Math Formula Reference

Formula reference for CSCA preparation

Algebra Basics

Name	Formula
Quadratic formula	$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ Applies when: a ≠ 0; real roots exist when b²−4ac ≥ 0 How the real exam asks it
Vieta's formulas	$x_1 + x_2 = -\frac{b}{a},\quad x_1 x_2 = \frac{c}{a}$ Applies when: x₁, x₂ are the roots of ax²+bx+c=0 with a ≠ 0 How the real exam asks it
Discriminant	$\Delta = b^2 - 4ac$ How the real exam asks it
Factoring	$a^2 - b^2 = (a+b)(a-b)$
Perfect square	$(a \pm b)^2 = a^2 \pm 2ab + b^2$
Sum/diff of cubes	$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$
Binomial theorem	$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

Sequences

Name	Formula
Arithmetic sequence	$a_n = a_1 + (n-1)d$ How the real exam asks it
Arithmetic sum	$S_n = \frac{n(a_1+a_n)}{2} = na_1 + \frac{n(n-1)d}{2}$
Geometric sequence	$a_n = a_1 \cdot r^{n-1}$
Geometric sum	$S_n = \frac{a_1(1-r^n)}{1-r},\quad r \neq 1$
Infinite geometric series	$S = \frac{a_1}{1-r},\quad \|r\| < 1$

Functions

Name	Formula
Exponent rules	$a^m \cdot a^n = a^{m+n},\quad \frac{a^m}{a^n} = a^{m-n}$
Logarithm rules	$\log(xy) = \log x + \log y$
Change of base	$\log_a b = \frac{\log_c b}{\log_c a}$ Applies when: a > 0, a ≠ 1, b > 0, c > 0, c ≠ 1 How the real exam asks it
Exp-log relation	$y = a^x \Longleftrightarrow x = \log_a y$
Composite function	$(f \circ g)(x) = f(g(x))$
Inverse function	$f(f^{-1}(x)) = x$

Trigonometry

Name	Formula
Pythagorean identity	$\sin^2\theta + \cos^2\theta = 1$ How the real exam asks it
Tangent	$\tan\theta = \frac{\sin\theta}{\cos\theta}$
Sum formula (sin)	$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$
Sum formula (cos)	$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$
Sum formula (tan)	$\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}$
Double angle (sin)	$\sin 2\theta = 2\sin\theta\cos\theta$
Double angle (cos)	$\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$
Double angle (tan)	$\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$
Half angle formulas	$\sin^2\frac{\theta}{2} = \frac{1-\cos\theta}{2},\quad \cos^2\frac{\theta}{2} = \frac{1+\cos\theta}{2}$
Auxiliary angle formula	$a\sin\theta + b\cos\theta = \sqrt{a^2+b^2}\sin(\theta+\varphi)$
Law of sines	$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$
Law of cosines	$c^2 = a^2 + b^2 - 2ab\cos C$

Derivatives

Name	Formula
Derivative definition	$f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$
Power rule	$(x^n)^{\prime} = nx^{n-1}$
Exponential derivative	$(e^x)^{\prime} = e^x,\quad (a^x)^{\prime} = a^x \ln a$
Logarithm derivative	$(\ln x)^{\prime} = \frac{1}{x}$
Trig derivatives	$(\sin x)^{\prime} = \cos x,\quad (\cos x)^{\prime} = -\sin x$
Product rule	$(fg)^{\prime} = f^{\prime}g + fg^{\prime}$
Quotient rule	$\left(\frac{f}{g}\right)^{\prime} = \frac{f^{\prime}g - fg^{\prime}}{g^2}$
Chain rule	$[f(g(x))]^{\prime} = f^{\prime}(g(x)) \cdot g^{\prime}(x)$

Analytic Geometry

Name	Formula
Distance formula	$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Midpoint formula	$M = \left(\frac{x_1+x_2}{2},\, \frac{y_1+y_2}{2}\right)$
Slope	$k = \frac{y_2-y_1}{x_2-x_1}$ Applies when: x₁ ≠ x₂ How the real exam asks it
Point-slope form	$y - y_0 = k(x - x_0)$ Applies when: Line through (x₀, y₀) with (existing) slope k How the real exam asks it
General form	$Ax + By + C = 0$
Point-line distance	$d = \frac{\|Ax_0+By_0+C\|}{\sqrt{A^2+B^2}}$
Circle equation	$(x-a)^2 + (y-b)^2 = r^2$ Applies when: Centre (a, b), radius r > 0 How the real exam asks it
Ellipse equation	$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a>b>0,\, c^2=a^2-b^2)$
Hyperbola equation	$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad (c^2=a^2+b^2)$
Parabola equation	$y^2 = 2px$
Eccentricity	$e = \frac{c}{a}$ Applies when: For standard-form ellipses and hyperbolas: ellipse 0 < e < 1 (c² = a² − b²); hyperbola e > 1 (c² = a² + b²) How the real exam asks it

