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Trigonometry

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Trigonometry PDF Notes, Important Questions and Formulas

Inverse Trigonometric Functions

INTRODUCTION:

Sin-1 x, cos-1 x, tan-1 x etc. represents angles or numbers whose values of sine, cosine and tangent is ‘x’, provided that the value in numerical form is smallest. These can be written as arc sin x, arc cos x etc. If two angles whose modulus is equal, in which one is positive and other is negative then we take positive sign.

 

DOMAIN & PRINCIPLE VALUE RANGE OF INVERSE TRIGONOMETRIC FUNCTIONS

 

S. No.

function

Domain

Principal value range

1

begin mathsize 12px style straight y equals sin to the power of negative 1 end exponent straight x end style  begin mathsize 12px style straight x element of left square bracket negative 1 comma 1 right square bracket end style   begin mathsize 12px style straight y element of left square bracket negative straight pi over 2 comma straight pi over 2 right square bracket end style

2

 begin mathsize 12px style straight y equals cos to the power of negative 1 end exponent straight x end style  begin mathsize 12px style straight x element of left square bracket 0 comma straight pi right square bracket end style 

 begin mathsize 12px style straight y element of left square bracket 0 comma straight pi right square bracket end style

 

3

 begin mathsize 12px style straight y equals tan to the power of negative 1 end exponent straight x end style   begin mathsize 12px style straight x element of straight R end style  begin mathsize 12px style straight y element of left parenthesis negative straight pi over 2 comma straight pi over 2 right parenthesis end style 

4

 begin mathsize 12px style straight y equals cot to the power of negative 1 end exponent straight x end style   begin mathsize 12px style straight x element of straight R end style

 begin mathsize 12px style straight y element of left square bracket 0 comma straight pi right square bracket end style

 

5

 begin mathsize 12px style straight y equals sec to the power of negative 1 end exponent straight x end style   begin mathsize 12px style straight x element of left parenthesis negative straight infinity comma negative 1 right square bracket union left square bracket 1 comma straight infinity right parenthesis end style  begin mathsize 12px style y element of left square bracket 0 comma pi right square bracket minus left curly bracket pi over 2 right curly bracket end style

6

 begin mathsize 12px style straight y equals cosec to the power of negative 1 end exponent straight x end style   begin mathsize 12px style straight x element of left parenthesis negative straight infinity comma negative 1 right square bracket union left square bracket 1 comma straight infinity right parenthesis end style

 begin mathsize 12px style y element of left square bracket negative pi over 2 comma pi over 2 right square bracket minus left curly bracket 0 right curly bracket end style

 

 

Inverse Trigonometric Functions

Key Concepts:

 

  1. There are two types of reasoning- deductive and inductive. Deductive reasoning was developed by Aristotle, Thales, and Pythagoras in the classical Period (600-300 B.C.).
  2. In deduction, given a statement to be proven, often called a conjecture or a theorem, valid deductive steps are derived and a proof may or may not be established. Deduction is the application of a general case to a particular case.
  3. Inductive reasoning depends on working with each case and developing a conjecture by observing incidence till each and every case is observed.
  4. Deductive approach is known as the ‘top-down approach’. The given theorem is narrowed down to a specific hypothesis, and then to an observation. Finally the hypothesis is tested with specific data to get the confirmation (or not) of the original theory.



  5. Mathematical reasoning is based on deductive reasoning. The classic example of deductive reasoning given by Aristotle is

    All men are mortal.

    Socrates is a man. 

    Socrates is mortal. 

  6. The basic unit involved in reasoning is the mathematical statement.
  7. A sentence is called a mathematically acceptable statement if it is either true or false but not both. A sentence which is both true and false simultaneously is called a paradox.
  8. Sentences which involve tomorrow, yesterday, here, there etc., i.e. variables are not statements.
  9. Sentences which express a request, a command or is simply a question are not statements.
  10. The denial of a statement is called the negation of the statement.
  11. Two or more statements joined by words such as ‘and’, ‘or’ are called compound statements. Each statement is called a component statement, where and’ ‘or’ are the connecting words.
  12. An ‘AND’ statement is true if each of the component statements is true, and it is false even if one of the component statements is false.
  13. An ‘OR’ statement is true when even one of its components is true; and it is false only when all its components are false.
  14. The word ‘OR’ can be used in two ways (i) Inclusive OR (ii) Exclusive OR. If only one of the two options is possible then the OR used is an exclusive OR.
    If any one of the two options or both the options are possible then the OR used is an inclusive OR.
  15. ‘There exists’ (∃) and ‘For all’ (∀) are called quantifiers.

 

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