CHAPTER:



CHAPTER: SINE AND COSINE RULES

Contents

1. The Sine Rule

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. C

The sine rule for the triangle is given by, b a

= =

A c B

Proof

In triangle ABC, CN is perpendicular to AB C

and AM is perpendicular to BC.

So, sin B = ( CN = CB sin B = a sin B

And sin A = ( CN = AC sin A = b sin A M

Thus, CN = a sin B = b sin A

( =

Similarly, AM = b sin C = c sin B A N B

( =

Hence, combining the two equations, we get = =

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. [AC = 9, BC = 7]

Solution

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.

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

( 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 y

The diagram shows triangle ABC on a coordinate plane C (b cos A, b sin A)

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 b a

= b2 cos2 A – 2bc cos A + c2 + b2 sin2 A

= b2 (cos2 A + sin2 A) + c2 – 2bc cos A

= b2 + c2 – 2bc cos 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

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

( 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, = =

Note: The sine rule is used to solve triangles when it is given:

two sides and a non-included angle, or,

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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