Introduction: Moving Beyond Right-Angled Triangles

Until now, you might have mainly used Pythagoras’ Theorem and SOH CAH TOA to solve triangle problems. These are fantastic tools, but they have one major limitation: they only work for right-angled triangles!

In this chapter, we are going to unlock the secrets of any triangle—whether it's acute, obtuse, or right-angled. These formulas are like the "Swiss Army Knife" of trigonometry. They are essential for navigating at sea, building bridges, and even calculating the heights of mountains.

Note: This chapter builds on your knowledge of "Sine, cosine and tangent up to 180 degrees". If you need a refresher on how the sine of an obtuse angle works, check out that chapter first!

Step 1: The "Secret Language" of Triangles

Before we look at the formulas, we must use a specific way of labeling triangles. If we don't, the formulas won't work!

  • We label the Angles with capital letters: \(A\), \(B\), and \(C\).
  • We label the Sides with lowercase letters: \(a\), \(b\), and \(c\).
  • The Rule: Side \(a\) must be exactly opposite Angle \(A\). Side \(b\) is opposite Angle \(B\), and side \(c\) is opposite Angle \(C\).

Quick Tip: Think of the angle as a flashlight; the side it shines its light on is its "partner" side with the same letter.

1. The Area of a Triangle

You probably know the old formula: \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\). But what if you don't know the vertical height?

If you know two sides and the included angle (the angle sandwiched between them), you can use this formula:

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

How to use it:
  1. Identify the two sides you know (let's call them \(a\) and \(b\)).
  2. Identify the angle between them (Angle \(C\)).
  3. Plug them into the formula!

Example: A triangle has sides of \(7 \text{ cm}\) and \(10 \text{ cm}\), and the angle between them is \(40^\circ\).
\(\text{Area} = \frac{1}{2} \times 7 \times 10 \times \sin(40^\circ) \approx 22.5 \text{ cm}^2\)

Key Takeaway: You need S-A-S (Side-Angle-Side) to find the area using this method.

2. The Sine Rule

The Sine Rule is used when we have "matching pairs" of sides and angles. It states that the ratio of a side to the sine of its opposite angle is constant for all three pairs.

The Formula:
\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)

When to use it:
  • Finding a Side: Use the version above when you know two angles and one side.
  • Finding an Angle: It is easier to flip the formula upside down: \(\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\). Use this when you know two sides and one opposite angle.

Don't worry if this seems tricky! Just remember: to use the Sine Rule, you must know at least one complete pair (an angle and its opposite side) and one other piece of information.

Common Mistake: Forgetting to use the opposite side. Always draw arrows from the angle to the side across from it to make sure you have the right pair!

3. The Cosine Rule

The Cosine Rule is a bit beefier than the Sine Rule. It's actually a more general version of Pythagoras' Theorem!

The Formula for a Side:
\(a^2 = b^2 + c^2 - 2bc \cos A\)

The Formula for an Angle (rearranged):
\(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\)

When to use it:
  • Case 1: Finding a Side. You know two sides and the angle between them (SAS).
  • Case 2: Finding an Angle. You know all three sides but no angles (SSS).

Step-by-Step Trick for the Angle Formula:
When finding Angle \(A\), the side you subtract (\(a^2\)) is always the side opposite the angle you are looking for. The other two sides (\(b\) and \(c\)) are the "arms" that make the angle.

Did you know? If Angle \(A\) is \(90^\circ\), then \(\cos(90^\circ) = 0\). This turns the Cosine Rule into \(a^2 = b^2 + c^2\), which is Pythagoras' Theorem! It all connects!

Summary: Which Rule Should I Use?

Choosing the right tool is half the battle. Use this checklist:

What do I know? What am I finding? Use this!
2 Sides and the Angle between them (SAS) Area \(\frac{1}{2} ab \sin C\)
1 Complete Pair (Angle & Side) + 1 other piece Side or Angle Sine Rule
2 Sides and the Angle between them (SAS) The 3rd Side Cosine Rule (Side version)
All 3 Sides (SSS) An Angle Cosine Rule (Angle version)

Top Tips for Exam Success

  • Calculator Mode: Always ensure your calculator is in DEG (Degrees) mode. If it’s in RAD or GRAD, your answers will be wrong!
  • Don't Round Too Early: Keep the full number in your calculator during multi-step problems. Only round your final answer (usually to 3 significant figures unless the question says otherwise).
  • Formula Sheet: In Specification B, these formulas are provided in the Appendix 4 Formulae sheet. You don't have to memorize them perfectly, but you must know how to label your triangle and plug in the numbers!
  • Check your answers: The longest side of a triangle is always opposite the largest angle. If your math says a \(20^\circ\) angle is opposite a \(50 \text{ cm}\) side while an \(80^\circ\) angle is opposite a \(10 \text{ cm}\) side, something has gone wrong!

Quick Review: Can you remember the SAS "sandwich" rule for Area? If you have two sides and the angle between them, you're ready to go!