Grade 12
  1. Algebra  1.1. Matrices  1.1.1. Definition and Types of Matrices  1.1.2. Operations on Matrices  1.1.3. Transpose of a matrix  1.1.4. Determinants and their properties  1.1.5. Inverse of a Matrix  1.1.6. Solving Linear Equations Using Matrices  1.1.7. Applications of Matrices  1.2. Complex Numbers  1.2.1. Definition and representation  1.2.2. Algebra of complex numbers  1.2.3. Modulus and argument  1.2.4. Polar and Exponential Forms  1.2.5. De Moivre's Theorem  1.2.6. Roots of complex numbers  1.2.7. Applications in Geometry  1.3. Vectors and 3D Geometry  1.3.1. Vector Algebra  1.3.2. Scalar and Vector Products  1.3.3. Equation of a Line in Space  1.3.4. Equation of a Plane  1.3.5. Distance between lines and planes  1.3.6. Angle between Lines and Planes
  2. Calculus  2.1. Limits and Continuity  2.1.1. Concept of Limits  2.1.2. Continuity of Functions  2.1.3. Types of Discontinuities  2.1.4. Left-hand and Right-hand Limits  2.2. Differentiation  2.2.1. Derivative as a Rate of Change  2.2.2. Techniques of Differentiation  2.2.3. Higher Order Derivatives  2.2.4. Applications of Derivatives  2.2.5. Tangents and Normals  2.2.6. Increasing and Decreasing Functions  2.3. Integration  2.3.1. Indefinite Integrals  2.3.2. Definite Integrals  2.3.3. Integration Techniques  2.3.4. Area under a curve  2.3.5. Applications of Integrals in Geometry  2.4. Differential Equations  2.4.1. Formation of Differential Equations  2.4.2. Order and Degree of Differential Equations  2.4.3. Solutions of Differential Equations  2.4.4. Applications of Differential Equations  2.4.5. Solving First-Order Linear Differential Equations
  3. Probability and Statistics  3.1. Probability  3.1.1. Conditional Probability  3.1.2. Bayes' Theorem  3.1.3. Random Variables  3.1.4. Probability Distributions  3.1.5. Binomial and Normal Distributions  3.2. Statistics  3.2.1. Mean and Standard Deviation  3.2.2. Variance and Covariance  3.2.3. Correlation and Regression  3.2.4. Hypothesis Testing
  4. Linear Programming  4.1. Graphical Method  4.1.1. Feasible and Infeasible Regions  4.1.2. Optimal solutions  4.1.3. Sensitivity Analysis in Graphical Method  4.2. Simplex Method  4.2.1. Formulation of Problems  4.2.2. Solving linear programming problems  4.2.3. Dual Simplex Method
  5. Relations and Functions  5.1. Types of Relations  5.1.1. Reflexive Relations  5.1.2. Symmetric Relations  5.1.3. Transitive Relations  5.1.4. Equivalence Relations  5.2. Types of Functions  5.2.1. Onto Functions  5.2.2. One-to-One Functions  5.2.3. Inverse Functions  5.2.4. Composite Functions  5.3. Inverse Trigonometric Functions  5.3.1. Principal Values in Inverse Trigonometric Functions  5.3.2. Properties and Graphs of Inverse Trigonometric Functions  5.3.3. Applications in Calculus
  6. Advanced Trigonometry  6.1. Trigonometric Equations  6.1.1. General solutions of equations  6.1.2. Transformation formulae  6.2. Hyperbolic Functions  6.2.1. Definitions and properties  6.2.2. Relationships between hyperbolic and trigonometric functions
  7. Mathematical Reasoning  7.1. Logical Reasoning  7.1.1. Statements and Logical Connectives  7.1.2. Tautologies and Contradictions  7.2. Proofs  7.2.1. Direct proof  7.2.2. Indirect Proof  7.2.3. Proof by contradiction  7.2.4. Proof by Mathematical Induction
  8. Applications of Mathematics  8.1. Applications of Calculus  8.1.1. Maxima and Minima Problems  8.1.2. Economics and Biology Applications  8.1.3. Rate of Change in Physics  8.2. Applications of Probability  8.2.1. Risk Analysis  8.2.2. Decision Making  8.2.3. Game Theory

Grade 12 ↓


Advanced Trigonometry


Trigonometry is a branch of mathematics that studies relationships involving side lengths and angles of triangles. In advanced trigonometry, we delve deeper into the properties and applications of trigonometric functions beyond the basic right triangle relationships.

