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

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 $\int _{a}^{b}y\,dx\ ({\mbox{for}}\ y\geq 0)$ 1. Area between a curve and the y-axis is $\int _{a}^{b}x\,dy\ ({\mbox{for}}\ x\geq 0)$ 2. 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. Appendix A: Formulae