Mathematics Intermediate Part I, Chapter 11

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Questions and Answers

If $a$ is any non-zero real number, what is its multiplicative inverse?

  • $\frac{1}{a}$ (correct)
  • $-a$
  • $-\frac{1}{a}$
  • $a^2$

Given $a, b \in R$, if $a = b \implies b = a$, what property is demonstrated?

  • Symmetric (correct)
  • Transitive
  • Reflexive
  • Trichotomy

The set {1, -1} possesses closure property with respect to which operation?

  • +
  • ÷
  • -
  • x (correct)

If $a < b$, which of the following statements is always true?

<p>$a &lt; b$ (B)</p> Signup and view all the answers

What is the additive identity in the set of complex numbers?

<p>(0, 0) (D)</p> Signup and view all the answers

What is the multiplicative identity in the set of complex numbers?

<p>(1, 0) (A)</p> Signup and view all the answers

What is the modulus of the complex number $z = a + ib$?

<p>$\sqrt{a^2 + b^2}$ (C)</p> Signup and view all the answers

Which operation is squaring a number considered as?

<p>Unary operation (C)</p> Signup and view all the answers

Given that $A = \phi$, what is $P(A)$?

<p>{\phi} (A)</p> Signup and view all the answers

Given that sets A and B are disjoint, which statement is true?

<p>$A \cap B = \phi$ (D)</p> Signup and view all the answers

Which of the following is NOT a binary operation?

<p>$\surd$ (A)</p> Signup and view all the answers

If $y = \sqrt{x}, x \geq 0$ is a function, what is its inverse?

<p>A parabola (A)</p> Signup and view all the answers

What is another name for an onto function?

<p>Surjective (C)</p> Signup and view all the answers

For the propositions $p$ and $q$, what is $(p \land q) \implies p$?

<p>Tautology (A)</p> Signup and view all the answers

A compound proposition which is always true is called what?

<p>Tautology (B)</p> Signup and view all the answers

Which of the following is true for any set $A$?

<p>$A \cap A = A$ (B)</p> Signup and view all the answers

What is a rectangular array of numbers enclosed by square brackets called?

<p>Matrix (C)</p> Signup and view all the answers

For a square matrix $A = [a_{ij}]{n \times n}$, what are the elements $a{11}, a_{22}, a_{33}, ..., a_{nn}$ called?

<p>Main diagonal (D)</p> Signup and view all the answers

If a matrix of order $m \times n$ is given, what is the order of its transpose?

<p>$n \times m$ (D)</p> Signup and view all the answers

Given that A is a matrix and n is an integer, what is equivalent to adding A to itself n times ($A + A + A + \dots$)?

<p>$nA$ (C)</p> Signup and view all the answers

If $Adj A$ = $\begin{bmatrix} -1 & -2\ 3 & -1 \end{bmatrix}$, then matrix $A^{-1}$ is?

<p>$\begin{bmatrix} -1/4 &amp; 1/2 \ -3/4 &amp; -1/4 \end{bmatrix}$ (B)</p> Signup and view all the answers

If $AX = B$, then $X$ is equal to which of the following?

<p>$A^{-1}B$ (A)</p> Signup and view all the answers

A square matrix $A$ is symmetric if what condition holds true?

<p>$A^t = A$ (D)</p> Signup and view all the answers

If $A$ is any square matrix, then $A + A^t$ is what type of matrix?

<p>Symmetric (A)</p> Signup and view all the answers

Which type of matrix possesses a multiplicative inverse?

<p>Non-singular (A)</p> Signup and view all the answers

In a homogeneous system of linear equations, what is the solution (0, 0, 0) called?

<p>Trivial solution (B)</p> Signup and view all the answers

The statement '$ax^2 + bx + c = 0$ will be quadratic' is true if what condition is met?

<p>$a \neq 0$ (B)</p> Signup and view all the answers

The imaginary solutions to all cube roots of unity are related in what way?

<p>Conjugate (D)</p> Signup and view all the answers

Flashcards

What is a matrix?

A rectangular array of numbers enclosed by square brackets.

What are rows?

In a matrix, these are the horizontal lines of numbers.

What are Columns?

In a matrix, these are the vertical lines of numbers.

