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

< A-level Mathematics‎ | OCR‎ | C2

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

Logarithmic Function

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

Change of Base Rule: ${\displaystyle \log _{a}x\,}$ can be written as ${\displaystyle {\frac {\log _{b}x}{\log _{b}a}}}$

Laws of Logarithmic Functions

When X and Y are positive.

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

Circles and Angles

Conversion of Degree Minutes and Seconds to a Decimal

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

Arc Length

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

Area of a Sector

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

Trigonometry

The Trigonometric Ratios Of An Angle

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

Important Trigonometric Values

You need to have these values memorized.

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

The Law of Cosines

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

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

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

The Law of Sines

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

Area of a Triangle

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

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

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

Trigonometric Identities

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

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

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.

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

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

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

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

Rules of Definite Integrals

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

Trapezium Rule

${\displaystyle \int _{a}^{b}y\,dx\approx {\frac {1}{2}}h\left\{\left(y_{0}+y_{n}\right)+2\left(y_{1}+y_{2}+\ldots +y_{n-1}\right)\right\}}$

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

Midpoint Rule

${\displaystyle \int _{a}^{b}f\left(x\right)\,dx\approx =h\left[f\left(x_{1}\right)+f\left(x_{2}\right)+\ldots +f\left(x_{n}\right)\right]}$

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

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

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