Vectors & Complex Numbers

Name	Formula
Vector addition	$\vec{a} + \vec{b} = (a_1+b_1,\, a_2+b_2)$
Dot product	$\vec{a} \cdot \vec{b} = \|\vec{a}\|\|\vec{b}\|\cos\theta = a_1b_1+a_2b_2$
Vector magnitude	$\|\vec{a}\| = \sqrt{x^2 + y^2}$ Applies when: Coordinate form shown for a plane vector (x, y) How the real exam asks it
Angle between vectors	$\cos\theta = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|\|\vec{b}\|}$
Perpendicularity condition	$\vec{a} \perp \vec{b} \Longleftrightarrow \vec{a} \cdot \vec{b} = 0$
Complex addition	$(a+bi) + (c+di) = (a+c) + (b+d)i$
Complex multiplication	$(a+bi)(c+di) = (ac-bd) + (ad+bc)i$
Complex modulus	$\|a+bi\| = \sqrt{a^2 + b^2}$

Probability & Statistics

Name	Formula
Classical probability	$P(A) = \frac{n(A)}{n(S)}$
Addition rule	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Multiplication rule	$P(A \cap B) = P(A) \cdot P(B\|A)$
Conditional probability	$P(B\|A) = \frac{P(A \cap B)}{P(A)}$
Permutation	$P(n,r) = \frac{n!}{(n-r)!}$
Combination	$C(n,r) = \frac{n!}{r!(n-r)!}$
Binomial distribution	$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$
Mean	$\bar{x} = \frac{\sum x_i}{n}$
Variance	$s^2 = \frac{\sum (x_i - \bar{x})^2}{n}$
Standard deviation	$s = \sqrt{s^2}$

Interactive Simulations

Deepen your understanding through visual animations

Trigonometric Functions & Unit Circle Interactive Demo

Drag a point on the unit circle to see real-time changes in sin, cos, tan values and waveforms.

\sin\theta\cos\theta

5 minBasicInteractive

Function Graph Transformations Interactive Demo

Adjust parameters with sliders to observe function graph translations, scaling, and reflections.

y = a \cdot f(bx + c) + d

5 minBasicInteractive

Set Operations Venn Diagram Interactive Demo

Click Venn diagram regions to visually understand union, intersection, complement, and difference operations.

A \cup BA \cap B

5 minBeginnerInteractive

Elementary Functions Comparison Interactive Demo

Compare exponential, logarithmic, and power functions. Adjust base and parameters to observe changes.

y = a^xy = \log_a x

5 minBasicInteractive

Inequality Solution Set Visualization

Visualize inequality solution sets on number lines and coordinate planes. Drag parameters to see changes.

ax + b > 0ax^2 + bx + c \leq 0

5 minBasicInteractive

Sequence Visualization: Arithmetic & Geometric

Dynamically display arithmetic and geometric sequence terms and partial sums. Adjust first term and common difference/ratio.

a_n = a_1 + (n-1)da_n = a_1 \cdot r^{n-1}

5 minBasicInteractive

Calculus Visualization: Derivatives & Integrals

Drag a point to observe tangent line slope (derivative) and see area under curve (integral) change in real-time.

f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\int_a^b f(x)\,dx

5 minIntermediateInteractive

Conic Sections Interactive Demo

Adjust parameters to observe ellipse, parabola, and hyperbola shapes. View foci and eccentricity.

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1y^2=2px

5 minIntermediateInteractive

Probability & Statistics Simulator

Roll dice to simulate frequency vs probability. Adjust parameters to observe normal distribution changes.

P(A) = \frac{n(A)}{n(S)}f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}

5 minBasicInteractive

Complex Number Plane & Operations Visualization

Visualize complex number addition and multiplication on the complex plane. Understand modulus and argument geometrically.

z = a + bi|z| = \sqrt{a^2+b^2}

5 minIntermediateInteractive

Vector Operations Interactive Demo

Visualize vector addition, subtraction, dot product, and scalar multiplication. Adjust vector components with sliders.

\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta|\vec{a}| = \sqrt{x^2+y^2}

5 minBasicInteractive

Solid Geometry 3D Shapes Explorer

Select 3D shapes, drag to rotate, adjust parameters to calculate volume and surface area in real-time.

V = lwhV = \pi r^2 h

5 minBasicInteractive

Space Coordinate System Interactive Demo

Place points in 3D coordinate system. Calculate spatial distance, midpoint, and section point coordinates.

d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)

5 minBasicInteractive

Ready to Practice?

Apply these formulas through practice questions