Addition formulae

Introduction to Addition Formulae

Welcome to one of the most useful chapters in Trigonometry! Up until now, you have likely worked with "standard" angles like \(30^\circ\), \(45^\circ\), and \(60^\circ\). But what happens if you need to find the exact value of \(\sin(75^\circ)\) or \(\cos(15^\circ)\) without reaching for a calculator?

The Addition Formulae (sometimes called Compound Angle formulae) allow us to break down a complex angle into the sum or difference of two simpler angles. This is a vital skill for solving equations and simplifying expressions in Further Pure Mathematics.

The Core Formulae

The good news is that these formulae are provided on your official formula sheet during the exam. However, you need to be very comfortable using them. There are six main versions you need to know:

Sine Formulae

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

Cosine Formulae

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

Tangent Formulae

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

Did you know? Notice the sign change in the Cosine formula. When you add the angles \((A+B)\), you subtract the terms on the right side. When you subtract the angles \((A-B)\), you add the terms. It’s a common "trap" for students!

Finding Exact Values

One of the most common ways these formulae are tested is by asking for an exact value. This means your answer should involve surds (like \(\sqrt{2}\) or \(\sqrt{3}\)) rather than decimals.

Example: Find the exact value of \(\sin(75^\circ)\)

Step 1: Split \(75^\circ\) into two standard angles that we know.
\(75^\circ = 45^\circ + 30^\circ\)

Step 2: Choose the correct formula.
Use \(\sin(A + B) = \sin A \cos B + \cos A \sin B\), where \(A = 45^\circ\) and \(B = 30^\circ\).

Step 3: Substitute the values.
\(\sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ\)

Step 4: Use your knowledge of exact trig ratios (from Section 10B).
\(\sin 45^\circ = \frac{\sqrt{2}}{2}\), \(\cos 30^\circ = \frac{\sqrt{3}}{2}\), \(\cos 45^\circ = \frac{\sqrt{2}}{2}\), \(\sin 30^\circ = \frac{1}{2}\)

Step 5: Simplify.
\((\frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2} \times \frac{1}{2}) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}\)

Double Angle Formulae

While the syllabus focuses on the basic addition formulae, you will often need to deal with "Double Angles" like \(\sin 2A\) or \(\cos 2A\). These are simply the addition formulae where \(B = A\).

• For Sine: \(\sin(A + A) = \sin A \cos A + \cos A \sin A\)
  \(\implies \sin 2A = 2 \sin A \cos A\)
• For Cosine: \(\cos(A + A) = \cos A \cos A - \sin A \sin A\)
  \(\implies \cos 2A = \cos^2 A - \sin^2 A\)

Quick Tip: You can use the identity \(\cos^2 A + \sin^2 A = 1\) to rewrite \(\cos 2A\) in different ways:
1. \(\cos 2A = 2 \cos^2 A - 1\)
2. \(\cos 2A = 1 - 2 \sin^2 A\)

Common Mistakes to Avoid

1. The "Distributive" Fallacy:
Many students mistakenly think \(\sin(A + B) = \sin A + \sin B\). This is incorrect. Always use the full formula from the sheet!

2. Wrong Signs:
Double-check the signs for the Cosine and Tangent formulae. The formula sheet is your best friend here—look at it every time to be sure.

3. Radians vs. Degrees:
The question might use radians (e.g., \(\frac{\pi}{4}\) instead of \(45^\circ\)). Ensure your calculator is in the correct mode and you are using the correct exact values.

Key Takeaways

• Addition Formulae help us evaluate or simplify trigonometric expressions involving sums or differences of angles.
• Exact values (surds) are required for angles like \(15^\circ, 75^\circ, 105^\circ\), etc.
• The Formula Sheet contains the basic addition formulae—don't try to guess them!
• Double Angle identities are derived directly from the addition formulae.
• Always keep the identity \(\cos^2 \theta + \sin^2 \theta = 1\) in mind; it often helps simplify results after using an addition formula.

Don't worry if this seems like a lot of algebra at first! Most exam questions follow a predictable pattern: identify the angles, substitute into the formula, and simplify the fractions. With a little practice, these marks will become some of the most reliable ones you get in the exam.