# O Level A Maths Standard Formula Sheet

GCE O Level Additional Mathematics Formula Sheet (Credits: SEAB)

This is the standard formula sheet given to you in any Additional Mathematics (A Maths) exams.

I strongly urge you to be familiar with the formulas provided. Tips about the usage of the formulas can be found below.

Tip

You should have a hard/soft copy readily available.

I used to have a copy in my file all the time. Here's a printable PDF version.

# 1. Algebra

## Quadratic Equation

This is the quadratic formula to find the roots/solutions of a quadratic equation:

$$\text{For the equation }ax^2 + bx + c = 0,$$

$$x={-b\pm \sqrt{b^2-4ac} \over 2a}$$

Notice the discriminant, b2 - 4ac, that appears within the square root.

Recall that it tells us about the "nature" of the roots of a quadratic equation, i.e. two real and distinct roots when b2 - 4ac > 0.

## Binomial expansion

$$(a+b)^n=a^n+\binom{n}{1}a^{n-1}b+\binom{n}{2}a^{n-2}b^2+...+\binom{n}{r}a^{n-r}b^r+ \ldots + b^n,$$

$$\text{where n is a positive integer and }\binom{n}{r}={n!\over r!(n-r)!}={n(n-1) \ldots (n-r+1)\over r!}$$

The general term or the (r+1)th term, can be found from the formula. It is actually:

$$T_{r+1} = \binom{n}{r}a^{n-r}b^r$$

# 2. Trigonometry

## Identities

These are the three Trigonometric identities that are commonly used:

$$\sin^2 A+\cos^2 A = 1$$

$$\sec^2 A = 1 + \tan^2 A$$

$$\text{cosec}^2 A = 1 + \cot^2 A$$

Sometimes you are required to manipulate the identity to suit the question. For example, we may need to make sin2 A the subject:

\begin{align} \sin^2 A + \cos^2 A & = 1 \\ \sin^2 A & = 1 - \cos^2 A \end{align}

## Compound Angle Formula (alternatively Addition & Subtraction Formula)

$$\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B$$

$$\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B$$

$$\tan (A \pm B) = {\tan A \pm \tan B \over 1\mp \tan A \tan B}$$

Some popular ways to utilize the formula:

\begin{align} \sin 3A & = \sin (2A + A) \\ & = \sin 2A \cos A + \cos 2A \sin A \\ \\ \cos (15^\circ) & = \cos (60^\circ - 45^\circ) \\ & = \cos 60^\circ \cos 45^\circ + \sin 60^\circ \sin 45^\circ \end{align}

## Double Angle Formula

$$\sin 2A = 2\sin A \cos A$$

$$\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$$

$$\tan 2A = {2\tan A \over 1 - \tan^2 A}$$

Sometimes  you are required to change the subject of the formula to suit the question's requirement.

The Half Angle formulas can be derived from the Double Angle formulas. For example:

$$\text{Let } A = {A \over 2},$$

\begin{align} \cos 2({A \over 2}) & = 2\cos^2 {A \over 2} - 1 \\ \cos A & = 2\cos^2 {A \over 2} - 1 \end{align}

## Formulae for ∆ABC

The following three formulas are actually from Mathematics (E Maths).

The first formula is Sine rule:

$${a \over \sin A}={b \over \sin B}={c \over \sin C}$$

The next formula is a variant of Cosine rule:

$$a^2 = b^2 + c^2 - 2bc\cos A$$

Finally, we have the formula to find the area of a triangle:

$$\text{∆} = {1 \over 2}bc\sin A$$

You can use the following triangle to visualize how to use the formula:

Triangle with angles a, b and c