Sine Rule and Area Rule

 

The Sine Rule

The sine rule allows us to find missing sides or angles in any non-right-angled triangle.

For any triangle:

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

\(a\}, \(b\), and $c$ are sides of the triangle

$A$, $B$, and $C$ are their opposite angles

 

 

When to Use It

You have a pair of an angle and its opposite side

You want to find a missing side or angle that also has a known opposite pair

 

Finding a Missing Side

To find side $a$:

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

Finding a Missing Angle

To find angle $A$:

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

Then use:

$$A = \sin^{-1}\left(\frac{a \cdot \sin B}{b}\right)$$

 

Ambiguous Case Warning

In some situations (especially when given two sides and an angle not between them), the triangle could be drawn in two different ways:

One angle is acute, the other is obtuse

Your calculator only returns the acute angle

Use: $\text{Obtuse angle} = 180\degree - \text{acute angle}$

 

Example: Using the Sine Rule

In triangle XYZ:

$XY = 10.5 cm$

$YZ = 14.2 cm$

Angle $XZY = 31\degree$

 

 

Step 1: Find angle X

Use:

$$\frac{\sin X}{14.2} = \frac{\sin 31\degree}{10.5}  \\ \sin X = \frac{14.2 \cdot \sin(31)}{10.5} \approx 0.6993 \\  X = \sin^{-1}(0.6993) \approx 44.4\degree$$

Step 2: Find angle Y

$$Y = 180\degree - 31\degree- 44.4\degree = 104.6\degree$$

 

Area of a Triangle Using Sine

To find the area of any triangle (not just right-angled), use:

$$\text{Area} = \frac{1}{2} ab \sin C$$

 

 

Where:

• $a$ and $b$ are two sides
• $C$ is the included angle

When to Use This Formula

You know two sides and the angle between them

It avoids having to use height directly

 

Worked Example: Area of a Triangle

In triangle DEF:

$DE = 5.2 m$

$EF = 7.1 m$

$\angle D = 64\degree$

 

 

Calculate the Area

$$\text{Area} = \frac{1}{2} \cdot 5.2 \cdot 7.1 \cdot \sin(64^\circ)$$

$$\text{Area} \approx 0.5 \cdot 5.2 \cdot 7.1 \cdot 0.8988 \approx 16.6 \text{ m}^2 \text{ (3 s.f.)}$$