# Math Blog

### Square Matrix | Definition of Square Matrix |Diagonal of Square Matrix

If square matrixes have n rows or columns then the matrix is called the square matrix of order n or an n-square matrix. Definition of Square Matrix: An n × n matrix is said to be a square matrix of order n. In other words when the number of rows and the number of columns in

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### Matrix | Definition of a Matrix | Examples of a Matrix | Elements

A rectangular array of mn elements aij into m rows and n columns, where the elements aij belongs to field F, is said to be a matrix of order m × n (or an m × n matrix) over the field F. Definition of a Matrix: A matrix is a rectangular arrangement or array of numbers

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### Joint Variation | Solving Joint Variation Problems and Application

One variable quantity is said to vary jointly as a number of other variable quantities, when it varies directly as their product. If the variable A varies directly as the product of the variables B, C and D, i.e., if.A ∝ BCD or A = kBCD (k = constant ), then A varies jointly

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### Indirect Variation | Inverse Variation | Inverse or Indirect Variation

When two variables change in inverse proportion it is called as indirect variation. In indirect variation one variable is constant times inverse of other. If one variable increases other will decrease, if one decrease other will also increase. This means that the variables

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### Direct Variation | Solving Direct Variation Word Problem

When two variables change in proportion it is called as direct variation. In direct variation one variable is constant times of other. If one variable increases other will increase, if one decrease other will also decease. This means that the variables change in a same ratio

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### Methods of Solving Simultaneous Linear Equations | Solved Examples

There are different methods for solving simultaneous linear Equations: 1. Elimination of a variable 2. Substitution 3. Cross-multiplication 4. Evaluation of proportional value of variables This topic is purely based upon numerical examples. So, let us solve some examples

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### Method of Cross Multiplication|Solve by Method of Cross Multiplication

The next method of solving linear equations in two variables that we are going to learn about is method of cross multiplication. Let us see the steps followed while soling the linear equation by method of cross multiplication: Assume two linear equation be A1 x + B1y + C1=

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### Properties of Angles of a Triangle |Sum of Three Angles of a Triangle

We will discuss about some of the properties of angles of a triangle. 1. The three angles of a triangle are together equal to two right angles. ABC is a triangle. Then ∠ZXY + ∠XYZ + ∠YZX = 180° Using this property, let us solve some of the examples. Solved examples

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### Geometrical Property of Altitudes|Altitudes of Triangle are Concurrent

The three altitudes of triangle are concurrent. The point at which they intersect is known as the orthocentre of the triangle. In the adjoining figure, the three altitudes XP, YQ and ZR intersect at the orthocentre O.

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### Medians and Altitudes of a Triangle |Three Altitudes and Three Medians

Here we will discuss about Medians and Altitudes of a Triangle. Median: The straight line joining a vertex of a triangle to the midpoint of the opposite side is called a median. A triangle has three medians. Here XL, YM and ZN are medians. A geometrical property of medians

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### Classification of Triangles on the Basis of Their Sides and Angles

Here we will discuss about classification of triangles on the basis of their sides and angles Equilateral triangle: An equilateral triangle is a triangle whose all three sides are equal. Here, XYZ is an equilateral triangle as XY = YZ = ZX. Isosceles triangle: An isosceles

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### Triangle | Exterior Opposite Angles|Interior Opposite Angles|Perimeter

A triangle is a plane figure bounded by three straight lines. A triangle has three sides and three angles, and each one of them is called an element of the triangle. Here, PQR is a triangle, its three sides are line segments PQ, QR and RP; ; ∠PQR, ∠QRP and ∠RPQ are its

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### Problem on Change the Subject of a Formula | Changing the Subject

We will solve different types of problems on change the subject of a formula. The subject of a formula is a variable whose relation with other variables of the context is sought and the formula is written in such a way that subject is expressed in terms of the other

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### Worksheet on Change of Subject | Change the Subject as Indicated

Practice the questions given in the worksheet on change of subject When a formula involving certain variables is known, we can change the subject of the formula. What is the subject in each of the following questions? Change the subject as indicated.

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### Worksheet on Framing a Formula | Framing Formulas | Frame an Equation

Practice the questions given in the worksheet on framing a formula. I. Frame a formula for each of the following statements: 1. The side ‛s’ of a square is equal to the square root of its area A.

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### Establishing an Equation | Framing a Formula | Framing Linear Equation

We will discuss here about establishing an equation. In a given context, the relation between variables expressed by equality (or inequality) is called a formula. When a formula is expressed by an equality, the algebraic expression is called an equation.

