Senior 4 Mathematics: Comprehensive Academic Summary

Fundamentals of Trigonometry

• Definitions: Trigonometry is defined as the measurement in triangles, derived from the Greek words $trigonon$ (triangle) and $metron$ (measure).
• Angular Measure:
  • Degree: The circumference of a circle is divided into $360$ equal parts; each part is $1^{\circ}$.
  • Radian: The unit of angular measure where an angle subtended from the centre of a unit circle produces an arc length of $r$. $1\,radian \approx 57.2958^{\circ}$.
  • Conversions: $\pi\,radians = 180^{\circ}$. To convert degrees to radians, multiply by $\frac{\pi}{180}$. To convert radians to degrees, multiply by $\frac{180}{\pi}$.
• The Unit Circle: A circle with centre $(0,0)$ and radius $1\,unit$. For a point $P(x,y)$ on the circle, $x = \cos(\theta)$ and $y = \sin(\theta)$.
• Fundamental Identity: $\cos^2(\theta) + \sin^2(\theta) = 1$.
• Trigonometric Ratios:
  • $\sin(\theta) = \frac{Opposite}{Hypotenuse}$
  • $\cos(\theta) = \frac{Adjacent}{Hypotenuse}$
  • $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{Opposite}{Adjacent}$
  • Reciprocal ratios: $\csc(\theta) = \frac{1}{\sin(\theta)}$, $\sec(\theta) = \frac{1}{\cos(\theta)}$, $\cot(\theta) = \frac{1}{\tan(\theta)}$.
• Reduction Formulae:
  • Complementary: $\sin(\frac{\pi}{2} - \theta) = \cos(\theta)$.
  • Supplementary: $\sin(\pi - \theta) = \sin(\theta)$; $\cos(\pi - \theta) = -\cos(\theta)$.
  • Opposite: $\sin(-\theta) = -\sin(\theta)$; $\cos(-\theta) = \cos(\theta)$.
• Triangle Rules:
  • Cosine Rule: $a^2 = b^2 + c^2 - 2bc\cos(A)$.
  • Sine Rule: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$.

Propositional and Predicate Logic

• Propositions: A declarative sentence that has a truth value (True or False), but not both.
• Logical Connectives:
  • Negation (NOT): $\neg p$ or $\sim p$. It reverses the truth value.
  • Conjunction (AND): $p \land q$. True only if both are true.
  • Disjunction (OR): $p \vee q$. True if at least one is true.
  • Implication (IF…THEN): $p \rightarrow q$. False only if the premise $p$ is true and conclusion $q$ is false.
  • Equivalence (IFF): $p \leftrightarrow q$. True only if both have the same truth value.
• Compound Statements:
  • Tautology: Always true regardless of variable values.
  • Contradiction: Always false regardless of variable values.
  • Contingency: Neither a tautology nor a contradiction.
• De Morgan’s Laws:
  • $\neg(p \land q) \iff \neg p \vee \neg q$
  • $\neg(p \vee q) \iff \neg p \land \neg q$
• Quantifiers:
  • Universal (\forall): "For all".
  • Existential (\exists): "There exists".

Binary Operations and Algebraic Structures

• Binary Operation: A calculation combining two elements (operands) of a set to produce another element of the set.
• Group: A non-empty set $G$ under operation $*$ satisfying:
  1. Closure: $\forall a, b \in G, (a * b) \in G$.
  2. Associativity: $\forall a, b, c \in G, a * (b * c) = (a * b) * c$.
  3. Identity: $\exists e \in G, a * e = e * a = a$.
  4. Inverse: $\forall a \in G, \exists a^{-1} \in G, a * a^{-1} = e$.
  • Abelian Group: A group that also satisfies commutativity: $a * b = b * a$.
• Rings/Fields: Rings involve two operations $(+, \cdot)$. A Field is a commutative ring with identity and inverses for all non-zero elements.
• Cayley Table: A grid used to describe the results of binary operations on finite sets.

