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

By the end of this module you will be expected to have learnt the following formulae:

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).

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).

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. ${\displaystyle b^{0}=1\,}$

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 ${\displaystyle {\frac {\log _{b}x}{\log _{b}a}}}$

When X and Y are positive.

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

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

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

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

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	${\displaystyle \tan \theta \,}$	${\displaystyle {\frac {Opposite}{Adjacent}}}$	${\displaystyle \arctan \theta \,}$	${\displaystyle \tan ^{-1}\theta \,}$	${\displaystyle x=\tan \ y\,}$

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	-

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

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

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

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

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

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

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

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

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

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

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

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

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

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 ${\displaystyle \int _{a}^{b}y\,dx\ ({\mbox{for}}\ y\geq 0)}$

1. Area between a curve and the y-axis is ${\displaystyle \int _{a}^{b}x\,dy\ ({\mbox{for}}\ x\geq 0)}$
2. Area between curves is ${\displaystyle \int _{a}^{b}{\begin{vmatrix}f\left(x\right)-g\left(x\right)\end{vmatrix}}dx}$

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

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

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

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

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