User:BalazsH/C4 Formulae

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A list of formulae that were introduced in the C4 module of the Mathematics OCR (MEI) course that may need to be used in the exam.

1 Not Given in Exam
2 Given in Exam Booklet
- 2.1 Trigonometry
- 2.2 Integration
  - 2.2.1 Trapezium Rule
  - 2.2.2 Volumes of revolution

[edit] Not Given in Exam

[edit] Trigonometry

$\sec^2x \equiv 1 + \tan^2x \,$

$\csc^2x \equiv 1 + \cot ^2x \,$

$\sin2x \equiv 2\sin x\cos x \,$

$\cos2x \equiv \cos^2x - \sin^2x \equiv 1 - 2\sin^2x \equiv 2\cos^2x - 1 \,$

$\tan2x \equiv \frac{2\tan x}{1-\tan^2x} \,$

$\sin^2x \equiv \frac{1}{2}(1 - \cos 2x)$

$\cos^2x \equiv \frac{1}{2}(1 + \cos 2x)$

[edit] Small Angle Approximations

$\lim_{x \to 0}\frac{x}{\sin x} = \lim_{x \to 0}\frac{\sin x}{x} = 1 \,$

$\sin x \approx x \,$

$\tan x \approx x \,$

$\cos x \approx 1 - \frac{x^2}{2} \,$

[edit] Differentiation

$\sin kx \Rightarrow k\cos kx \,$

$\cos kx \Rightarrow -k\sin kx \,$

$e^{kx} \Rightarrow ke^{kx} \,$

[edit] Integration

$\cos kx \Rightarrow \frac{1}{k}\sin kx + c \,$

$\sin kx \Rightarrow -\frac{1}{k}\cos kx + c \,$

[edit] Vectors

$\begin{bmatrix} x\\ y\\ z\\ \end{bmatrix} \bullet \begin{bmatrix} a\\ b\\ c\\ \end{bmatrix} = xa + yb + zc$

[edit] Parametric Equations

Circle centre (0,0) and radius r: $x = r\cos\theta \,$ and $y = r\sin\theta \,$
Circle centre (a,b) and radius r: $x = a + r\cos\theta \,$ and $y = b + r\sin\theta \,$
$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \,$ provided that $\frac{dx}{dt} \neq 0 \,$
Ellipse centre (0,0) with major axis 2a and minor axis 2b: $x = a\cos\theta \,$ and $y = b\sin\theta \,$
Parabola with line of symmetry the x-axis: $x = at^2 \,$ and $y = 2at \,$
Rectangular hyperbola xy = c^2: $x = ct \,$ and $y = \frac{c}{t} \,$

[edit] Algebra

$\frac{px + q}{(ax + b)(cx + d)} \equiv \frac{A}{ax + b} + \frac{B}{cx + d} \,$

$\frac{px^2 + qr + r}{(ax + b)(cx^2 + d)} \equiv \frac{A}{ax + b} + \frac{Bx + C}{cx^2 + d} \,$

$\frac{px^2 + qx + r}{(ax + b)(cx + d)} \equiv \frac{A}{ax + b} + \frac{B}{cx + d} + \frac{C}{(cx + d)^2} \,$

[edit] Given in Exam Booklet

[edit] Trigonometry

$\sin(\theta \pm \phi) = \sin \theta \cos \phi \pm \cos \theta \sin \phi \,$

$\cos(\theta \pm \phi) = \cos \theta \cos \phi \mp \sin \theta \sin \phi\,$

$\tan(\theta \pm \phi) = \frac{\tan \theta \pm \tan \phi}{1 \mp \tan \theta \tan \phi}$

$\sin\alpha \pm \sin\beta = 2\sin\left(\frac{\alpha \pm \beta}{2}\right)\cos\left(\frac{\alpha \mp \beta}{2}\right)$

$\cos\alpha + \cos\beta = 2\cos\left(\frac{\alpha + \beta}{2}\right)\cos\left(\frac{\alpha - \beta}{2}\right)$

$\cos\alpha - \cos\beta = -2\sin\left(\frac{\alpha + \beta}{2}\right)\sin\left(\frac{\alpha - \beta}{2}\right)$