Podcast
Questions and Answers
If $a$ is any non-zero real number, what is its multiplicative inverse?
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?
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?
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?
If $a < b$, which of the following statements is always true?
What is the additive identity in the set of complex numbers?
What is the additive identity in the set of complex numbers?
What is the multiplicative identity in the set of complex numbers?
What is the multiplicative identity in the set of complex numbers?
What is the modulus of the complex number $z = a + ib$?
What is the modulus of the complex number $z = a + ib$?
Which operation is squaring a number considered as?
Which operation is squaring a number considered as?
Given that $A = \phi$, what is $P(A)$?
Given that $A = \phi$, what is $P(A)$?
Given that sets A and B are disjoint, which statement is true?
Given that sets A and B are disjoint, which statement is true?
Which of the following is NOT a binary operation?
Which of the following is NOT a binary operation?
If $y = \sqrt{x}, x \geq 0$ is a function, what is its inverse?
If $y = \sqrt{x}, x \geq 0$ is a function, what is its inverse?
What is another name for an onto function?
What is another name for an onto function?
For the propositions $p$ and $q$, what is $(p \land q) \implies p$?
For the propositions $p$ and $q$, what is $(p \land q) \implies p$?
A compound proposition which is always true is called what?
A compound proposition which is always true is called what?
Which of the following is true for any set $A$?
Which of the following is true for any set $A$?
What is a rectangular array of numbers enclosed by square brackets called?
What is a rectangular array of numbers enclosed by square brackets called?
For a square matrix $A = [a_{ij}]{n \times n}$, what are the elements $a{11}, a_{22}, a_{33}, ..., a_{nn}$ called?
For a square matrix $A = [a_{ij}]{n \times n}$, what are the elements $a{11}, a_{22}, a_{33}, ..., a_{nn}$ called?
If a matrix of order $m \times n$ is given, what is the order of its transpose?
If a matrix of order $m \times n$ is given, what is the order of its transpose?
Given that A is a matrix and n is an integer, what is equivalent to adding A to itself n times ($A + A + A + \dots$)?
Given that A is a matrix and n is an integer, what is equivalent to adding A to itself n times ($A + A + A + \dots$)?
If $Adj A$ = $\begin{bmatrix} -1 & -2\ 3 & -1 \end{bmatrix}$, then matrix $A^{-1}$ is?
If $Adj A$ = $\begin{bmatrix} -1 & -2\ 3 & -1 \end{bmatrix}$, then matrix $A^{-1}$ is?
If $AX = B$, then $X$ is equal to which of the following?
If $AX = B$, then $X$ is equal to which of the following?
A square matrix $A$ is symmetric if what condition holds true?
A square matrix $A$ is symmetric if what condition holds true?
If $A$ is any square matrix, then $A + A^t$ is what type of matrix?
If $A$ is any square matrix, then $A + A^t$ is what type of matrix?
Which type of matrix possesses a multiplicative inverse?
Which type of matrix possesses a multiplicative inverse?
In a homogeneous system of linear equations, what is the solution (0, 0, 0) called?
In a homogeneous system of linear equations, what is the solution (0, 0, 0) called?
The statement '$ax^2 + bx + c = 0$ will be quadratic' is true if what condition is met?
The statement '$ax^2 + bx + c = 0$ will be quadratic' is true if what condition is met?
The imaginary solutions to all cube roots of unity are related in what way?
The imaginary solutions to all cube roots of unity are related in what way?
Flashcards
What is a matrix?
What is a matrix?
A rectangular array of numbers enclosed by square brackets.
What are rows?
What are rows?
In a matrix, these are the horizontal lines of numbers.
What are Columns?
What are Columns?
In a matrix, these are the vertical lines of numbers.
What is the order of A?
What is the order of A?
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What does a_{ij} represent?
What does a_{ij} represent?
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When is a matrix real?
When is a matrix real?
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What is a row vector?
What is a row vector?
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What is a column vector?
What is a column vector?
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What is a square matrix?
What is a square matrix?
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What is a rectangular matrix?
What is a rectangular matrix?
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What is the main diagonal?
What is the main diagonal?
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What is a scalar matrix?
What is a scalar matrix?
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What is a Symmetric matrix
What is a Symmetric matrix
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What is a Skew-symmetric matrix?
What is a Skew-symmetric matrix?
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When does inverse exist?
When does inverse exist?
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What is a Theorem?
What is a Theorem?
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What is a conditional equation?
What is a conditional equation?
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What is a proper fraction?
What is a proper fraction?
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What is a Rational fraction?
What is a Rational fraction?
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What is Squaring?
What is Squaring?
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What is an Irrational number?
What is an Irrational number?
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What is a Venn diagram?
What is a Venn diagram?
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What is an inverse function?
What is an inverse function?
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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=2cos3θ cosθ
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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