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### Rectangular Cartesian Co-ordinates | Abscissa | Ordinate | Oblique Co-ordinate

What is Rectangular Cartesian Co-ordinates? Let O be a fixed point on the plane of this page; draw mutually perpendicular straight line XOX’ and YOY’ through O. Clearly, these lines divide the plane

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### What is Co-ordinate Geometry? | Analytical Geometry| Cartesian Co-ordinate

What is co-ordinate geometry? The subject co-ordinate geometry is that particular branch of mathematics in which geometry is studied with the help of algebra. This branch of mathematics was fir

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### Table of Tangents and Cotangents | Natural Tangents and Natural Cotangents

We will discuss here the method of using the table of tangents and cotangents. This table shown below is also known as the table of natural tangents and natural cotangents. Using the table we can

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### Table of Sines and Cosines |Trigonometric Table|Table of Natural sines & cosines

We will discuss here the method of using the table of sines and cosines: The above table is also known as the table of natural sines and natural cosines. Using the table we can find the values

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### Problems on Properties of Triangle | Angle Properties of Triangles

We will solve different types of problems on properties of triangle. 1. If in any triangle the angles be to one another as 1 : 2 : 3, prove that the corresponding sides are 1 : √3 : 2.

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### Properties of Triangle Formulae | Triangle Formulae | Properties of Triangle

We will discuss the list of properties of triangle formulae which will help us to solve different types of problems on triangle.

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### Law of Tangents |The Tangent Rule|Proof of the Law of Tangents|Alternative Proof

We will discuss here about the law of tangents or the tangent rule which is required for solving the problems on triangle. In any triangle ABC,

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### Area of a Triangle | ∆ = ½ bc sin A | ∆ = ½ ca sin B | ∆ = ½ ab sin C

If ∆ be the area of a triangle ABC, Proved that, ∆ = ½ bc sin A = ½ ca sin B = ½ ab sin C That is, (i) ∆ = ½ bc sin A (ii) ∆ = ½ ca sin B (iii) ∆ = ½ ab sin C

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### The Law of Cosines | The Cosine Rule | Cosine Rule Formula | Cosine Law Proof

We will discuss here about the law of cosines or the cosine rule which is required for solving the problems on triangle. In any triangle ABC, Prove that, (i) b$$^{2}$$

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### Proof of Projection Formulae | Projection Formulae | Geometrical Interpretation

The geometrical interpretation of the proof of projection formulae is the length of any side of a triangle is equal to the algebraic sum of the projections of other sides upon it. In Any Triangle

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### Projection Formulae | a = b cos C + c cos B | b = c cos A + a cos C

Projection formulae is the length of any side of a triangle is equal to the sum of the projections of other two sides on it. In Any Triangle ABC, (i) a = b cos C + c cos B

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### Theorem on Properties of Triangle | p/sin P = q/sin Q = r/sin R = 2K

Proof the theorem on properties of triangle p/sin P = q/sin Q = r/sin R = 2K. Proof: Let O be the circum-centre and R the circum-radius of any triangle PQR. Let O be the circum-centre and R

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### The Law of Sines | The Sine Rule | The Sine Rule Formula | Law of sines Proof

We will discuss here about the law of sines or the sine rule which is required for solving the problems on triangle. In any triangle the sides of a triangle are proportional to the sines

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###### Jan 20, 2018

In trigonometry we will discuss about the different properties of triangles. We know any triangle has six parts, the three sides and the three angles are generally called the elements of the triangle.

### Problems on Inverse Trigonometric Function | Inverse Circular Function Problems

We will solve different types of problems on inverse trigonometric function. 1. Find the values of sin (cos$$^{-1}$$ 3/5)

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### Principal Values of Inverse Trigonometric Functions |Different types of Problems

We will learn how to find the principal values of inverse trigonometric functions in different types of problems. The principal value of sin$$^{-1}$$ x for x > 0, is the length of the arc of a unit

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### Inverse Trigonometric Function Formula | Inverse Circular Function Formula

We will discuss the list of inverse trigonometric function formula which will help us to solve different types of inverse circular or inverse trigonometric function.

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### 3 arctan(x) | 3 tan$$^{-1}$$ x |3 tan inverse x | Inverse Trigonometric Function

We will learn how to prove the property of the inverse trigonometric function 3 arctan(x) = arctan($$\frac{3x - x^{3}}{1 - 3 x^{2}}$$) or, 3 tan$$^{-1}$$ x = tan$$^{-1}$$

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### 3 arccos(x) | 3 cos$$^{-1}$$ x |3 cos inverse x | Inverse Trigonometric Function

We will learn how to prove the property of the inverse trigonometric function 3 arccos(x) = arccos(4x$$^{3}$$ - 3x) or, 3 cos$$^{-1}$$ x = cos$$^{-1}$$ (4x$$^{3}$$ - 3x)

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### 3 arcsin(x) | 3 sin$$^{-1}$$ x |3 sin inverse x | Inverse Trigonometric Function

We will learn how to prove the property of the inverse trigonometric function 3 arcsin(x) = arcsin(3x - 4x$$^{3}$$) or, 3 sin$$^{-1}$$ x = sin$$^{-1}$$ (3x - 4x$$^{3}$$)

