Understanding Sine and Cosine Rules

Sine Rule

The sine rule establishes a relationship between the sides and angles of any triangle. It can be written in two equivalent ways:

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

Or alternatively:

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

This powerful rule is useful when you know either:

• Two angles and one side (AAS or ASA)
• Two sides and the angle opposite one of them (SSA)

💡 Think of the sine rule as matching pairs - each side with its opposite angle. The rule works for any triangle, not just right-angled ones!

Sine Rule Example

Let's see how to apply the sine rule to find an unknown side length:

Given a triangle with:

• A side length of 15 units
• The angle opposite this side is 65°
• Another angle in the triangle is 31°

We can rearrange the sine rule to find the unknown side x:

$\frac{15}{\sin 65°} = \frac{x}{\sin 31°}$

Multiplying both sides by $\sin 31°$:

$x = \frac{15 \times \sin 31°}{\sin 65°} = 26.40$

The unknown side length is 26.40 units.

Cosine Rule Example

The cosine rule is perfect when you have:

• Three sides and need an angle
• Two sides and the included angle (to find the third side)

In this example, we have a triangle with sides 15.5 cm, 21.8 cm, and 23.6 cm, and we need to find angle θ.

The cosine rule states: $\cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab}$

Substituting our values: $\cos(\theta) = \frac{15.5^2 + 23.6^2 - 21.8^2}{2 \times 15.5 \times 23.6} = \frac{321.97}{731.6} = 0.4400...$

Taking the inverse cosine: $\theta = \cos^{-1}(0.4400) = 63.89°$

🔍 Remember: The cosine rule is especially useful when you don't have enough information to use the sine rule directly!