two sets of answers.
Note :
When there are two possible values for an angle obtained by using sine rule, we can find out which value is
correct by:
a) using a scale drawing of the triangle
b) verifying a known side or angle using the two possible values
c) using the rule that the largest angle is opposite the longest side and the smallest angle is opposite the shortest
side.
In some cases, both answers may be correct, depending on the question.
Since there is sometimes an ambiguous case when using the sine rule, always try to use cosine rule whenever
possible.
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Example 1.3 (Additional Maths Vol 1)
In triangle ABC,  A = 300 , a = 3 cm and b = 5 cm. Find the possible values of AB,  B and  C.
[6.0 cm, 56.40, 93.60 or 2.7 cm, 123.60 , 26.40 ]
Solution
Example 1.4 (Additional Maths Vol 1)
ABC is a triangle in which  C = 720 , AC = 7 cm , AB = 12 cm. Find  A,  B and BC.
[74.30 , 33.70, 12.1 cm]
Solution
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 2 The Cosine Rule
The cosine rule of a triangle ABC is given by
a2 = b2 + c2 – 2bc cos A
b2 = a2 + c2 – 2ac cos B
c2 = a2 + b2 – 2ab cos C
Proof
The diagram shows triangle ABC on a coordinate plane
with AB on the positive x-axis and A at the origin.
Then point B  (c, 0) and point C  (b cos A, b sin A).
Hence to find length of BC, i.e. a, we have
a2 = (b cos A – c)2 + (b sin A – 0)2
= b2 cos2 A – 2bc cos A + c2 + b2 sin2 A
= b2 (cos2 A + sin2 A) + c2 – 2bc cos A
= b2 + c2 – 2bc cos A
y
C (b cos A, b sin A)
b
a
x
A
c
B (c, 0)
By placing B at the origin, we would get
b2 = a2 + c2 – 2ac cos B
By placing C at the origin, we would get
c2 = a2 + b2 – 2ab cos C
Note: The cosine rule is used to solve a triangle when it is given
a) three sides, or
b) two sides and the included angle.
Example 2.1
In triangle ABC, a = 6.5cm, b = 5 cm and c = 7 cm. Calculate all the angles of the triangle. [630 ,430, 740 ]
Solution
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Example 2.2
Given that ABC is a triangle in which a = 15 cm, b = 10 cm and c = 7 cm, find angle A. [1230 ]
Solution
Example 2.3
ABC is a triangle in which a = 13 cm, b = 8 cm and c = 12 cm. Calculate the largest and smallest angles.
[largest  A = 780 , smallest  B = 370 ]
Solution
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 3 Miscellaneous Examples
Example 3.1 (SAJC 01/1/7)
In triangle ABC,  ABC is 400, AB is 10 and AC is 8. Find the two possible angles for  BAC.
Assuming that  ABC and AB remains unchanged, what is the range of values that AC can take such that there
will be two possible values for  BAC? [86.50 or 13.50, 6.43 < AC < 10]
Solution
SUMMARY (Sine and Cosine Rule)
The sine rule for the triangle is given by,
a
b
c
=
=
sin A sin B sin C
Note: The sine rule is used to solve triangles when it is given:
a) two sides and a non-included angle, or,
b) two angles and one side.
The Sine Rule : Ambiguous case
The ambiguous case occurs when 2 sides and 1 angle of a triangle are given with the angle opposite the shorter
side. There is no ambiguity if the given angle is obtuse or the side opposite the given angle is greater than the
other side. In the ambiguous case, there will be two distinct triangles satisfying the given data with the result
that we get two sets of answers.
Note :
When there are two possible values for an angle obtained by using sine rule, we can find out which value is
correct by:
a) using a scale drawing of the triangle
b) verifying a known side or angle using the two possible values
c) using the rule that the largest angle is opposite the longest side and the smallest angle is opposite the shortest
side.
Since there is sometimes an ambiguous case when using the sine rule, always try to use cosine rule whenever
possible.
The cosine rule of a triangle ABC is given by
a2 = b2 + c2 – 2bc cos A
b2 = a2 + c2 – 2ac cos B
c2 = a2 + b2 – 2ab cos C
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