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

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

Dividing and Factoring Polynomials[edit]

Remainder Theorem[edit]

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[edit]

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[edit]

The Laws of Exponents[edit]

  1. b^xb^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[edit]

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[edit]

When X and Y are positive.

  • \log_bXY = \log_bX + \log_bY\,
  • \log_b \frac{X}{Y} = \log_bX - \log_bY\,
  • \log_b X^k = k \log_bX\,

Circles and Angles[edit]

Conversion of Degree Minutes and Seconds to a Decimal[edit]

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

Arc Length[edit]

s= \theta r\, Note: θ need to be in radians

Area of a Sector[edit]

Area = \frac{1}{2}r^2 \thetaNote: θ need to be in radians.


The Trigonometric Ratios Of An Angle[edit]

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[edit]

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[edit]

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[edit]

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

Area of a Triangle[edit]

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

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

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

Trigonometric Identities[edit]

\sin ^2 \theta + \cos ^2 \theta = 1 \,

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


Integration Rules[edit]

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 \ne -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[edit]

  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 \ge 0)
  5. Area between a curve and the y-axis is \int_{a}^{b} x\, dy\ ( \mbox{for}\ x \ge 0)
  6. Area between curves is \int_{a}^{b}\begin{vmatrix} f\left(x\right) - g\left(x\right) \end{vmatrix} dx

Trapezium Rule[edit]

\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[edit]

\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.

Dividing and Factoring Polynomials / Sequences and Series / Logarithms and Exponentials / Circles and Angles / Integration

Appendix A: Formulae