Sine and cosine rules

Introduction: Beyond the Right Angle

Welcome! Up until now, you’ve likely used SOHCAHTOA to solve triangle problems. That’s a fantastic tool, but it only works for right-angled triangles. In this chapter, we are going to unlock the "superpowers" of trigonometry: the Sine Rule and the Cosine Rule. These rules allow you to find missing sides and angles in any triangle, no matter its shape!

Don’t worry if trigonometry has felt a bit like a maze in the past. We’ll break this down into simple steps, show you how to spot which rule to use, and point out the common traps to avoid.

1. The Golden Rule: Labeling Your Triangle

Before we touch a calculator, we must label our triangle correctly. If the labeling is wrong, the formulas won't work!

1. We use Uppercase letters (A, B, C) for the Angles.
2. We use lowercase letters (a, b, c) for the Sides.
3. The Match-up: Side \(a\) must be directly opposite Angle \(A\). Side \(b\) is opposite Angle \(B\), and side \(c\) is opposite Angle \(C\).

Analogy: Think of the angle as a flashlight. The side it shines its light on is its "partner" side. They share the same letter!

Quick Review: Always check your diagram. Is side \(a\) across from angle \(A\)? If yes, you’re ready to go!

2. The Sine Rule: The Rule of "Matching Pairs"

The Sine Rule is all about relationships between sides and their opposite angles. Use this when you have "matching pairs."

The Formula

To find a side, put the sides on top:
\( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

To find an angle, flip it over to make the algebra easier:
\( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \)

When do I use it?

You use the Sine Rule when you know one full pair (a side and its opposite angle) plus one other piece of information.

Step-by-Step: Finding a Side

1. Identify your "known pair" (e.g., side \(b\) and angle \(B\)).
2. Identify what you want to find (e.g., side \(a\)) and its opposite angle (\(A\)).
3. Set up the equation: \( \frac{a}{\sin A} = \frac{b}{\sin B} \).
4. Multiply both sides by \( \sin A \) to solve for \(a\).

The "Ambiguous Case" (Be Careful!)

Did you know? Sometimes, given two sides and a non-included angle (SSA), there are actually two possible triangles that could be drawn! This is called the ambiguous case.

When you use \( \sin^{-1} \) on your calculator to find an angle, it gives you an acute angle (less than 90°). However, there might be an obtuse angle (between 90° and 180°) that also works.

The Trick: To find the second possible angle, just do: \( 180^\circ - \text{your first answer} \).
Example: If your calculator says 40°, the other possibility is \( 180 - 40 = 140^\circ \). Check if 140° + your other known angle is less than 180°. If it is, you have two valid triangles!

Key Takeaway: Use the Sine Rule when you have pairs. If you are looking for an angle and have SSA, check if a second "obtuse" triangle is possible.

3. The Cosine Rule: The "Pythagoras Plus" Rule

If you don’t have enough pairs for the Sine Rule, the Cosine Rule is your best friend. It looks a bit like Pythagoras’ Theorem but with an extra bit at the end.

The Formula

To find a side:
\( a^2 = b^2 + c^2 - 2bc \cos A \)

To find an angle (rearranged):
\( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \)

When do I use it?

1. SAS (Side-Angle-Side): You know two sides and the angle between them.
2. SSS (Side-Side-Side): You know all three sides and want to find an angle.

Common Mistake to Avoid!

In the formula \( a^2 = b^2 + c^2 - 2bc \cos A \), many students accidentally calculate \( (b^2 + c^2 - 2bc) \) first and then multiply by \( \cos A \). Don't do this!
Remember BODMAS/BIDMAS: You must multiply \( 2 \times b \times c \times \cos A \) together before subtracting them from \( b^2 + c^2 \).

Key Takeaway: The Cosine Rule is for when the Sine Rule fails. Think of it as the "Sandwich Rule" (for SAS) or the "All-Sides Rule" (for SSS).

4. Area of a Triangle

Forget "half base times height" for a moment. In A-Level Maths, we often don't have the vertical height. Instead, we use trigonometry!

The Formula

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

In plain English: The area is half times one side times the other side times the sine of the angle between them.

Memory Aid: To use this, you need a "side-angle-side" sandwich. If you have the two bread slices (sides) and the filling (angle), you can find the area!

Quick Review: Always ensure the angle you use is the one "included" between the two sides you've picked.

5. Real-World Application: Bearings

OCR exam questions often mix these rules with bearings. If you see a question about a ship or a plane, it’s likely a trigonometry problem in disguise!

Bearing Basics:

1. Always measured from North.
2. Always measured clockwise.
3. Always written as three digits (e.g., 045° instead of 45°).

Top Tip: When dealing with bearings, draw North lines at every point on your diagram. This usually creates parallel line rules (like alternate or co-interior angles) that help you find the internal angles of your triangle.

Summary: Which Rule When?

Stuck on which rule to use? Follow this checklist:

1. Is it a right-angled triangle?
- Yes: Use SOHCAHTOA or Pythagoras.
- No: Move to step 2.

2. Do I have a "matching pair" (side and opposite angle)?
- Yes: Use the Sine Rule.
- No: Move to step 3.

3. Do I have SAS (two sides and the angle between) or SSS (all sides)?
- Yes: Use the Cosine Rule.

4. Am I looking for Area?
- Use \( \frac{1}{2} ab \sin C \).

Don't worry if this seems tricky at first! The most common errors are simply calculator mistakes. Always double-check that your calculator is in Degrees (D) mode, not Radians (R), unless the question specifically asks for radians!