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+c2−2bccosA

$b^2=a^2+c^2-2ac\cos B$b2=a2+c2−2accosB

$c^2=a^2+b^2-2ab\cos C$c2=a2+b2−2abcosC

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+c2−2bccosA

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.

Round your answer to one decimal place.

QUESTION 2

Find the value of angle $w$w in degrees.

Round your answer to two decimal places.

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.

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
Which law should be used to start solving the triangle?
the law of sines
A
the law of cosines
B

Outcomes

10.T.IT.1

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0° and 90°. Values (with proofs) of the trigonometric ratios of 30°, 45° and 60°. Relationships between the ratios.

Solve problems involving either sine or cosine rule