What is the order of A?

If A has m rows and n columns, this describes its size.

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What does a_{ij} represent?

a_{ij} represents the entry in the ith row and jth column.

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When is a matrix real?

A matrix is real if all its elements are...

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What is a row vector?

A matrix with only one row.

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What is a column vector?

A matrix with only one column.

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What is a square matrix?

A matrix where the number of rows equals the number of columns.

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What is a rectangular matrix?

A matrix where the number of rows does not equal the number of columns.

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What is the main diagonal?

The diagonal from the top-left corner to the bottom-right corner.

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What is a scalar matrix?

For a square matrix A, if all a_{ij} = 0 for i ≠ j and a_{ij} = k (non-zero) for i = j, then A is called a...

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What is a Symmetric matrix

A matrix where A^t = A .

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What is a Skew-symmetric matrix?

A matrix where A^t = -A .

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When does inverse exist?

Multiplicative inverse of a 2x2 matrix

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What is a Theorem?

A statement that is proven to be true through a mathematical argument.

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What is a conditional equation?

An equation to be true for all values of the variable.

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What is a proper fraction?

A fraction where the degree of P(x) is less than the degree of Q(x).

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What is a Rational fraction?

A fraction which can be expressed as an algebraic function.

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What is Squaring?

A process by which the number, quantity or term is multiplied by itself.

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What is an Irrational number?

A number that is not rational

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What is a Venn diagram?

A diagram that uses shapes to represent sets and their relationships.

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What is an inverse function?

A function where each element of the range is associated with one element of its domain

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Study Notes

  • Mathematics: Intermediate Part I, Chapter 11, Important Multiple Choice Questions.

Contents Overview

  • Chapter 1 deals with the Number System
  • Chapter 2 covers Sets, Functions, and Groups
  • Chapter 3 is about Matrices and Determinants
  • Chapter 4 focuses on Quadratic Equations
  • Chapter 5 discusses Partial Fractions
  • Chapter 6 covers Sequences and Series
  • Chapter 7 explores Permutations, Combinations, and Probability
  • Chapter 8 is about Mathematical Induction and the Binomial Theorem
  • Chapter 9 deals with the Fundamentals of Trigonometry
  • Chapter 10 covers Trigonometric Identities
  • Chapter 11 discusses Trigonometric Functions and Their Graphs
  • Chapter 12 focuses on the Application of Trigonometry
  • Chapter 13 is about Inverse Trigonometric Functions
  • Chapter 14 covers the Solution of Trigonometric Equations

Chapter 1: Number System

  • For any complex number z, the absolute value |z| is always equal to |−z|, |−z̄|, and z̄.
  • Numbers that can be written in the form of p/q, where p, q ∈ Z, and q ≠ 0, are rational numbers
  • A decimal that has a finite number of digits in its decimal part is called a terminating decimal
  • 5.333.... is a rational number
  • π is an irrational number
  • 22/7 is an irrational number
  • The multiplicative inverse of '0' is not defined
  • If 'a' is any non-zero real number, then its multiplicative inverse is 1/a
  • For all a ∈ R, a = a is a Reflexive Property
  • For all a, b ∈ R, a = b ⇒ b = a is called a Symmetric Property
  • The golden rule of fractions states that for k ≠ 0, a/b = (ka)/(kb)
  • The set {1, -1} possesses a closure property with respect to multiplication
  • If a < b, then a/b < 1
  • The additive identity in the set of complex numbers is (0, 0)
  • The multiplicative identity in the set of complex numbers is (1, 0)
  • The modulus of z = a + ib is √(a² + b²)
  • i¹³ equals i
  • The multiplicative inverse of (4, -7) is (4/65, 7/65)
  • (0.3)(0.5) = -0.15
  • (-i)² = -1
  • √3 is an irrational number
  • The product of √-2 × √-2 = -2
  • The imaginary part of the complex number (b, a) is 'a'
  • If z = -1 - i, then z̄ = -1 + i
  • The property 7.8 + (-7.8) = 0 demonstrates the additive inverse
  • If x ≠ 0, then the multiplicative inverse of x is 1/x
  • (-i)¹⁵ = i
  • If z₁ and z₂ are complex numbers, then |z₁ + z₂| ≤ |z₁| + |z₂|
  • A recurring decimal represents a rational number
  • Every recurring decimal is a rational number
  • The imaginary part of (-2 + 3i³) is -3
  • The product of two conjugate complex numbers is a real number
  • (0, 1)³ is equal to -i
  • If n is an even integer, then iⁿ is equal to ±1
  • Factors of 3(x² + y²) are √3(x + iy)(x - iy)
  • The real part of (2 + i)/i is 1

