x To The Power Of Four

X to the power of four defines that the variable x has the power value four. Generally the variables come under the linear algebra category. The problems with the variable x with the power four gives the brief introduction to this topic. The power deals with multiplying the particular variable with the times mention in the power value. For example x to the power four multiplies the variable x four times. Following examples gives the brief idea about the power value.

Problems on x to the power four:

 

Example 1:

Calculate the variable x with the given power value.

x ⁴ – 8x = 0

Solution:

Given
x ⁴ – 8x = 0

Take x as common factor.
x (x ³  – 8) = 0

For the product x (x ³   – 8) to be equal to zero

x = 0 or x ³   – 8 = 0

Solve the above simple equations to get the solutions.
x = 0

or

x ³ = 8

X = ±2

Example 2:

Calculate the variable x with the given power value.

f(x) = x ⁴ – 108x + 1

Solution:

The given function is

f(x) = x ⁴ – 108x + 1

The first derivative f ‘ is given by

f ‘(x) = 4 x ³ – 108

Example 3:

Calculate the variable x with the given power value.

f(x) = x ⁵ – 8 x ⁴  + 10

Solution:

The given function is

f(x) = x ⁵ – 8 x ⁴  + 10

The first derivative f ‘ is given by

f ‘(x)= 5x ⁴ – 8(4 x ³ )

f ‘(x) = 5x ⁴ – 32 x ³

Example 4:

Calculate the variable x with the given power value.

x ⁵ – 16x = 0

Solution:

Given
x ⁵ – 16x = 0

Take x as common factor.
x (x ⁴  – 16) = 0

For the product x (x ³   – 8) to be equal to zero

x = 0 or x ⁴   – 16 = 0

Solve the above simple equations to get the solutions.
x = 0

or

x ⁴ = 16

X = ±2

 

Practice problems on x to the power four:

 

1) Calculate the variable x with the given power value.

x ⁴ – 27x = 0

Answer: X = +-3

2) Calculate the variable x with the given power value.

f(x) = x ⁵ – 4 x ⁴  + 5

Answer: f ‘(x) = 5x ⁴ – 16 x ³

How To Write Scientific Questions

In subjects, such the same as astronomy, physics, chemistry, biology and engineering, we appear crosswise extremely large numbers and very small numbers.

For example, the distance of sun from earth is about 92,900,000 miles.

If m is a natural number and a is a real number, then a^m means the product of m numbers each equal to a: that is `a^(m)` = `axxaxxa……m` factors. Here a is called the base and m, the power or exponent or index. The notation a^m is read as a to the power m or a raised to m.

For example, `a^5=axxaxxaxxaxxa.`

Using some laws in writing Scientific questions:

• `a^(m)xxa^n =a^mxxa^n`
• `a^m/a^n =a^(m-n),a!=0 m> n`
• `(a^m)^n = a^(mn)`               (Power law)
• `a^mxxb^m = (axxb)^m`   (Combination law).  When `a!=0` , we define `1/a^m` as `a^-m` and define `a^0 =1`

Now using the laws of indices, any positive real number can be written in the form a*10^n, where 1 ≤ a ≤ 10 and n is an integer. For example

(i)   7.32 = 7.32 `xx` `10^0`

(ii)  11.2 = 1.12 * 10 =1.12 * `10^1`

(iii)  226 = 2.26 *100=2.26 `xx` `10^2`

(iv)  92900000 = 9.29 * 10000000 = 9.29 `xx` `10^7`

(v)   0.000000537 = 5.37 `xx` `10^-7`

(vi)   0.0000000279 = 2.79 `xx` `10^-8` .

Here after, by a number, we shall mean a positive number only. We again mentain that, when a number is written in scientific notation `axx10^n` , the integral part of the number, a iisa digit from 1 to 9 and the power of 10 is an integer(positive, negative or zero).

I am planning to write more post on Prime and Composite Number with example, . Keep checking my blog.

We also observe that converting a given number into the scientific notation, if the decimal point is moved r places to the left, then this movement is compensated by the factot `10^r` ; and if the decimal point is moved r places to the right, then this movement is compensated by the factor `10^-r`

Some examples of write scientific questions:

Let us see some examples of write scientific questions:

Example 1:

How to write following numbers in scientific questions:

(i)   7493

(ii)  105001

Solution:

(i)    7493 = `7.493xx10^3`

(ii)   105001 = `1.05001xx10^5`

Example 2:

How to write the following numbers in scientific questions:

(i)    0.00567

(ii)   0.0002079

(iii)  0.000001024

Solution:

(i)   0.00567 = `5.67xx10^-3`

(ii)  0.0002079 = `2.079xx10^-4`

(iii)  0.000001024 =` 1.024xx10^-6`

These are examples of write scientific questions

Euclid’s Division Lemma

In mathematics, Euclid’s lemma is most important lemma as regards divisibility and prim numbers. In simplest form, lemma states that a prime number that divides a product of two integers have to divide one of the two integers. This key fact requires a amazingly sophisticated proof and is a wanted step in the ordinary proof of the fundamental theorem of arithmetic.

Euclid’s division lemma

 

• Euclid’s division lemma, state that for a few two positive integers ‘a’ and ‘b’ we can obtain two full numbers ‘q’ and ‘r’ such that

•  `a=bxxq+r`

• Euclid’s division lemma can be used to:
  Find maximum regular factor of any two positive integers and to show regular properties of numbers.
• Finding Highest Common Factor (HCF) using Euclid’s division lemma:
  Suppose, we hold two positive integers a and b such that a is greater than b. Apply Euclid’s division lemma to specified integers a and b to find two full numbers q and r such that, a is equal to b multiplied by q plus r.

