# 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

* $b^{x}b^{y}=b^{x+y}\,$
* ${\frac {b^{x}}{b^{y}}}=b^{x-y}$
* $\left(b^{x}\right)^{y}=b^{xy}$
* $a^{n}b^{n}=\left(ab\right)^{n}\,$
* $\left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}$
* $b^{-n}={\frac {1}{b^{n}}}$
* $b^{\frac {c}{x}}=\left({\sqrt[{x}]{b}}\right)^{c}$ where c is a constant
* $b^{1}=b\,$
* $b^{0}=1\,$

### 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. $\int x^{n}\,dx={\frac {1}{n+1}}x^{n+1}+C,\ (n\neq -1)$
$\int kx^{n}\,dx=k\int x^{n}\,dx$
$\int \left\{f^{'}(x)+g^{'}(x)\right\}\,dx=f(x)+g(x)+C$
$\int \left\{f^{'}(x)-g^{'}(x)\right\}\,dx=f(x)-g(x)+C$

### Rules of Definite Integrals

* $\int _{a}^{b}f\left(x\right)\ dx=F\left(b\right)-F\left(a\right)$, F is the anti derivative of f such that F' = f
* $\int _{a}^{b}f\left(x\right)\ dx=-\int _{b}^{a}f\left(x\right)\ dx$
* $\int _{a}^{a}f\left(x\right)\ dx=0$
* Area between a curve and the x-axis is $\int _{a}^{b}y\,dx\ ({\mbox{for}}\ y\geq 0)$
* Area between a curve and the y-axis is $\int _{a}^{b}x\,dy\ ({\mbox{for}}\ x\geq 0)$
* 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 $x_{i}={\frac {1}{2}}\left[\left(a+\left\{i-1\right\}h\right)+\left(a+ih\right)\right]$
Dividing and Factoring Polynomials / Sequences and Series / Logarithms and Exponentials / Circles and Angles / Integration