Area of triangle; sine and cosine rules

Welcome to Trigonometry Part 1: Non-Right-Angled Triangles

Hello! In your earlier studies, you likely mastered SOH CAH TOA and Pythagoras' Theorem. Those are fantastic tools, but they have one major limitation: they only work for right-angled triangles.

In this chapter, we are "unlocking" trigonometry for all triangles. Whether a triangle is skinny, fat, or tilted, the Sine Rule, Cosine Rule, and the Trig Area Formula will allow you to find every side and angle with confidence. These tools are vital for everything from architecture and satellite navigation to understanding the physics of forces.

1. The Area of a Triangle

You probably remember 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 find the area directly.

The Formula

The area of any triangle is given by:
\( \text{Area} = \frac{1}{2} ab \sin C \)

When to use it

Think of this as the SAS rule (Side-Angle-Side). You need:
1. Two side lengths (\(a\) and \(b\)).
2. The included angle (\(C\))—this is the angle tucked right between the two sides you know.

Analogy: Think of the two sides like the blades of a pair of scissors. To know how much "space" the scissors cover, you need to know the length of the blades and how wide you've opened them (the angle).

Step-by-Step Process:

1. Label your known sides as \(a\) and \(b\).
2. Identify the angle \(C\) trapped between them.
3. Plug the numbers into the formula: \( \frac{1}{2} \times a \times b \times \sin(C) \).
4. Check your calculator is in Degree mode (unless the question uses radians!).

Common Mistake: Using an angle that isn't "sandwiched" between the two sides. If the angle is elsewhere, you might need to use the Sine or Cosine rule first to find the correct angle!

Quick Takeaway: If you have a "side-angle-side" sandwich, you can find the area instantly with \( \frac{1}{2} ab \sin C \).

2. The Sine Rule

The Sine Rule is all about matching pairs. It links the ratio of a side length to the sine of 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: \( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \)

Memory Tip: Always put the thing you are trying to find on the top of the fraction. It makes the algebra much easier!

When to use it

Use the Sine Rule when you have a matching pair (a side and its opposite angle) plus one other piece of information.

Step-by-Step to find a side:

1. Identify your matching pair (e.g., you know side \(a\) and angle \(A\)).
2. Identify the angle opposite the side you want to find (e.g., angle \(B\) is opposite unknown side \(b\)).
3. Set up the equation: \( \frac{b}{\sin B} = \frac{a}{\sin A} \).
4. Multiply both sides by \( \sin B \) to get \( b \) on its own.

Did you know? The Sine Rule is used by surveyors to measure distances across rivers or between mountains without ever having to cross them!

Quick Takeaway: Look for "opposite pairs." If you have one full pair and half of another, the Sine Rule is your best friend.

3. The Cosine Rule

Don't worry if the Sine Rule doesn't fit—the Cosine Rule usually will! Think of the Cosine Rule as "Pythagoras 2.0." It’s a bit longer, but it’s incredibly powerful.

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

There are two specific scenarios for the Cosine Rule:
1. SAS (Side-Angle-Side): You have two sides and the angle between them, and you want the third side.
2. SSS (Side-Side-Side): You have all three sides and want to find an angle.

Analogy: The Cosine Rule is like a "correction" for Pythagoras. If the angle \(A\) is exactly \(90^\circ\), \( \cos 90^\circ = 0 \), so the whole end part (\( 2bc \cos A \)) vanishes, leaving you with \( a^2 = b^2 + c^2 \). It literally turns into Pythagoras' Theorem!

Common Mistake: Order of Operations

When calculating \( a^2 = b^2 + c^2 - 2bc \cos A \), many students accidentally do \( (b^2 + c^2 - 2bc) \times \cos A \). Don't do this! The \( 2bc \) is glued to the \( \cos A \). Treat it like this: \( (b^2 + c^2) - (2bc \cos A) \).

Quick Takeaway: No matching pairs? Use the Cosine Rule. Use the "Side" version for SAS and the "Angle" version for SSS.

4. Working with Bearings

In the MEI H640 syllabus, you are often asked to apply these rules to bearings. This is just a fancy way of giving you the angles in a real-world context.

Three Rules for Bearings:

1. They are always measured from North.
2. They are always measured clockwise.
3. They are always written with three digits (e.g., \(045^\circ\) instead of \(45^\circ\)).

Pro-Tip: When solving bearing problems, always draw a "North arrow" at every point (town, ship, or plane) in your diagram. This helps you find alternate or interior angles to use in your Sine or Cosine rules.

Summary: Which Tool Should I Pick?

Stuck at the start of a question? Follow this simple checklist:

1. Is it a Right-Angled Triangle? Use SOH CAH TOA or Pythagoras.
2. Do I have a matching "Side-Angle Pair"? Use the Sine Rule.
3. Do I have SAS (a side-angle-side sandwich)? Use the Cosine Rule (for the side) or the Area Formula (for the area).
4. Do I have all three sides (SSS)? Use the Cosine Rule (for the angle).

Don't worry if this seems tricky at first! The more triangles you solve, the faster you'll start "seeing" which rule to use before you even pick up your pen.