Topics
Mathematical Logic
- Statements and Truth Values in Mathematical Logic
- Logical Connectives
- Tautology, Contradiction, and Contingency
- Quantifier, Quantified and Duality Statements in Logic
- Negations of Compound Statements
- Converse, Inverse, and Contrapositive
- Algebra of Statements
- Application of Logic to Switching Circuits
- Overview of Mathematical Logic
Matrices
- Elementry Transformations
- Adjoint & Inverse of Matrix
- Application of Matrices
- Overview of Matrices
Trigonometric Functions
- Trigonometric Equations and Their Solutions
- Solutions of Triangle>Polar Co-Ordinates
- Solving a Triangle>Solving a Triangle
- Basics of Inverse Trigonometric Functions
- Graphs of Inverse Trigonometric Functions
- Domain, Range & Principal Value
- Properties of Inverse Trigonometric Functions
- Overview of Trigonometric Functions
Pair of Straight Lines
Vectors
- Overview of Vectors
- Basic Concepts of Vector Algebra
- Types of Vectors in Algebra
- Algebra of Vectors > Scalar Multiplication
- Algebra of Vectors > Addition & Subtraction of Two Vectors
- Algebra of Vectors > Addition & Subtraction of Two Vectors
- Collinearity and Coplanarity of Vectors
- Vectors in Coordinate Geometry
- Components of Vector in Algebra
- Section Formula in Vector Algebra
- Product of Two Vectors > Scalar (Dot) Product
- Product of Two Vectors > Vector (Cross) Product
- Direction Ratios, Direction Cosine & Direction Angles in Vector
- Scalar Triple Product
Line and Plane
Linear Programming
Differentiation
- Introduction & Derivatives of Some Standard Functions
- Derivatives of Composite Functions
- Geometrical Meaning of Derivative
- Derivative of Inverse Function
- Logarithmic Differentiation
- Derivative of Implicit Functions
- Derivatives of Functions in Parametric Forms
- Higher Order Derivatives
- Overview of Differentiation
Applications of Derivatives
- Applications of Derivatives in Geometry
- Derivatives as a Rate Measure
- Approximations
- Rolle's Theorem
- Lagrange's Mean Value Theorem (LMVT)
- Increasing and Decreasing Functions
- Maxima and Minima
- Overview of Applications of Derivatives
Indefinite Integration
Definite Integration
- Definite Integral as Limit of Sum
- Integral Calculus
- Methods of Evaluation and Properties of Definite Integral
- Overview of Definite Integration
Application of Definite Integration
- Application of Definite Integration
- Area Bounded by Two Curves
- Overview of Application of Definite Integration
Differential Equations
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
- Formation of Differential Equations
- Methods of Solving Differential Equations> Homogeneous Differential Equations
- Methods of Solving Differential Equations>Linear Differential Equations
- Applications of Differential Equation
- Solution of a Differential Equation
- Overview of Differential Equations
Probability Distributions
- Random Variables
- Probability Distribution of Discrete Random Variables
- Probability Distribution of a Continuous Random Variable
- Variance of a Random Variable
- Expected Value and Variance of a Random Variable
- Overview of Probability Distributions
Binomial Distribution
Introduction
Inverse trigonometric functions are used to find the angle when the value of a trigonometric ratio is known. Since trigonometric functions are not one-one over their full domains, their domains are restricted so that inverse functions can be defined properly. This is why domain, range, and principal value are central ideas in this chapter.
Definition: Inverse trigonometric Function
The inverse trigonometric functions are the inverse forms of trigonometric functions after suitable domain restriction. They are written as:
-
\[\sin^{-1} x\]
-
\[\cos^{-1} x\]
-
\[\tan^{-1} x\]
-
\[\cot^{-1} x\]
-
\[\sec^{-1} x\]
-
\[\csc^{-1} x\]
Important:
- \[\sin^{-1} x\]
does not mean 1/sinx. It means the angle whose sine is x.
Definition: principal value
The value returned by an inverse trigonometric function is called its principal value. It is the unique angle chosen from the standard restricted interval for that function.
Standard domain and principal value ranges
| Inverse Trigonometric Function | Domain | Range (Principal Value) |
|---|---|---|
| sin⁻¹ x | [-1, 1] | [-π/2, π/2] |
| cos⁻¹ x | [-1, 1] | [0, π] |
| cosec⁻¹ x | R − (-1, 1) | [-π/2, π/2] − {0} |
| sec⁻¹ x | R − (-1, 1) | [0, π] − {π/2} |
| tan⁻¹ x | R | (-π/2, π/2) |
| cot⁻¹ x | R | (0, π) |
Key Points: Domain, Range & Principal Value
-
Inverse trigonometric functions give angles corresponding to known trigonometric values.
-
Their domains are restricted because ordinary trigonometric functions are not one-one on full domains.
-
Principal value means the standard angle selected from a fixed interval.
Graph for Time Series
Trend line by semi-averages
Magazine subscribers, 1976–1983. Step through the five stages to fit a trend line graphically and read off the rate of increase.
Step 1: Plot the data
Plot the 8 yearly subscriber counts on a Year vs Subscribers chart. The zigzag captures every year-to-year wobble — useful, but noisy. The publisher wants the underlying direction, not the noise.
