Welcome to Compound Angle Formulae!

In your previous trigonometry work, you learned how to deal with single angles like \(\sin(\theta)\) or \(\tan(x)\). But what happens when we want to find the sine of two angles added together, like \(\sin(A + B)\)?

In this chapter, we are going to unlock the secrets of Compound Angle Formulae (sometimes called Addition Formulae). These are powerful tools that let us break down complex angles into simpler parts. Whether you are aiming for an A* or just trying to wrap your head around the basics, these notes will guide you through the "must-know" identities for your OCR MEI Mathematics B (H640) course.

Don't worry if this seems like a lot of symbols at first! Once you see the patterns, it becomes much easier—almost like learning the rules of a new game.

1. The Addition and Subtraction Formulae

The core of this chapter is six main formulae. These allow us to expand trigonometric functions of the sum or difference of two angles.

The Sine Formulae

\(\sin(A + B) = \sin A \cos B + \cos A \sin B\)
\(\sin(A - B) = \sin A \cos B - \cos A \sin B\)

Memory Aid: For Sine, the sign stays the Same (plus stays plus, minus stays minus). The terms "mix" together: Sin-Cos-Cos-Sin.

The Cosine Formulae

\(\cos(A + B) = \cos A \cos B - \sin A \sin B\)
\(\cos(A - B) = \cos A \cos B + \sin A \sin B\)

Memory Aid: For Cosine, the sign is the Contrary (plus becomes minus, minus becomes plus). The terms stay with their "friends": Cos-Cos and Sin-Sin.

The Tangent Formulae

\(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\)
\(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\)

Step-by-Step Example: Finding Exact Values
Suppose you need to find the exact value of \(\sin(75^\circ)\) without a calculator.
1. Identify: We know exact values for \(30^\circ, 45^\circ\), and \(60^\circ\).
2. Break it down: \(75^\circ = 45^\circ + 30^\circ\).
3. Apply: Use \(\sin(A + B) = \sin A \cos B + \cos A \sin B\).
4. Substitute: \(\sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ)\).
5. Calculate: \((\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2})(\frac{1}{2}) = \frac{\sqrt{6} + \sqrt{2}}{4}\).

Common Mistake to Avoid:
Never assume that \(\sin(A + B)\) is the same as \(\sin A + \sin B\). It’s a very common trap! If you ever forget, try it with numbers: \(\sin(30 + 30) = \sin(60) = 0.866\), but \(\sin 30 + \sin 30 = 0.5 + 0.5 = 1\). They are definitely not the same!

Key Takeaway: Compound angle formulae are used to break complex angles into "friendly" ones like \(30^\circ, 45^\circ,\) and \(60^\circ\).

2. Double Angle Formulae

What happens if the two angles are the same? If we let \(A = \theta\) and \(B = \theta\), we get the Double Angle Formulae.

Sine Double Angle

\(\sin(2\theta) = 2\sin\theta\cos\theta\)

Cosine Double Angle

This one is special because there are three ways to write it. They are all equal, but choosing the right one can save you a lot of work!
1. \(\cos(2\theta) = \cos^2\theta - \sin^2\theta\)
2. \(\cos(2\theta) = 2\cos^2\theta - 1\)
3. \(\cos(2\theta) = 1 - 2\sin^2\theta\)

Tangent Double Angle

\(\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}\)

Quick Review: Which Cosine version should I use?
- If your equation only has \(\cos\theta\), use the \(2\cos^2\theta - 1\) version.
- If your equation only has \(\sin\theta\), use the \(1 - 2\sin^2\theta\) version.
- If you are trying to cancel terms, look for the version that helps most!

Did you know?
The double angle formulae are used in computer graphics and video game programming to calculate how light reflects off curved surfaces at different angles!

Key Takeaway: Double angle formulae are just a specific case of compound angles where the two angles are identical.

3. The R-Formula Form: \(a\cos\theta \pm b\sin\theta\)

Sometimes you’ll see an expression like \(3\cos\theta + 4\sin\theta\). It’s hard to solve equations in this form because there are two different trig functions. The trick is to turn them into one single wave.

We can write \(a\cos\theta \pm b\sin\theta\) as:
\(R\cos(\theta \mp \alpha)\) or \(R\sin(\theta \pm \alpha)\)

How to find R and \(\alpha\):

1. Calculate \(R\): Use Pythagoras! \(R = \sqrt{a^2 + b^2}\). This is the maximum value (the height of the wave).
2. Calculate \(\alpha\): Use \(\tan \alpha = \frac{\text{coefficient of sine}}{\text{coefficient of cosine}}\) (usually). It's best to expand the formula you've chosen and compare coefficients to be sure.

Analogy: Imagine two people pushing a swing at different times. The "R" value tells you the total height the swing reaches, and "\(\alpha\)" tells you the delay (or shift) in the swing's timing compared to a standard wave.

Quick Review Box: Max and Min
The function \(f(\theta) = R\sin(\theta + \alpha)\) has:
- Maximum value: \(R\)
- Minimum value: \(-R\)
This is very helpful for finding the range of a function or the highest point on a graph.

Key Takeaway: The R-formula collapses two different trig terms into one, making equations much easier to solve.

4. Solving Equations and Proving Identities

Now that you have these tools, the exam will ask you to use them to solve equations or prove that one side of an expression equals another.

Top Tips for Proofs:

  • Start with the more complex side: It's usually easier to "break down" a big expression than to "build up" a small one.
  • Look for Double Angles: If you see a \(2\theta\), try expanding it immediately.
  • Convert everything to Sin and Cos: If you're stuck with \(\tan\), \(\sec\), or \(\cot\), turning them back to basics often reveals the path.
  • Watch your signs: Especially with \(\cos(A + B)\)—remember it becomes a minus sign!

Example Goal: Solve \(\sin 2\theta = \cos \theta\) for \(0 \le \theta \le 360^\circ\).
1. Use the double angle: \(2\sin\theta\cos\theta = \cos\theta\).
2. Don't divide by \(\cos\theta\)! Instead, move it to one side: \(2\sin\theta\cos\theta - \cos\theta = 0\).
3. Factorise: \(\cos\theta(2\sin\theta - 1) = 0\).
4. Set each part to zero: \(\cos\theta = 0\) or \(\sin\theta = 0.5\).
5. Find your angles!

Summary of Chapter 14:
Compound and double angle formulae are not just "more identities to learn"—they are the bridge that connects different parts of trigonometry. By mastering these expansions, you can handle almost any trigonometric equation or proof the OCR MEI H640 exam throws at you. Keep practicing the expansions, and they will become second nature!