Welcome to the World of Non-Right-Angled Triangles!
Up until now, you’ve probably spent a lot of time working with right-angled triangles using SOH CAH TOA. But what happens when the triangle doesn't have a nice 90° corner? Don't worry! In this chapter, we are going to learn three powerful tools—the Sine Rule, the Cosine Rule, and a new Area Formula—that allow you to solve any triangle that exists. Whether you're navigating a ship using bearings or designing a roof, these rules are your best friends.
1. Getting the Basics Right: Labeling
Before we look at the formulas, we must label our triangles correctly. If we don't, the math won't work!
- We use Capital Letters (\(A, B, C\)) for the angles.
- We use lowercase letters (\(a, b, c\)) for the sides.
- The Rule: Side \(a\) must be directly opposite angle \(A\). Side \(b\) is opposite angle \(B\), and side \(c\) is opposite angle \(C\).
Imagine an arrow shooting out from Angle A; the side it hits is lowercase a!
2. 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 angle between them, you can use this:
\( \text{Area} = \frac{1}{2} ab \sin C \)
When to use it?
Think of the "Sandwich Rule". You need two sides and the angle "sandwiched" right in the middle of them (this is often called SAS: Side-Angle-Side).
Example:
If side \(a = 8\text{cm}\), side \(b = 11\text{cm}\), and the angle between them \(C = 35^\circ\):
\( \text{Area} = \frac{1}{2} \times 8 \times 11 \times \sin(35^\circ) \approx 25.2\text{cm}^2 \)
Quick Review: No vertical height? No problem! Just make sure the angle you use is the one touching both sides you know.
3. The Sine Rule
The Sine Rule is all about Pairs. It creates a relationship between a side and its opposite angle.
The Formula:
To find a Side: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
To find an Angle: \( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \)
When to use it?
You use this when you have a "Known Pair" (a side and its opposite angle) and one other piece of information.
Analogy: It’s like a buddy system—every side needs its angle buddy!
Step-by-Step for Finding a Side:
- Identify your "complete pair" (e.g., you know side \(a\) and angle \(A\)).
- Identify your "incomplete pair" (e.g., you know angle \(B\) but want side \(b\)).
- Set up the equation: \( \frac{b}{\sin B} = \frac{a}{\sin A} \).
- Multiply both sides by \(\sin B\) to get \(b\) on its own.
Common Mistake: Make sure your calculator is in Degrees (DEG) mode, not Radians! If your answers look weirdly small, check the screen for a little 'D'.
4. The Cosine Rule
The Cosine Rule is the "heavy lifter." It’s slightly more complex, but it works when the Sine Rule fails.
The Formula:
To find a Side: \( a^2 = b^2 + c^2 - 2bc \cos A \)
To find an Angle: \( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \)
When to use it?
- SAS: You have two sides and the angle between them (and you want the third side).
- SSS: You have all three sides and you want to find an angle.
Memory Aid:
Does the first part look familiar? \( a^2 = b^2 + c^2 \) is just Pythagoras! The \( -2bc \cos A \) is just the "tail" we add because the triangle isn't right-angled.
Step-by-Step for Finding an Angle:
Don't worry if this seems tricky at first—the formula looks long, but it's just plugging in numbers! If you are finding Angle A, make sure you subtract side a (the opposite side) on the top of the fraction.
Key Takeaway: If you have a "Pair," use Sine. If you have a "Sandwich" (SAS) or all sides (SSS), use Cosine.
5. Bearings and Real-World Context
In your exams, these problems often involve Bearings. Bearings are just a fancy way of giving directions.
- Always measured from North.
- Always measured Clockwise.
- Always written with three digits (e.g., 045° instead of 45°).
Did you know? Pilots and sailors don't use "Left" or "Right" because it depends on which way you are facing. They use bearings because North is the same for everyone!
Strategy for Bearing Problems:
- Draw a North line at every point (A, B, C).
- Use Z-angles (alternate angles) or C-angles (co-interior angles) between the North lines to find missing internal angles of your triangle.
- Once you have the internal angles, use your Sine or Cosine rules as usual.
Quick Review Box: Which Rule do I need?
- Finding Area? Use \( \frac{1}{2} ab \sin C \).
- Have a matching pair of side/angle? Use Sine Rule.
- No pairs, but have SAS or SSS? Use Cosine Rule.
- Stuck on a bearing? Draw the North lines and look for "Z" shapes!
You've got this! Practice labeling your triangle first, and the formulas will start to fall into place.