Trigonometric functions and their definitions

The elementary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions are traditionally defined as the ratios of the sides of a right triangle:

sin(θ) = (frac{text{opposite}}{text{hypotenuse}})
cos(θ) = (frac{text{adjacent}}{text{hypotenuse}})
tan(θ) = (frac{text{opposite}}{text{adjacent}})

The inverses of these functions are cosecant (csc), secant (sec), and cotangent (cot):

csc(θ) = (frac{1}{sin(θ)})
sec(θ) = (frac{1}{cos(θ)})
cot(θ) = (frac{1}{tan(θ)})

Unit circle and radian measure

The unit circle is a fundamental part of trigonometry. It is a circle with radius one, centered at the origin in the coordinate plane. Angles in radians are an alternative way to measure angles, where the angle is defined as the length of the arc that subtends the angle at the center of a circle with radius one.

1 radian is approximately equal to 57.2958 degrees. An angle measured in radians is given by:

Angle (in radians) = (frac{text{Arc length}}{text{Radius}})

1r=1θ

Trigonometric identities

Trigonometric identities are equations that are true for any value of the variable where both sides of the equation are defined. These identities are useful in simplifying expressions or solving equations.

  • Pythagorean identity
    sin²(θ) + cos²(θ) = 1
    1 + tan²(θ) = sec²(θ)
    1 + cot²(θ) = csc²(θ)
  • Angle sum and difference identities
    sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β)
    cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β)
    tan(α ± β) = (frac{tan(α) ± tan(β)}{1 ∓ tan(α)tan(β)})

Example: Proving an identity

Let us consider the identity sin(2θ) = 2sin(θ)cos(θ).

  1. Start with the angle sum identity for sine: sin(α + β) = sin(α)cos(β) + cos(α)sin(β)
  2. Set α = β = θ :
    3. sin(θ + θ) = sin(θ)cos(θ) + cos(θ)sin(θ)
  3. It's simple: sin(2θ) = 2sin(θ)cos(θ)

Graphing trigonometric functions

The graphs of trigonometric functions are periodic. This means that they repeat values at regular intervals or periods. Understanding the basic shape of these graphs helps in understanding their behavior.

Sine and cosine graphs

Sine and cosine functions have amplitudes (the height of their peaks), periods (the length of time before they start repeating), and phases (the shift along the x-axis).

XYChord

Tangent graph

The tangent function has a period of π and has vertical asymptotes, where the function is undefined.

XYtangent line

Inverse trigonometric functions

The inverse trigonometric functions are used to find the angle corresponding to a given trigonometric ratio. For example, if sin(θ) = 0.5, then θ = sin -1 (0.5).

  • sin -1 (x) where -1 ≤ x ≤ 1
  • cos -1 (x) for -1 ≤ x ≤ 1
  • tan -1 (x) for all real numbers

Solving trigonometric equations

Solving trigonometric equations often involves using identities and algebraic manipulation to find the angles that satisfy the equation.

Example: Solve 2sin(θ) - 1 = 0 for 0 ≤ θ < 2π

  1. Add 1 to both sides:
    2sin(θ) = 1
  2. Divide by 2:
    sin(θ) = 0.5
  3. Find θ:
    θ = sin -1 (0.5)
  4. Possible solution: θ = (frac{π}{6}) or (frac{5π}{6})

Applications of trigonometry

Trigonometry is widely used in various fields such as physics, engineering, astronomy and architecture. Here are some applications:

  • Physics: Calculating models of wave motion, oscillations, and periodic phenomena.
  • Engineering: Design and analysis of mechanical structures, electrical circuits, and signal processing.
  • Astronomy: Measuring distances between stars and understanding celestial mechanics.
  • Architecture: Designing roofs, bridges and other structures that involve angles and distances.

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