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. $b^x{b}^y=b^{x+y}$
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.

• ${\log }_{b}XY={\log }_{b}X+{\log }_{b}Y$
• ${\log }_b\frac{X}{Y}={\log }_{b}X-{\log }_{b}Y$
• ${\log }_b{X}^k=k{\log }_{b}X$

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

$\int k{x}^{n}dx=k\int x^{n}dx$

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

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

Rules of Definite Integrals

1. $\underset{a}{\overset{b}{\int }}f\left(x\right)dx=F\left(b\right)-F\left(a\right)$, F is the anti derivative of f such that F' = f
2. $\underset{a}{\overset{b}{\int }}f\left(x\right)dx=-\underset{b}{\overset{a}{\int }}f\left(x\right)dx$
3. $\underset{a}{\overset{a}{\int }}f\left(x\right)dx=0$
4. Area between a curve and the x-axis is $\underset{a}{\overset{b}{\int }}ydx(\text{for}y\ge 0)$

1. Area between a curve and the y-axis is $\underset{a}{\overset{b}{\int }}xdy(\text{for}x\ge 0)$
2. Area between curves is $\underset{a}{\overset{b}{\int }}\left|\begin{array}{c}f\left(x\right)-g\left(x\right)\end{array}\right|dx$

Trapezium Rule

$\underset{a}{\overset{b}{\int }}ydx\approx \frac{1}{2}h\{(y_0+y_n)+2(y_1+y_2+\dots +y_{n-1})\}$

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

Midpoint Rule

$\underset{a}{\overset{b}{\int }}f\left(x\right)dx\approx =h[f\left(x_1\right)+f\left(x_2\right)+\dots +f\left(x_n\right)]$

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

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