Chapter 2: Sets, Functions, and Groups

  • If x ∈ L ∪ M, then x ∈ L or x ∈ M or both
  • The total number of subsets that can be formed from the set {x, y, z} is 8
  • If x ∈ B' = U – B, then x ∉ B and x ∈ U
  • If L ∪ M = L ∩ M, then L is equal to M
  • A set is a collection of objects which are well-defined and distinct
  • The set of odd numbers between 1 and 9 are {3, 5, 7}
  • Every recurring non-terminating decimal represents Q̄ or irrational numbers
  • A diagram which represents a set is called a Venn diagram
  • R - {0} i.e real numbers excluding zero, is a group w.r.t the binary operation of multiplication
  • In a proposition, if p ↔ q then q → p is called the converse of p → q
  • If A and B are disjoint sets, then A ∩ B = φ (empty set)
  • ~p → q is called the converse of p → q
  • If A ⊆ B then n(A ∩ B) = n(A)
  • The inverse of any element of a group is unique
  • Every function is a Relation
  • The number of all subsets of a set having three elements is 8
  • The graph of a linear function is a straight line
  • The identity element in the set of complex numbers (C, .) is (1, 1)
  • Squaring a number is a unary operation
  • For sets, N ⊂ Z ⊂ Q ⊂ R is a true statement.
  • The set of integers is a group w.r.t Addition
  • If A = φ, then P(A) = {φ}
  • If A and B are disjoint sets then A ∩ B = φ
  • The set of non-zero real numbers w.r.t multiplication is a group.
  • Division is not a binary operation.
  • If y = √x, where x ≥ 0, is a function, then its inverse is a parabola.
  • An onto function is also called surjective.
  • A (1–1) function is also called Injective or one-to-one
  • For the propositions p and q, (p ∧ q) → p is a tautology.
  • For the propositions p and q, p → (p ∨ q) is a tautology.
  • If set A has 2 elements and set B has 4 elements, then the number of elements in A × B is 8
  • The inverse of a line is not defined correctly in the options
  • The function f = {(x, y), y = x} is an Identity function
  • If y = √x, x ≥ 0 is a function, then its inverse is a parabola.
  • The truth set of a tautology is a Universal set
  • A compound proposition which is always true is called a Tautology
  • A compound proposition which is always neither true nor false is called a contingency.
  • A compound proposition which is always wrong is called a contradiction.
  • If S = {}, then the order of set S is 1.
  • The symbol used to denote negation of a proposition is '~'.
  • If B ⊆ A, then n(B – A) is equal to 0

