 Hong Kong
Stage 4 - Stage 5

# Sine and Cosine Rule

Lesson

We have already seen the Sine Rule, and Cosine Rule in action.

## Sine Rule

The sine rule (or the law of sines) states that the ratio of the sine of any angle to the length of the side opposite that angle is the same for all three angles of a triangle

Sine Rule

$\frac{\sin A}{a}$sinAa$\frac{\sin B}{b}=\frac{\sin C}{c}$sinBb=sinCc

It can be used, for example, when two angles and a side are known, such as in triangulation - a process for finding the location of a point.

## Cosine Rule

In trigonometry the cosine rule relates the lengths of the sides and the cosine of one of its angles.

The Law of Cosines is useful in finding:

• the third side of a triangle when you know two sides and the angle between them
• the angles of a triangle when you know all three sides ABC is a triangle with side lengths $BC=a$BC=a , $CA=b$CA=b and $AB=c$AB=c and the opposite angles of the sides are respectively angle $A$A, angle $B$B and angle $C$C.

Law of cosines

$a^2=b^2+c^2-2bc\cos A$a2=b2+c22bccosA

$b^2=a^2+c^2-2ac\cos B$b2=a2+c22accosB

$c^2=a^2+b^2-2ab\cos C$c2=a2+b22abcosC

Notice that Pythagoras' Theorem $a^2=b^2+c^2$a2=b2+c2 makes an appearance in the Cosine Rule: $a^2=b^2+c^2-2bc\cos A$a2=b2+c22bccosA

The questions in this set will require you to choose whether you are needing to use the Sine Rule or the Cosine Rule, (or indeed other aspects of trigonometry).

When to use which?

Use the Sine Rule

• When two angles and a side are known and you wish to find the other side
• When two sides and an angle are known and you wish to find the other angle

Use the Cosine Rule

• When you want to find the third side of a triangle when you know two sides and the angle between them
• When you want to find the angles of a triangle when you know all three sides

#### Worked Examples

##### QUESTION 1

Calculate the length of $y$y in metres. ##### QUESTION 2

Find the value of angle $w$w in degrees. ##### QUESTION 3

$\triangle ABC$ABC consists of angles $A$A, $B$B and $C$C which appear opposite sides $a$a, $b$b and $c$c respectively. Consider the case where the measures of $a$a, $c$c and $A$A are given.

1. Which of the following is given?

$SSA$SSA: Two sides and an angle

A

$SAS$SAS: Two sides and the included angle

B

$SAA$SAA: two angles and a side

C

$ASA$ASA: two angles and the side between them

D

$SSS$SSS: Three sides

E
2. Which law should be used to start solving the triangle?

the law of sines

A

the law of cosines

B