The Set of Real Numbers

• Subsets: Natural Numbers ($\mathbb{N}$), Integers ($\mathbb{Z}$), Rational Numbers ($\mathbb{Q}$), and Irrational Numbers ($\mathbb{I}$) form the Real Numbers ($\mathbb{R}$).
• Absolute Value: $|x|$ represents the distance from zero. $|x| = x$ if $x \geq 0$ and $-x$ if $x < 0$.
• Powers and Radicals:
  • Product: $(a^m)(a^n) = a^{m+n}$.
  • Quotient: $\frac{a^n}{a^m} = a^{n-m}$.
  • Surds: Irrational numbers expressed with a radical sign $\sqrt{}$.
• Logarithms: The power to which $10$ must be raised to produce a number $A$ ($\log_{10}A$).
  • Rules: $\log(AB) = \log(A) + \log(B)$; $\log(A^n) = n\log(A)$.

Equations and Inequalities

• Linear Equations: Polynomials of degree $1$ in the form $ax + b = 0$.
• Quadratic Equations: Form $ax^2 + bx + c = 0$.
  • Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
  • Roots: Sum $S = -\frac{b}{a}$, Product $P = \frac{c}{a}$.
• Inequalities: Solved using sign diagrams. When multiplying/dividing by a negative number, the inequality sign reverses.
• Simultaneous Equations: Systems of two variables solved via substitution, elimination, Cramer’s method, or graphical methods.

Numerical Functions and Limits

• Function Concepts: A rule assigning every element of Domain $A$ to a unique element in Range $B$.
  • Composition: $(g \circ f)(x) = g(f(x))$.
  • Parity: Even if $f(-x) = f(x)$; Odd if $f(-x) = -f(x)$.
• Limits: The value a function approaches as the input approaches a point.
  • One-sided: $\lim_{x \to a^+} f(x)$ (right) and $\lim_{x \to a^-} f(x)$ (left).
  • Squeeze Theorem: If $f(x) \leq g(x) \leq h(x)$ and $\lim f = \lim h = L$, then $\lim g = L$.
  • Indeterminate Forms: Cases like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ requiring simplification or rationalisation.
• Continuity: $f(x)$ is continuous at $x_0$ if $\lim_{x \to x_0} f(x) = f(x_0)$.
• Asymptotes: Lines approached by curves. Vertical (denominator is zero), Horizontal (limit at $\pm \infty$), and Oblique (quotient of long division).

Differentiation

• Definition: The instantaneous rate of change or slope of the tangent line.
  • First Principles: $f'(x) = \lim_{h \to 0} \frac{f(x+1) - f(x)}{h}$.
• Rules:
  • Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$.
  • Product Rule: $(uv)' = u'v + uv'$.
  • Quotient Rule: $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$.
  • Chain Rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.
• Applications:
  • Stationary Points: Where $f'(x) = 0$.
  • Inflection Points: Where concavity changes ($f''(x) = 0$).
  • Kinematics: Velocity $v(t) = \frac{ds}{dt}$, Acceleration $a(t) = \frac{dv}{dt}$.

Vectors and Linear Transformations

• Vectors: Quantities with magnitude and direction.
  • Magnitude: $|AB| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
  • Dot Product: $a \cdot b = |a||b|\cos(\theta) = a_x b_x + a_y b_y$.
• Linear Transformations: Functions $T: V \to W$ satisfying $T(u+v) = T(u)+T(v)$ and $T(\lambda u) = \lambda T(u)$.
  • Geometric: Includes reflection (symmetry about axes), rotation (by angle $\theta$), and scaling.
• Matrices: Rectangular arrays $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$.
  • Determinant: $\det(A) = ad - bc$.
  • Inverse: $A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.

Statistics and Combinatorics

• Descriptive Statistics:
  • Central Tendency: Mean ($\bar{x} = \frac{\sum f x}{\sum f}$), Median (middle value), Mode (highest frequency).
  • Dispersion: Range, Variance ($s^2 = \frac{\sum(x-\bar{x})^2}{n}$), and Standard Deviation ($s$).
  • Coefficient of Variation: $C.V. = \frac{s}{\bar{x}} \times 100\%$.
• Combinatorics:
  • Factorial: $n! = n \times (n-1) \times ... \times 1$.
  • Permutations: Order matters. $P(n,r) = \frac{n!}{(n-r)!}$.
  • Combinations: Order does not matter. $C(n,r) = \frac{n!}{r!(n-r)!}$.
• Probability: The measure of an event $E$ occurring.
  • $P(E) = \frac{n(E)}{n(S)}$.
  • Complement: $P(E') = 1 - P(E)$.
  • Sum Law: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.