Chapter 3: Matrices and Determinants

  • A rectangular array of numbers enclosed by square brackets is called a Matrix
  • The horizontal lines of numbers in a matrix are called Rows
  • The vertical lines of numbers in a matrix are called Columns
  • If a matrix A has m rows and n columns, then the order of A is m × n
  • The element aij of any matrix A is present in the ith row and jth column
  • A matrix A is called real if all aij are Real numbers
  • If a matrix A has only one row, it is called a Row vector
  • If a matrix A has only one column, it is called a Column vector
  • If a matrix A has the same number of rows and columns, it is called a Square matrix
  • If a matrix A has different numbers of rows and columns, it is called a Rectangular matrix
  • For a square matrix A = [aij]n×n, the elements A₁₁, A₂₂, A₃₃...ann are on the Main diagonal
  • For a square matrix A = [aij], if all aij = 0, i ≠ j and all aij = k (non-zero) for i = j, then A is called a Scalar Matrix
  • For a square matrix A = [aij], if all aij = 0, i ≠ j and at least aij ≠ 0, i = j, then A is called a Diagonal matrix
  • The matrix [7] is a Square matrix
  • If A is a matrix of order m × n, then the matrix of order n × m is the Transpose of A
  • For any two matrices A and B, (A + B)ᵗ = Aᵗ + Bᵗ
  • Let A be a matrix and n is an integer; then A + A + A + ⋯ to n terms = nA
  • If the order of A is m × n and the order of B is n × q, then the order of AB is m × q
  • If adj A = [[-1, -2], [3, -4]], then matrix A = [[-4, 2], [-3, -1]]
  • If A is a non-singular matrix, then A⁻¹ = (1/|A|) adjA
  • If AX = B, then X = A⁻¹B
  • The inverse of a matrix exists if it is non-singular.
  • For any matrix A, it is always true that |A| = |Aᵗ|
  • If all entries of a square matrix of order 3 are multiplied by k, then the value of |kA| is equal to k³|A|
  • For a non-singular matrix A, it is true that (A⁻¹)⁻¹ = A
  • For any non-singular matrices A and B, it is true that (AB)⁻¹ = B⁻¹A⁻¹
  • A square matrix A = [aij] for which aij = 0, i > j, then A is called Upper triangular
  • A square matrix A = [aij] for which aij = 0, i < j, then A is called Lower triangular
  • Any matrix A is called singular if |A| = 0
  • R (Real Numbers) is a field
  • A square matrix A is symmetric if Aᵗ = A
  • A square matrix A is skew-symmetric if Aᵗ = -A
  • The main diagonal elements of a skew-symmetric matrix must be 0
  • The main diagonal elements of a skew-hermitian matrix must be any complex number
  • In echelon form of a matrix, the first non-zero entry is called the Leading entry
  • The additive inverse of a matrix exists only if it is any matrix of order m × n
  • The multiplicative inverse of a matrix exists only if it is a non-singular matrix
  • The number of non-zero rows in echelon form of a matrix is called the Rank of the matrix
  • If A is any square matrix, then A + Aᵗ is a symmetric matrix.
  • If A is any square matrix, then A² must be symmetric if A is symmetric.
  • In a homogenous system of linear equations, the solution (0,0,0) is a Trivial solution
  • If AX = 0, then X = 0
  • If the system of linear equations has no solution at all, then it is called an Inconsistent system
  • If the system x + λy = 4; 2x + 4y = −3 does not possess a unique solution, then λ=2.
  • The inverse of a Unit matrix is a Unit matrix
  • The transpose of a row matrix is a column matrix

Chapter 4: Quadratic Equation

  • The equation ax² + bx + c = 0 will be quadratic if a ≠ 0.
  • The solution set of the equation x² − 4x + 4 = 0 is {2}.
  • The quadratic formula for solving the equation ax² + bx + c = 0; a ≠ 0 is x = (-b ± √(b² - 4ac)) / (2a).
  • To convert ax²ⁿ + bxⁿ + c = 0 (a ≠ 0) into quadratic form, the correct substitution is y = xⁿ.
  • The equation in which the variable occurs as an exponent is called an Exponential equation.
  • To convert 4^(1+x) + 4^(1-x) = 10 into a quadratic equation, the substitution is y = 4^x.
  • The equations involving radical expressions of the variable are called Radical equations.
  • The cube roots of unity are 1, (-1+√(-3))/2, (-1-√(-3))/2.
  • The sum of all cube roots of 64 is 0.
  • The product of the cube roots of -1 is 1.
  • 16w⁸ + 16w⁴ = -16
  • The sum of all four fourth roots of unity is 0.
  • The product of all four fourth roots of unity is -1.
  • The sum of all four fourth roots of 16 is 0.
  • The complex cube roots of unity are Conjugate to each other.
  • The complex cube roots of unity are reciprocals of each other, and additive inverses.
  • The complex fourth roots of unity are squares of each other.
  • If the sum of all cube roots of unity is equal to x² + 1, then x is equal to ±i
  • If the product of all cube roots of unity is equal to p² + 1, then p is equal to 0.
  • The expression x² + 1/x - 3 is not a polynomial.
  • If f(x) is divided by x - a, then dividend = (Divisor)(Quotient) + Remainder.
  • If f(x) is divided by x - a, by the remainder theorem, then the remainder is f(a).
  • The polynomial (x – a) is a factor of f(x) if and only if f(a) = 0
  • x - 2 is a factor of x² - kx + 4, if k is 4.
  • If x = -2 is the root of kx⁴ – 13x² + 36 = 0, then k = -2.
  • x + a is a factor of xⁿ + aⁿ when n is any odd integer.
  • x - a is a factor of xⁿ – aⁿ when n is any integer.
  • The sum of roots of ax² – bx - c = 0 is (a ≠ 0) b/a.
  • The sum of roots of ax² + bx - c = 0 is (a ≠ 0) -b/a.
  • If 2 and -5 are roots of a quadratic equation, then the equation is x² + 3x − 10 = 0.
  • If α and β are the roots of 3x² – 2x + 4 = 0, then the value of α + β is 2/3.
  • If roots of ax² + bx + c = 0, (a ≠ 0) are real, then Disc≥ 0
  • If roots of ax² + bx + c = 0, (a ≠ 0) are equal, then Disc= 0
  • The graph of a quadratic equation is a Parabola
  • ω²⁸ + ω²⁹ + 1 = 0
  • Synthetic division is a process of division
  • The degree of a quadratic equation is 2.
  • Basic techniques for solving quadratic equations are finding factors.
  • If 16ω⁴ + 16ω⁸=16.

