Mastering Addition Formulae in Trigonometry

In A-Level Mathematics, you will frequently encounter situations where trigonometric functions involve the sum or difference of two angles, such as $\sin(A + B)$ or $\cos(A - B)$. A common misconception is that $\sin(A + B) = \sin A + \sin B$, but this is mathematically incorrect. To handle these expressions, we use the addition formulae (also known as compound angle formulae).

Mastering these identities is essential for your exams. They allow you to simplify complex expressions, prove trigonometric identities, and solve equations that would otherwise be impossible to tackle. This guide will walk you through the core formulae and show you how to apply them effectively.

The Sine Addition Formulae

The sine addition formulae allow us to expand the sine of a sum or difference of two angles. The identities are:

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

Note that for sine, the sign in the middle of the expansion matches the sign inside the brackets.

The Cosine Addition Formulae

The cosine addition formulae are slightly different. The identities are:

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

Crucially, notice that the sign in the expansion is the opposite of the sign inside the brackets. If you are adding angles, you subtract the product of the sines.

The Tangent Addition Formulae

While you can derive these by dividing the sine expansion by the cosine expansion, it is much faster to memorise the tangent identities:

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

Worked Example 1: Exact Values

Find the exact value of $\sin(75^\circ)$.

We can write $75^\circ$ as $(45^\circ + 30^\circ)$. Using the sine addition formula:

$$\sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$$

Substituting the exact values:

$$\sin(75^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)$$ $$\sin(75^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}$$

Worked Example 2: Proving Identities

Show that $\cos(x + 90^\circ) = -\sin x$.

Using the cosine addition formula $\cos(A + B) = \cos A \cos B - \sin A \sin B$:

$$\cos(x + 90^\circ) = \cos x \cos 90^\circ - \sin x \sin 90^\circ$$

Since $\cos 90^\circ = 0$ and $\sin 90^\circ = 1$:

$$\cos(x + 90^\circ) = \cos x (0) - \sin x (1) = -\sin x$$

Common Mistakes

1. The Sign Flip Error: Forgetting that the cosine addition formula uses a minus sign for a sum, and a plus sign for a difference. Always double-check your signs.
2. Distributive Fallacy: Assuming $\sin(A + B) = \sin A + \sin B$. This is a fundamental error that will lead to incorrect results in almost every part of the question.
3. Angle Confusion: Mixing up the angles $A$ and $B$ when substituting into the formula. Keep your variables clearly defined before you start the expansion.
4. Calculator Reliance: In many A-Level papers, you are required to show algebraic steps. Relying on a calculator to find values without showing the expansion will result in lost marks.

Frequently Asked Questions

Do I need to memorise these formulae? Most A-Level exam boards provide these in the formula booklet, but memorising them will significantly increase your speed and confidence during the exam.

How do these relate to double angle formulae? Double angle formulae are simply a special case of addition formulae where $A = B$. For example, $\sin(2A) = \sin(A + A) = \sin A \cos A + \cos A \sin A = 2\sin A \cos A$.

When should I use these in an equation? Use them whenever you have a mixture of compound angles and single angles, or when you need to simplify an expression to solve for $x$.

Conclusion

Addition formulae are a cornerstone of A-Level trigonometry. By understanding how to expand and manipulate these expressions, you gain the tools to solve complex problems with ease. To see these concepts in action with step-by-step visualisations, head over to MathInstructor AI and generate a free animated lesson on this topic today.