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Jul 29, 2014
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.
Continue reading "Theorems on Properties of Triangle | p/sin P = q/sin Q = r/sin R = 2K"
Jul 27, 2014
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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Jul 27, 2014
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,
Continue reading "Law of Tangents |The Tangent Rule|Proof of the Law of Tangents|Alternative Proof"
Jul 24, 2014
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\(^{2}\)
Continue reading "The Law of Cosines | The Cosine Rule | Cosine Rule Formula | Cosine Law Proof"
Jul 17, 2014
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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Jul 16, 2014
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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Jul 16, 2014
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
Continue reading "Projection Formulae | a = b cos C + c cos B | b = c cos A + a cos C"
Jul 16, 2014
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
Continue reading "The Law of Sines | The Sine Rule | The Sine Rule Formula | Law of sines Proof"
Jul 14, 2014
Properties of Triangles | Semi-perimeter| Circum-circle|Circum-radius|In-radius
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.
Continue reading "Properties of Triangles | Semi-perimeter| Circum-circle|Circum-radius|In-radius "
Jul 13, 2014
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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Jul 13, 2014
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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Jul 12, 2014
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
Continue reading "Principal Values of Inverse Trigonometric Functions |Different types of Problems"
Jul 10, 2014
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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Jul 07, 2014
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\(^{-1}\)
Continue reading "3 arctan(x) | 3 tan\(^{-1}\) x |3 tan inverse x | Inverse Trigonometric Function"
Jul 07, 2014
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)
Continue reading "3 arccos(x) | 3 cos\(^{-1}\) x |3 cos inverse x | Inverse Trigonometric Function"
Jul 07, 2014
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\(^{3}\))
Continue reading "3 arcsin(x) | 3 sin\(^{-1}\) x |3 sin inverse x | Inverse Trigonometric Function"
Jul 07, 2014
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(\(\frac{2x}{1 + x^{2}}\))
Continue reading "2 arctan(x) | 2 tan\(^{-1}\) x | 2 tan inverse x |Inverse Trigonometric Function"
Jul 07, 2014
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).
Continue reading "2 arccos(x) | 2 cos\(^{-1}\) x | 2 cos inverse x |Inverse Trigonometric Function"
Jul 07, 2014
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\(\sqrt{1 - x^{2}}\))
Continue reading "2 arcsin(x) | 2 sin\(^{-1}\) x | 2 sin inverse x |Inverse Trigonometric Function"
Jul 04, 2014
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 + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))
Continue reading "arccos(x) - arccos(y) | cos^-1 x - cos^-1 y | Inverse Trigonometric Function"
Jul 04, 2014
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 - \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))
Continue reading "arccos(x) + arccos(y) | cos^-1 x + cos^-1 y | Inverse Trigonometric Function"
Jul 04, 2014
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\(\sqrt{1 - x^{2}}\))
Continue reading "arcsin x - arcsin y |sin\(^{-1}\) x - sin\(^{-1}\) y|sin inverse x-sin inverse y"
Jul 04, 2014
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\(^{-1}\)
Continue reading "arccot(x) - arccot(y) | cot^-1 x - cot^-1 y | Inverse Trigonometric Function"
Jul 04, 2014
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\(^{-1}\)
Continue reading "arccot(x) + arccot(y) | cot^-1 x + cot^-1 y | Inverse Trigonometric Function "
Jul 03, 2014
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\(\sqrt{1 - x^{2}}\))
Continue reading "arcsin(x) + arcsin(y) |sin\(^{-1}\) x+sin\(^{-1}\) y|sin inverse x+sin inverse y"
Jul 01, 2014
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
Continue reading "arctan(x) + arctan(y) + arctan(z) | tan^-1 x + tan^-1 y + tan^-1 z |Inverse Trig"
Jul 01, 2014
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
Continue reading "arctan x - arctan y | tan^-1 x - tan^-1 y | Inverse Trigonometric Function"
Jul 01, 2014
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\(^{-1}\)
Continue reading "arctan(x) + arctan(y) = arctan(\(\frac{x + y}{1 - xy}\)) | tan^-1 x + tan^-1 y"
Jun 30, 2014
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 = \(\frac{π}{2}\)).
Continue reading "arcsec x + arccsc x = π/2 | arcsec(x) + arccsc(x) = \(\frac{π}{2}\) | Examples"
Jun 30, 2014
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 = \(\frac{π}{2}\)).
Continue reading "arctan x + arccot x = π/2 | arctan(x) + arccot(x) = \(\frac{π}{2}\) | Examples"
Jun 28, 2014
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
Continue reading "arcsin x + arccos x = π/2 | arcsin(x) + arccos(x) = \(\frac{π}{2}\) | Examples"
Jun 27, 2014
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.
Continue reading "General and Principal Values of cot\(^{-1}\) x | Inverse Circular Functions"
Jun 24, 2014
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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Jun 24, 2014
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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Jun 23, 2014
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.
Continue reading "General and Principal Values of tan\(^{-1}\) x | Inverse Circular Functions"
Jun 23, 2014
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.
Continue reading "General and Principal Values of cos\(^{-1}\) x | Inverse Circular Functions"
Jun 22, 2014
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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Jun 22, 2014
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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Jun 20, 2014
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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Jun 20, 2014
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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Jun 20, 2014
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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Jun 18, 2014
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
Continue reading "Cos Theta Equals 0 | General Solution of the Equation cos θ = 0 | cos θ = 0"
Jun 17, 2014
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.
Continue reading "Tan Theta Equals Tan Alpha | General Solution of tan θ = tan ∝ |General Solution"
Jun 17, 2014
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.
Continue reading "Cos Theta Equals Cos Alpha | General Solution of cos θ = cos ∝ |General Solution"
Jun 17, 2014
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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Jun 16, 2014
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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Jun 16, 2014
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.
Continue reading "Sin Theta Equals 1 | General Solution of the Equation sin θ = 1 | sin θ = 1"
Jun 15, 2014
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 ).
Continue reading "a cos Theta Plus b sin Theta Equals c |General Solution of a cos θ + b sin θ = c"
Jun 14, 2014
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.
Continue reading "Sin Theta Equals Sin Alpha | General Solution of sin θ = sin ∝ |General Solution"
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