Chapter 5: Partial Fractions

  • An open sentence formed by using the sign "=" is called an Equation.
  • If an equation is true for all values of the variable, then it is called an identity.
  • (x + 3)(x + 4) = x² + 7x + 12 is an identity.
  • The quotient of two polynomials P(x)/Q(x) is called a rational fraction
  • A fraction P(x)/Q(x), Q(x) ≠ 0 is a proper fraction if the degree of P(x) <the Degree of Q(x).
  • A mixed form of a fraction has a polynomial and a proper fraction.
  • When a rational fraction is separated into partial fractions, then the result is always an Identity.
  • The number of partial fractions x³/((x + 1)(x² − 1)) is 4.
  • Conditional equation 2x + 3 = 0 holds when x is equal to -3/2
  • x³ - 6x² + 8x is a reducible factor
  • A quadratic factor which cannot be written as a product of linear factors with real coefficients is called an irreducible factor.
  • 9x²/(x³-1) is a proper fraction.

Chapter 6: Sequence and Series

  • An arrangement of numbers according to some definite rule is called a Sequence
  • A Sequence is also known as a Progression
  • A sequence is a function whose domain is N.
  • A sequence whose range is R (set of real numbers) is called a Real Sequence
  • If aₙ = {n + (−1)ⁿ}, then a₁₀ = 12
  • The last term of an infinite sequence doesn't exist
  • The next term of the sequence 1, 2, 12, 40, ... is not defined
  • A sequence {aₙ} in which aₙ - Aₙ₋₁ is the same number for all n ∈ N, n > 1 is called an A.P (Arithmetic Progression)
  • If the nth term of an A.P is 3n – 1, then the 10th term is 29
  • If aₙ₋₁, aₙ, aₙ₊₁ are in A.P, then aₙ is A.M (Arithmetic Mean)
  • The arithmetic mean between √2 and 3√2 is 2√2
  • The sum of terms of a sequence is called a Series
  • No term of a G. P is 0
  • A.M between c and d is (c+d)/2
  • If A, G, and H are Arithmetic, Geometric and Harmonic means between two +ive numbers then A >=G >=H
  • The general term of a G. P is aₙ = arⁿ⁻¹

