The Sine Rule and Cosine Rule are important tools for solving triangles that don't have a right angle. They each have their own uses, but they also work well together.

The Sine Rule

The Sine Rule says that the lengths of the sides of a triangle compared to the sines of their opposite angles are equal. You can write it like this:

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

This rule is really helpful when you know:

• Two angles and one side (called AAS or ASA)
• Two sides and a non-included angle (called SSA)

Example: In triangle ABC, if you know $A = 40°$, $B = 70°, and side a = 10$, you can use the Sine Rule to find side $b$.

The Cosine Rule

The Cosine Rule connects the sides of a triangle to the cosine of one of its angles. You can express it like this:

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

This rule is helpful for:

• Finding a side when you have two sides and the angle between them (called SAS)
• Finding an angle when you know all three sides (called SSS)

Example: If you have triangle ABC with sides $a = 5$, $b = 7$, and angle $C = 60°$, you can use the Cosine Rule to find side $c$.

Conclusion

In short, the Sine Rule is great for figuring out relationships between angles and sides. Meanwhile, the Cosine Rule is best when you need to combine angles and sides. By learning both rules, you can confidently solve all kinds of triangle problems.