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### 2 arctan(x) | 2 tan$$^{-1}$$ x | 2 tan inverse x |Inverse Trigonometric Function

We will learn how to prove the property of the inverse trigonometric function, 2 arctan(x) = arctan($$\frac{2x}{1 - x^{2}}$$) = arcsin($$\frac{2x}{1 + x^{2}}$$)

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### 2 arccos(x) | 2 cos$$^{-1}$$ x | 2 cos inverse x |Inverse Trigonometric Function

We will learn how to prove the property of the inverse trigonometric function 2 cos$$^{-1}$$ x = cos$$^{-1}$$ (2x$$^{2}$$ - 1) or, 2 arccos(x) = arccos(2x$$^{2}$$ - 1).

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### 2 arcsin(x) | 2 sin$$^{-1}$$ x | 2 sin inverse x |Inverse Trigonometric Function

We will learn how to prove the property of the inverse trigonometric function 2 arcsin(x) = arcsin(2x$$\sqrt{1 - x^{2}}$$) or, 2 sin$$^{-1}$$ x = sin$$^{-1}$$ (2x$$\sqrt{1 - x^{2}}$$)

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### arccos(x) - arccos(y) | cos^-1 x - cos^-1 y | Inverse Trigonometric Function

We will learn how to prove the property of the inverse trigonometric function arccos(x) - arccos(y) = arccos(xy + $$\sqrt{1 - x^{2}}$$$$\sqrt{1 - y^{2}}$$)

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### arccos(x) + arccos(y) | cos^-1 x + cos^-1 y | Inverse Trigonometric Function

We will learn how to prove the property of the inverse trigonometric function arccos (x) + arccos(y) = arccos(xy - $$\sqrt{1 - x^{2}}$$$$\sqrt{1 - y^{2}}$$)

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### arcsin x - arcsin y |sin$$^{-1}$$ x - sin$$^{-1}$$ y|sin inverse x-sin inverse y

We will learn how to prove the property of the inverse trigonometric function arcsin (x) - arcsin(y) = arcsin (x $$\sqrt{1 - y^{2}}$$ - y$$\sqrt{1 - x^{2}}$$)

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### arcsin(x) + arcsin(y) |sin$$^{-1}$$ x+sin$$^{-1}$$ y|sin inverse x+sin inverse y

We will learn how to prove the property of the inverse trigonometric function arcsin (x) + arcsin(y) = arcsin (x $$\sqrt{1 - y^{2}}$$ + y$$\sqrt{1 - x^{2}}$$)

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### arccot(x) - arccot(y) | cot^-1 x - cot^-1 y | Inverse Trigonometric Function

We will learn how to prove the property of the inverse trigonometric function arccot(x) - arccot(y) = arccot($$\frac{xy + 1}{y - x}$$) (i.e., cot$$^{-1}$$ x + cot$$^{-1}$$ y = cot$$^{-1}$$

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### arccot(x) + arccot(y) | cot^-1 x + cot^-1 y | Inverse Trigonometric Function

We will learn how to prove the property of the inverse trigonometric function arccot(x) + arccot(y) = arccot($$\frac{xy - 1}{y + x}$$) (i.e., cot$$^{-1}$$ x - cot$$^{-1}$$ y = cot$$^{-1}$$

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### arctan(x) + arctan(y) + arctan(z) | tan^-1 x + tan^-1 y + tan^-1 z |Inverse Trig

We will learn how to prove the property of the inverse trigonometric function arctan(x) + arctan(y) + arctan(z) = arctan$$\frac{x + y + z – xyz}{1 – xy – yz – zx}$$ (i.e., tan$$^{-1}$$ x

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### arctan x + arccot x = π/2 | arctan(x) + arccot(x) = $$\frac{π}{2}$$ | Examples

We will learn how to prove the property of the inverse trigonometric function arctan(x) + arccot(x) = $$\frac{π}{2}$$ (i.e., tan$$^{-1}$$ x + cot$$^{-1}$$ x = $$\frac{π}{2}$$).

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### arctan x - arctan y | tan^-1 x - tan^-1 y | Inverse Trigonometric Function

We will learn how to prove the property of the inverse trigonometric function arctan(x) - arctan(y) = arctan($$\frac{x - y}{1 + xy}$$) (i.e., tan$$^{-1}$$ x - tan$$^{-1}$$ y

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### arctan(x) + arctan(y) = arctan($$\frac{x + y}{1 - xy}$$) | tan^-1 x + tan^-1 y

We will learn how to prove the property of the inverse trigonometric function arctan(x) + arctan(y) = arctan($$\frac{x + y}{1 - xy}$$), (i.e., tan$$^{-1}$$ x + tan$$^{-1}$$ y = tan$$^{-1}$$

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