Chapter 7: Permutation, Combination and Probability

  • 20P₃= 6840
  • If ⁿP₂ = 30, then n = 6
  • The number of diagonals in a 10-sided figure is calculated as ¹⁰C₂ - 10
  • ⁿC₀ = 1
  • Number of arrangements of the world "MATHEMATICS" can be made ¹¹!/((3,2,1,1,1,1,1) * (2,2,1,1,1,1,1))
  • If n is negative, then n! is not defined.
  • ⁿ⁻¹Cᵣ + ⁿ⁻¹Cᵣ₋₁ = ⁿCᵣ
  • The number of signals that can be given by 5 flags of different colors, using 3 at a time is 60.
  • ⁿPₙ = factorial 'n'
  • If an event A can occur in p ways and event B can occur by q ways, the, the numbers of ways of both events occur is p.q
  • If "ⁿC₁₂ = ⁿC₁₃, then the value of n = 20
  • The probability of non-occurrence of an event E is equal to 1-P(E).
  • For independent events, P(A ∩ B) = P(A).P(B)
  • If A card is drawn from a deck of 52 playing cards, then the probability of getting an ace card is 4/13.
  • If four persons want to sit on circular sofa, then the total number of ways are 6
  • In the playing of two teams A and B, is team A does not lose the probability of is ½.
  • Let S = {1, 2, 3, ..., 10} the probability that a number is divided by 4 is 3/10.
  • If rolling a tie, then probability of getting 3 or getting 5 is 1/3
  • If a coin is tossed 5 times, then n(S) is equal to 32
  • The number of ways for sitting 4 persons in atrain on a straight sofa is 24.
  • Sample space for tossing a coin is {H,T}.
  • If an event always occurs, then it is called a Certain event
  • If E is a certain even, then P(E) =1
  • In a permutation, when nPr or P(n,r), then n>=r
  • Non occurrence of an event E is denoted by E^C

Chapter 8: Mathematical Induction and Binomial Theorem

  • The statement 4ⁿ + 3 + 4 is true when: n ≥ 2
  • The number of terms in the expansion of (a + b)ⁿ are: n+1
  • Middle term/s in the expansion of (a – 3x)¹⁴ is/are : T₇ & T₈
  • The coefficient of the last term in the expansion of (2 – x)⁷ is : (-1)^7 = -7
  • (ⁿC₀) + (ⁿC₁) + (ⁿC₂) + ... + (ⁿCₙ) is equal to: 2²ⁿ⁻¹
  • 1 + x + x² + x³ + ... = (1-x)⁻¹ if |x| < -1
  • The method of induction was given by Francesco from 1494-1575

Chapter 9: Fundamentals of Trigonometry

  • Two rays with a common starting point form an Angle
  • The common starting point of the two rays is called the Vertex
  • If the rotation of an angle is counter clockwise, then the angle is positive.
  • If the initial ray OA rotates in an anti-clockwise direction in such a way that it coincides with itself, the angle then formed is 360°
  • One rotation in an anti-clock wise direction is equal to 360°
  • If the rotation of the angle is clockwise, then the angle is negative.
  • A straight line angle is equal to π radian.
  • One right angle is equal to π rotation.
  • 1° is equal to 60 minutes.
  • 1° is equal to 3600 seconds
  • 60th part of 1° is equal to minutes.
  • 60th part of 1 is equal to 1".
  • 3600th part of 1° is equal to 1"
  • The system of angular measurement in which the angle is measured in radian (cicular measure) is called Circular System
  • The Sexagesimal system is also called the English System
  • The radian is a constant angle
  • One sign of the angle is that of the quadrantan in which the terminal sides of the angle falls
  • The relation between the length of an arc of a circle and the circular measure of it central angle is The point (0,1) lies on the terminal side of the angle is 90°

Chapter 10: Trigonometric Identities

  • The distance between the points A(3,8) & B(5,6) is √8
  • The fundamental law of trigonometry is, cos(α - β) = cosαcosβ + sinαsinβ
  • The set of sine and cosine of the angle lies in the close interval of [-1,1]
  • sin(π/2−θ)=cosθ
  • tan (π/2−θ)=cotθ
  • The simplest form is cos4θ + cos2θ=2cos⁡3θ cos⁡θ

Chapter 11: Trigonometric Function and their graphs

  • The period of trigonometric ration lies in [-1,1]
  • The smallest+ive number which when added to the original circular measure of the angle gives the same value of the function is called Period
  • For Trigonometric series, the value of 2cos2θ sinθ is = sin3θ

Chapter 12: Application of Trigonometry

  • To solve an oblique triangle, we use : Law of Sine
  • To solve an oblique triangles when three sides are given, we can use :Law of cosine

Chapter 13: Inverse Trigonometric Functions

  • Inverse of a function exist only if it is : √(1-1) function
  • The following are trigonometric expressions but not identities

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