Secondary Math Tuition Punggol

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  • Principal values of inverse trigonometric functions

    To understand principal values of inverse trigonometric functions, let’s break it down step by step:

    1. Why Do We Need Inverse Trig Functions?

    Trigonometric functions (like \( \sin \theta \), \( \cos \theta \), \( \tan \theta \)) take an angle and give a number. Inverse trig functions (like \(\sin^{-1} x\), \(\cos^{-1} x \), \(\tan^{-1} x \)) do the opposite: they take a number and return an angle.

    2. The Problem: Periodicity

    Trig functions are periodic, meaning they repeat their values. For example:

    • \( \sin 30^\circ = \sin 150^\circ = 0.5 \)
    • \( \cos 0^\circ = \cos 360^\circ = 1 \)

    This means there are infinitely many angles that give the same trig value. But a function can only have one output for each input. So, how do we define inverse trig functions?

    3. Solution: Restrict the Domain (Principal Values)

    To make inverse trig functions work, we restrict their range (the angles they can output) to a specific interval called the principal value. This ensures each input gives exactly one angle.

    Principal Value Ranges:

    FunctionPrincipal Value RangeWhy?
    \( \sin^{-1}x \)\([- \frac{\pi}{2}, \frac{\pi}{2}] \)Covers all outputs from \(-1 \) to \(1 \) and is centred around 0.
    \( \cos^{-1}x \)\([0, \pi] \)Covers all outputs from \(-1 \) to \(1 \) and includes all quadrants.
    \( \tan^{-1}x \)\( (- \frac{\pi}{2}, \frac{\pi}{2}) \)Avoids vertical asymptotes and covers all real numbers.

    4. Key Takeaways

    • Principal values are the “main” angles returned by inverse trig functions.
    • Calculators use these ranges to give a single answer (e.g., typing \( \sin^{-1}(0.5) \) gives \( \frac{\pi}{6} \)).

  • Partial Fractions Made Easy: Step-by-Step Guide to Decomposing Rational Expressions

    Express \( \frac{(x+2)^2}{x^2(x-2)} \) as the sum of 3 partial fractions.

    Take note that \( x^2 \) is a repeated factor

    \( \begin{aligned} \frac{(x+2)^2}{x^2(x-2)}=\frac{x^2+4 x+4}{x^2(x-2)} \\ \text { Let } \frac{x^2+4 x+4}{x^2(x-2)} =\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x-2} \\ \quad x^2+4 x+4 =A x(x-2)+B(x-2)+C x^2\end{aligned}\)

    Find values of \( A \) and \( B \) by substituting suitable values of \( x \)

    To find \( B \), sub \( x = 0 \)

    To find \( C \), sub \( x = 2 \)

    Now that we know \( B \) and \( C \), sub \( x = 1 \) to find \( A \)

  • Mastering the Remainder Factor Theorem: Your Ultimate Guide to Polynomial Division and Factorization

    (i) Find the value of \(a\) and of \(b\) for which \(2 x^2+3 x-2\) is a factor of \( 2 x^4+3 x^3+a\left(x^2+x\right)+b \)

    (ii) Using the values of \(a\) and \(b\) found in part (i), solve the equation \(2 x^4+3 x^3+a\left(x^2+x\right)+b=0 \)

    (i) If \(2 x^2+3 x-2\) is a factor of \( 2 x^4+3 x^3+a\left(x^2+x\right)+b \), the factors of \(2 x^2+3 x-2\) are also factors of \( 2 x^4+3 x^3+a\left(x^2+x\right)+b \)

    Factorise \(2 x^2+3 x-2\) to get \( (2x-1)(x+2) \)

    Apply factor theorem

    Let \( f(x) = 2 x^4+3 x^3+a\left(x^2+x\right)+b \) ,

    \( f(0.5) =0 \) , since \( (2x-1) \) is a factor

    \( 2 + 3a +4b = 0 \)

    \( f(-2) =0 \) , since \( (x+2) \) is a factor

    \( 8 + 2a + b = 0 \)

    Solving simultaneously, \(a = – 6 \), \( b = 4 \)

    (ii) To solve \(2 x^4+3 x^3-6\left(x^2+x\right)+4=0 \)

    Perform long division

    The quotient of \( \left(2x^4 +3x^3 −6(x^2+x)+4 \right) \div \left(2x^2+3x−2 \right)\) is:\(x^2−2 \)

    Hence \( \left(2x^2+3x−2 \right) \left(x^2−2 \right) = 0 \)

  • O-Level Math Syllabus: What You Need to Know for Success

    Summary of the O-Level Mathematics syllabus (Syllabus Code 4052), broken down into sections and suggested levels of study:

    1. Number and Algebra

    1. Numbers and Their Operations (Sec 1):
      • Prime factorization, HCF, LCM, indices, and standard form.
      • Ordering numbers on a number line and approximation.
    2. Ratio, Proportion, and Percentage (Sec 1):
      • Simplifying ratios, map scales, direct/inverse proportions, and percentage calculations.
    3. Rate and Speed (Sec 1):
      • Concepts of average speed and unit conversion.
    4. Algebraic Expressions and Formulae (Sec 1 & Sec 2):
      • Simplification, factorization, expansion, and algebraic fractions.
      • Changing the subject of a formula and finding nth terms in patterns.
    5. Functions and Graphs (Sec 2 & Sec 3):
      • Linear and quadratic functions, graph plotting, and properties of curves.
    6. Equations and Inequalities (Sec 2 & Sec 3):
      • Solving linear, quadratic, and fractional equations.
      • Formulating and solving real-world problems.
    7. Set Language and Notation (Sec 3):
      • Venn diagrams, union/intersection, and complement of sets.
    8. Matrices (Sec 4):
      • Basic matrix operations, scalar multiplication, and interpreting data.

    2. Geometry and Measurement

    1. Angles, Triangles, and Polygons (Sec 1 & Sec 2):
      • Properties of angles, parallel lines, polygons, and constructions.
    2. Congruence and Similarity (Sec 2 & Sec 3):
      • Similar triangles, scale drawings, and ratios of areas and volumes.
    3. Properties of Circles (Sec 3):
      • Tangent properties, angle properties in circles, and equal chords.
    4. Pythagoras’ Theorem and Trigonometry (Sec 2 & Sec 3):
      • Applications of Pythagoras’ theorem and trigonometric ratios.
      • Sine and cosine rules, and solving 2D/3D problems involving angles.
    5. Mensuration (Sec 1 to Sec 4):
      • Calculating areas, perimeters, volumes, and surface areas of various shapes.
      • Sector areas and arc lengths (including radian measure).
    6. Coordinate Geometry (Sec 2 & Sec 3):
      • Gradient, length of a line segment, and equations of straight lines.
    7. Vectors in Two Dimensions ( Sec 4):
      • Operations with vectors, position vectors, and geometric applications.

    3. Statistics and Probability

    1. Data Handling and Analysis (Sec 1 to Sec 4):
      • Collecting, representing, and interpreting data through various graphs.
      • Measures of central tendency (mean, median, mode) and spread (range, standard deviation).
    2. Probability (Sec 1 & Sec 4):
      • Probability of single and combined events, including tree diagrams and addition/multiplication rules.

    Suggested Progression

    • Sec 1: Foundational topics like numbers, algebra basics, and simple geometry.
    • Sec 2: Intermediate algebra, trigonometry basics, and expanded geometry concepts.
    • Sec 3: Advanced algebra, trigonometry, coordinate geometry, and probability/statistics introduction.
    • Sec 4: Consolidation of complex topics (e.g., vectors, matrices, probability, advanced trigonometry).