# Math Blog

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

How to proof the theorems 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.

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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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### 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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### 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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### 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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### 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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###### Jul 14, 2014

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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### General Values of Inverse Trigonometric Functions | Inverse Circular Functions

We will learn how to find the general values of inverse trigonometric functions in different types of problems. 1. Find the general values of sin$$^{-1}$$ (- √3/2)

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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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### 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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### 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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### 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 - 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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### arcsec x + arccsc x = π/2 | arcsec(x) + arccsc(x) = $$\frac{π}{2}$$ | Examples

We will learn how to prove the property of the inverse trigonometric function arcsec(x) + arccsc(x) = $$\frac{π}{2}$$ (i.e., sec$$^{-1}$$ x + csc$$^{-1}$$ x = $$\frac{π}{2}$$).

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

We will learn how to prove the property of the inverse trigonometric function arcsin(x) + arccos(x) = $$\frac{π}{2}$$. Proof: Let, sin$$^{-1}$$ x = θ Therefore, x = sin θ

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### General and Principal Values of cot$$^{-1}$$ x | Inverse Circular Functions

How to find the general and principal values of cot$$^{-1}$$ x? Let cot θ = x (- ∞ < x < ∞) then θ = cot$$^{-1}$$ x. Here θ has infinitely many values.

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### General and Principal Values of sec$$^{-1}$$ x | General values of arc sec x

How to find the general and principal values of sec$$^{-1}$$ x? Let sec θ = x ( I x I ≥ 1 i.e., x ≥ 1 or, x ≤ - 1 ) then θ = sec - 1x . Here θ has infinitely many values.

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### General and Principal Values of csc$$^{-1}$$ x | Inverse Circular Functions

How to find the general and principal values of ccs$$^{-1}$$ x? Let csc θ = x ( I x I ≥ 1 i.e., x ≥ 1 or, x ≤ - 1 ) then θ = csc - 1x . Here θ has infinitely many values.

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### General and Principal Values of tan$$^{-1}$$ x | Inverse Circular Functions

How to find the general and principal values of tan$$^{-1}$$ x? Let tan θ = x (- ∞ < x < ∞) then θ = tan$$^{-1}$$ x. Here θ has infinitely many values.

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### General and Principal Values of cos$$^{-1}$$ x | Inverse Circular Functions

How to find the general and principal values of cos$$^{-1}$$ x? Let cos θ = x where, (- 1 ≤ x ≤ 1) then θ = cos$$^{-1}$$ x. Here θ has infinitely many values.

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### General and Principal Values of sin$$^{-1}$$ x | Inverse Trigonometric Functions

What are the general and principal Values of sin$$^{-1}$$ x? What is sin$$^{-1}$$ ½? We know that sin (30°) = ½.

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### Inverse Trigonometric Functions | Inverse Circular Functions | Introduction

We will discuss here about Inverse trigonometric Functions or inverse circular functions. The inverse of a function f: A ⟶B exists if and only if f is one-one onto (i.e., bijection) and given by

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### General solution of Trigonometric Equation |Solution of a Trigonometric Equation

We will learn how to find the general solution of trigonometric equation of various forms using the identities and the different properties of trig functions.

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### Problems on Trigonometric Equation | Trigonometric Equation Formulas

We will learn how to solve different types of problems on trigonometric equation containing one or many trig functions. First we need to solve the trigonometric function (if required) and

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### Trigonometric Equation using Formula | Problems on Trigonometric Equation

We will learn how to solve trigonometric equation using formula. Here we will use the following formulas to get the solution of the trigonometric equations. (a) If sin θ = 0 then θ = nπ

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### Cos Theta Equals 0 | General Solution of the Equation cos θ = 0 | cos θ = 0

How to find the general solution of the equation cos θ = 0? Prove that the general solution of cos θ = 0 is θ = (2n + 1) π/2, n ∈ Z

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### Tan Theta Equals Tan Alpha | General Solution of tan θ = tan ∝ |General Solution

How to find the general solution of an equation of the form tan θ = tan ∝? Prove that the general solution of tan θ = tan ∝ is given by θ = nπ +∝, n ∈ Z.

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### Cos Theta Equals Cos Alpha | General Solution of cos θ = cos ∝ |General Solution

How to find the general solution of an equation of the form cos θ = cos ∝? Prove that the general solution of cos θ = cos ∝ is given by θ = 2nπ ± ∝, n ∈ Z.

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### Tan Theta Equals 0 | General Solution of the Equation tan θ = 0 | tan θ = 0

How to find the general solution of the equation tan θ = 0? Prove that the general solution of tan θ = 0 is θ = nπ, n ∈ Z.

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### Sin Theta Equals 0 | General Solution of the Equation sin θ = 0 | sin θ = 0

How to find the general solution of the equation sin θ = 0? Prove that the general solution of sin θ = 0 is θ = nπ, n ∈ Z Solution:

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### Sin Theta Equals 1 | General Solution of the Equation sin θ = 1 | sin θ = 1

How to find the general solution of an equation of the form sin θ = 1? Prove that the general solution of sin θ = 1 is given by θ = (4n + 1)π/2, n ∈ Z.

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### a cos Theta Plus b sin Theta Equals c |General Solution of a cos θ + b sin θ = c

Trigonometric equations of the form a cos theta plus b sin theta equals c (i.e. a cos θ + b sin θ = c) where a, b, c are constants (a, b, c ∈ R) and |c| ≤ √(a^2 + b^2 ).

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### Sin Theta Equals Sin Alpha | General Solution of sin θ = sin ∝ |General Solution

How to find the general solution of an equation of the form sin θ = sin ∝? Prove that the general solution of sin θ = sin ∝ is given by θ = nπ + (-1)^n ∝, n ∈ Z.

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