Formula Sheets
📐 1. Algebra
Quadratic Equation
For the equation \(ax^2 + bx + c = 0\):
\(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Binomial Theorem
\((a + b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{r}a^{n-r}b^r + \cdots + b^n\)
where \(\binom{n}{r} = \dfrac{n!}{(n-r)!r!}\).
Arithmetic Series
\(u_n = a + (n-1)d\)
\(S_n = \dfrac{n}{2}(a + l) = \dfrac{n}{2}[2a + (n-1)d]\)
Geometric Series
\(u_n = ar^{n-1}\)
\(S_n = \dfrac{a(1 - r^n)}{1 - r} \quad (r \neq 1)\)
\(S_\infty = \dfrac{a}{1 - r} \quad (|r| < 1)\)
📐 2. Trigonometry
Identities
\(\sin^2 A + \cos^2 A = 1\)
\(\sec^2 A = 1 + \tan^2 A\)
\(\cosec^2 A = 1 + \cot^2 A\)
\(\tan A = \dfrac{\sin A}{\cos A}\)
\(\cot A = \dfrac{\cos A}{\sin A}\)
Sine Rule
\(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\)
Cosine Rule
\(a^2 = b^2 + c^2 - 2bc\cos A\)
Area of Triangle
\(\text{Area} = \dfrac{1}{2}bc\sin A\)
Exact Values
| \(\theta\) |
\(0^\circ\) |
\(30^\circ\) |
\(45^\circ\) |
\(60^\circ\) |
\(90^\circ\) |
| \(\sin \theta\) |
0 |
\(\frac{1}{2}\) |
\(\frac{\sqrt{2}}{2}\) |
\(\frac{\sqrt{3}}{2}\) |
1 |
| \(\cos \theta\) |
1 |
\(\frac{\sqrt{3}}{2}\) |
\(\frac{\sqrt{2}}{2}\) |
\(\frac{1}{2}\) |
0 |
| \(\tan \theta\) |
0 |
\(\frac{1}{\sqrt{3}}\) |
1 |
\(\sqrt{3}\) |
undefined |
📐 3. Coordinate Geometry
Circle
\((x - a)^2 + (y - b)^2 = r^2\)
Centre: \((a, b)\), Radius: \(r\)
Distance Between Two Points
\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
Midpoint
\(M = \left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)\)
Gradient
\(m = \dfrac{y_2 - y_1}{x_2 - x_1}\)
Parallel Lines
\(m_1 = m_2\)
Perpendicular Lines
\(m_1 \times m_2 = -1\)
📐 4. Calculus
Derivatives
\(\dfrac{d}{dx}(x^n) = nx^{n-1}\)
\(\dfrac{d}{dx}(\sin x) = \cos x\)
\(\dfrac{d}{dx}(\cos x) = -\sin x\)
\(\dfrac{d}{dx}(\tan x) = \sec^2 x\)
\(\dfrac{d}{dx}(e^x) = e^x\)
\(\dfrac{d}{dx}(\ln x) = \dfrac{1}{x}\)
Product Rule
\(\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}\)
Quotient Rule
\(\dfrac{d}{dx}\left(\dfrac{u}{v}\right) = \dfrac{v\dfrac{du}{dx} - u\dfrac{dv}{dx}}{v^2}\)
Chain Rule
\(\dfrac{dy}{dx} = \dfrac{dy}{du} \times \dfrac{du}{dx}\)
Integrals
\(\int x^n \, dx = \dfrac{x^{n+1}}{n+1} + C \quad (n \neq -1)\)
\(\int \dfrac{1}{x} \, dx = \ln|x| + C\)
\(\int e^x \, dx = e^x + C\)
\(\int \sin x \, dx = -\cos x + C\)
\(\int \cos x \, dx = \sin x + C\)
\(\int (ax+b)^n \, dx = \dfrac{(ax+b)^{n+1}}{a(n+1)} + C \quad (n \neq -1)\)
\(\int \sin(ax+b) \, dx = -\dfrac{1}{a}\cos(ax+b) + C\)
\(\int \cos(ax+b) \, dx = \dfrac{1}{a}\sin(ax+b) + C\)
📐 5. Kinematics
\(v = \dfrac{ds}{dt}\)
\(a = \dfrac{dv}{dt} = \dfrac{d^2s}{dt^2}\)
Constant Acceleration (SUVAT)
\(v = u + at\)
\(s = ut + \dfrac{1}{2}at^2\)
\(v^2 = u^2 + 2as\)
📐 6. Circular Measure
\(s = r\theta\)
\(A = \dfrac{1}{2}r^2\theta\)