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

Logarithmic Function

The inverse of $y=b^{x}\,$$y=b^{x}\,$ is $x=b^{y}\,$$x=b^{y}\,$ which is equivalent to $y=\log _{b}x\,$$y=\log _{b}x\,$

Change of Base Rule: $\log _{a}x\,$$\log _{a}x\,$ can be written as ${\frac {\log _{b}x}{\log _{b}a}}$${\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\,$$\log _{b}XY=\log _{b}X+\log _{b}Y\,$
• $\log _{b}{\frac {X}{Y}}=\log _{b}X-\log _{b}Y\,$$\log _{b}{\frac {X}{Y}}=\log _{b}X-\log _{b}Y\,$
• $\log _{b}X^{k}=k\log _{b}X\,$$\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}}$$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\,$$s=\theta r\,$ Note: θ need to be in radians

Area of a Sector

$Area={\frac {1}{2}}r^{2}\theta$$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 \,$$\cos \theta \,$	${\frac {Adjacent}{Hypotenuse}}$${\frac {Adjacent}{Hypotenuse}}$	$\arccos \theta \,$$\arccos \theta \,$	$\cos ^{-1}\theta \,$$\cos ^{-1}\theta \,$	$x=\cos \ y\,$$x=\cos \ y\,$
Sine	$\sin \theta \,$$\sin \theta \,$	${\frac {Opposite}{Hypotenuse}}$${\frac {Opposite}{Hypotenuse}}$	$\arcsin \theta \,$$\arcsin \theta \,$	$\sin ^{-1}\theta \,$$\sin ^{-1}\theta \,$	$x=\sin \ y\,$$x=\sin \ y\,$
Tangent	$\tan \theta \,$$\tan \theta \,$	${\frac {Opposite}{Adjacent}}$${\frac {Opposite}{Adjacent}}$	$\arctan \theta \,$$\arctan \theta \,$	$\tan ^{-1}\theta \,$$\tan ^{-1}\theta \,$	$x=\tan \ y\,$$x=\tan \ y\,$

Important Trigonometric Values

You need to have these values memorized.

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

The Law of Cosines

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

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

$c^{2}=a^{2}+b^{2}-2ab\cos \gamma \,$$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 }}$${\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}}bc\sin \alpha \,$

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

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

Trigonometric Identities

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

$tan\theta ={\frac {\sin \theta }{\cos \theta }}$$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\neq -1)$$\int x^{n}\,dx={\frac {1}{n+1}}x^{n+1}+C,\ (n\neq -1)$

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

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