A-level Mathematics/OCR/C2/Appendix A: Formulae

By the end of this module you will be expected to have learnt the following formulae:

Dividing and Factoring Polynomials

Remainder Theorem

If you have a polynomial f(x) divided by x - c, the remainder is equal to f(c). Note if the equation is x + c then you need to negate c: f(-c).

The Factor Theorem

A polynomial f(x) has a factor x - c if and only if f(c) = 0. Note if the equation is x + c then you need to negate c: f(-c).

Formula For Exponential and Logarithmic Function

The Laws of Exponents

1. $b^xb^y = b^{x+y}\,$
2. $\frac{b^x}{b^y} = b^{x-y}$
3. $\left(b^x\right)^y = b^{xy}$
4. $a^n b^n = \left(ab\right)^n\,$
5. $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
6. $b^{-n}=\frac{1}{b^n}$
7. $b^ \frac {c}{x} = \left( \sqrt[x] b \right)^c$ where c is a constant
8. $b^1=b\,$
9. $b^0=1\,$

Logarithmic Function

The inverse of $y = b^x\,$ is $x = b^y \,$ which is equivalent to $y = \log_b x\,$

Change of Base Rule: $\log_a x\,$ can be written as $\frac { \log_b x}{ \log_b a}$

Laws of Logarithmic Functions

When X and Y are positive.

• $\log_bXY = \log_bX + \log_bY\,$
• $\log_b \frac{X}{Y} = \log_bX - \log_bY\,$
• $\log_b X^k = k \log_bX\,$

Circles and Angles

Conversion of Degree Minutes and Seconds to a Decimal

$X + \frac{Y}{60}+ \frac{Z}{3600}$ where X is the degree, y is the minutes, and z is the seconds.

Arc Length

$s= \theta r\,$ Note: θ need to be in radians

Area of a Sector

$Area = \frac{1}{2}r^2 \theta$Note: θ need to be in radians.

Trigonometry

The Trigonometric Ratios Of An Angle

Function	Written	Defined	Inverse Function	Written	Equivalent to
Cosine	$\cos \theta\,$	$\frac{Adjacent}{Hypotenuse}$	$\arccos \theta\,$	$\cos ^{-1} \theta\,$	$x = \cos\ y\,$
Sine	$\sin \theta\,$	$\frac{Opposite}{Hypotenuse}$	$\arcsin \theta\,$	$\sin ^{-1} \theta\,$	$x = \sin\ y\,$
Tangent	$\tan \theta\,$	$\frac{Opposite}{Adjacent}$	$\arctan \theta\,$	$\tan ^{-1} \theta\,$	$x = \tan\ y\,$

Important Trigonometric Values

You need to have these values memorized.

$\theta\,$	$rad\,$	$\sin \theta\,$	$\cos \theta\,$	$\tan \theta\,$
$0^\circ$	0	0	1	0
$30^\circ$	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$45^\circ$	$\frac{\pi}{4}$	$\frac{ \sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1\,$
$60^\circ$	$\frac{\pi}{3}$	$\frac{ \sqrt{3}}{2}$	$\frac{\sqrt{1}}{2}$	$\sqrt{3}$
$90^\circ$	$\frac{\pi}{2}$	1	0	-

The Law of Cosines

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

$b^2=a^2 + c^2 - 2ac \cos \beta \,$

$c^2=a^2 + b^2 - 2ab \cos \gamma \,$

The Law of Sines

$\frac {a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac {c}{\sin \gamma}$

Area of a Triangle

$Area = \frac{1}{2}bc \sin \alpha \,$

$Area = \frac{1}{2}ac \sin \beta \,$

$Area = \frac{1}{2}ab \sin \gamma \,$

Trigonometric Identities

$\sin ^2 \theta + \cos ^2 \theta = 1 \,$

$tan \theta = \frac{\sin \theta}{\cos \theta}$

Integration

Integration Rules

The reason that we add a + C when we compute the integral is because the derivative of a constant is zero, therefore we have an unknown constant when we compute the integral.

$\int x^n\, dx = \frac{1}{n+1} x^{n+1} + C,\ (n \ne -1)$

$\int kx^n\, dx = k \int x^n\, dx$

$\int \left\{ f^'(x) + g^'(x)\right\}\, dx = f(x) + g(x) + C$

$\int \left\{ f^'(x) - g^'(x)\right\}\, dx = f(x) - g(x) + C$

Rules of Definite Integrals

1. $\int_{a}^{b} f \left ( x \right )\ dx = F \left ( b \right ) - F \left ( a \right )$, F is the anti derivative of f such that F' = f
2. $\int_{a}^{b} f \left ( x \right )\ dx = - \int_{b}^{a} f \left ( x \right )\ dx$
3. $\int_{a}^{a} f \left ( x \right )\ dx = 0$
4. Area between a curve and the x-axis is $\int_{a}^{b} y\, dx\ ( \mbox{for}\ y \ge 0)$
5. Area between a curve and the y-axis is $\int_{a}^{b} x\, dy\ ( \mbox{for}\ x \ge 0)$
6. Area between curves is $\int_{a}^{b}\begin{vmatrix} f\left(x\right) - g\left(x\right) \end{vmatrix} dx$

Trapezium Rule

$\int_a^b y \,dx \approx \frac{1}{2} h \left \{\left (y_0 + y_n \right ) + 2\left (y_1 + y_2 + \ldots + y_{n-1} \right ) \right\}$

Where: $h = \frac{b-a}{n}$

Midpoint Rule

$\int_a^b f \left (x \right ) \,dx \approx = h \left [ f \left (x_1 \right ) + f \left (x_2 \right ) + \ldots + f \left (x_n \right )\right ]$

Where: $h = \frac{b-a}{n}$ n is the number of strips.

and $x_i = \frac{1}{2} \left [ \left( a +\left \{i - 1 \right \} h \right) + \left (a + ih \right) \right]$

This is part of the C2 (Core Mathematics 2) module of the A-level Mathematics text.

Dividing and Factoring Polynomials / Sequences and Series / Logarithms and Exponentials / Circles and Angles / Integration

Appendix A: Formulae