The Sine Rule (HSC SSCE Mathematics Standard): Revision Notes

The Sine Rule

Introduction to non-right-angled triangles

The sine rule is a powerful tool that allows us to work with triangles that don't contain a right angle. When we have a non-right-angled triangle, we can still find unknown sides and angles using this special relationship.

Naming sides and angles in a triangle

In any triangle, we follow a standard naming convention. Each side is named with a lowercase letter that corresponds to the angle opposite to it.

What is the sine rule?

The sine rule describes a special relationship between the sides of a triangle and the sines of their opposite angles. We can express this relationship in words as: "Side $a$ divided by the sine of angle $A$ equals side $b$ divided by the sine of angle $B$, which also equals side $c$ divided by the sine of angle $C$."

Deriving the sine rule

We can prove the sine rule by constructing a perpendicular line. Let's draw a vertical line of length $h$ from vertex $C$ down to point $D$ on the base of the triangle.

In the right-angled triangle $\triangle BCD$, we can write:

$\sin B = \frac{h}{a}$

Therefore: $h = a \sin B$

Similarly, in the right-angled triangle $\triangle ACD$:

$\sin A = \frac{h}{b}$

Therefore: $h = b \sin A$

Since both expressions equal $h$, they must equal each other:

$a \sin B = b \sin A$

Dividing both sides by $\sin A \times \sin B$ gives us:

$\frac{a}{\sin A} = \frac{b}{\sin B}$

By constructing a perpendicular line from a different vertex, we can show in the same way that:

$\frac{a}{\sin A} = \frac{c}{\sin C}$

Combining these results gives us the complete sine rule.

The sine rule formula

The sine rule can be written in two different forms, depending on whether you need to find a side or an angle.

When to use the sine rule

You can use the sine rule in a non-right-angled triangle when you know:

• Two sides and an angle opposite one of those sides, or
• Two angles and one side

Finding an unknown side

Let's look at an example where we need to find an unknown side.

Finding an unknown angle

Now let's see how to find an unknown angle using the sine rule.

Exam tips

Remember!