• ‘r’ value is verified. If r is equal to zero then b is the HCF of the known numbers.
• If r is not equal to zero, apply Euclid’s division lemma to the latest divisor b and remainder r.
• Maintain this process till remainder r becomes zero. Value of  divisor b in that case is the HCF of two given numbers.
  Euclid’s division algorithm can be used to find some regular properties of numbers.

I am planning to write more post on Calculate Harmonic Mean with example,Prime and Composite Number. Keep checking my blog.

 

Example

 

Euclid’s lemma in plain language says: If a number N is a multiple of a prime number p, and N = a · b, then at least one of a and b must be a multiple of p. Say,

N=56

p=7

`N=14xx4`

Then either

`x*7=14`

or

`x*7=4`

Obviously, in this case, 7 divides 14 (x = 2).

Properties of determinants

  In this page we are going to discuss about properties of determinants .Let a matrix A = [a_ij] be a square matrix. For every matrix A, can be associated with a determinant which is formed by exactly the same elements of the matrix A. Such determinant formed is denoted by the symbol det A or |A|.

The determinant of a square matrix will always be scalar.

For example,

Let a matrix,    `A = [[5,2],[7,3]]`

`|A| = |[5,2],[7,3]|`

Thus,  The value of  `|[5,2],[7,3]| = 5 X 3 – 7 X 2 = 15 – 14 = 1`

Also , We need to know the following definitions to find the determinant of a matrix which is of order three or more .

Minors:

Let |A| = |aij| be a determinant of order n.

The determinant obtained by removing the i^th row and j^th column is called the minor of element a_ij and is denoted by M_ij.

Co-factors:

The co-factor of the element a_ij is (-1)^i+j times its minor a_ij. The co factor of an element is denoted by the its corresponding letter written in capitals.

Co factor of a_ij = A_ij = (-1)^i+j M_ij 

The value of the determinant ∆ of a 3 X 3 matrix is given in general by,

            ∆ = a _i1 A _i1 +a _i2 A _i2 +a _i3 A _i3      where i `in`  {1,2,3}

or        ∆ = a _1j A _1j +a _2j A _2j +a _3j A _3j      Where j `in`  {1,2,3}

 

Properties of Determinants

 

    Below are the properties of determinants  –

• In a determinant, If the rows and columns are inter-changed, then the value remain same.                                                              

 

`|[x,y,z],[a,b,c],[1,2,3]| = |[x,a,1],[y,b,2],[z,c,3]|`                                                              

• In a determinant, If any two rows (or any two columns) are same, the value of the determinant is always zero.                      

  `|[x,y,z],[x,y,z],[1,2,3]| = 0`

• If any two rows (or any two columns) in a determinant are interchanged, then the  value of the determinant is (-1) times the value of the original determinant.                                                                                                                                                                          

  `|[x,y,z],[a,b,c],[1,2,3]| =-|[a,b,c],[x,y,z],[1,2,3]|`                                                                                                                                                                                                                                                                                  

• In a determinant, If every element of one row (or one column) is multiplied by a number k, then the value of the new determinant is k times the value of the original determinant.                                                                                                                                                                              

   `|[kx,ky,kz],[a,b,c],[1,2,3]| =k |[x,y,z],[a,b,c],[1,2,3]|`                                                                                                                                              

• In a determinant, if to any row or to any column, a multiple of another row or another column is added, then the value of the determinant remains the unaltered. 

  I am planning to write more post on systems of linear equations in two variables with example, Horizontal Lines. Keep checking my blog.
                                                                                                                                                                                              

  `|[x,y,z],[a,b,c],[1,2,3]|=|[x+ka,y+kb,z+kc],[a,b,c],[1,2,3]|`

• In a determinant,  If some or all the elements of a row (or a column) are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two or more determinants.

 

            ∆=`|[a+l,b+m,c+n],[x,y,z],[1,2,3]|` =   `|[a,b,c],[x,y,z],[1,2,3]|`+ `|[l,m,n],[x,y,z],[1,2,3]|`

• The sum of the products of the elements in any row (or any column) with their corresponding co factors is equal to  the value of the original determinant.

                      Example : ∆ = a _i1 A _i1 +a _i2 A _i2 +a _i3 A _i3      where i = 1, 2 and 3               

• The sum of the products of the elements in any row (or any column) and the co factors of the corresponding elements of any other row (or any other column) is zero.  

              Example: For a matrix of order 3,    a₁₁A₂₁ + a₁₂A₂₂ + A₁₃A₂₃ = 0 .

• The value of a determinant of a square zero matrix is  zero.                                                                           `|[0,0,0],[0,0,0],[0,0,0]|=0`

• If any row (or any column) has all entries as zeros then the determinant is zero.

                `|[x,y,z],[0,0,0],[1,2,3]|=0`

• The  value of the determinant of a triangular matrix is obtained by the product of elements in the main diagonal.                            `|[x,1,2],[0,y,3],[0,0,z]|`= `xyz`

• The value of the determinant of a diagonal matrix is equal to the product of elements in its diagonal.          

 

           `xyz`

Using the above properties of determinants, it is easy to solve many equations.

Quadrilaterals and Symmetry : 2

Quadrilateral is a type of polygon with four sides and four vertices’s or corners. The quadrilaterals are may be convex or concave. The entire convex quadrilateral covers the plane by continual revolving around the midpoints of their ends.

Types of quadrilaterals and symmetry :-

Some unique quadrilateral used are

• Parallelogram
• Rhombus
• Rectangle
• Square
• Kite
• Trapezium
• Isosceles trapezium.

Introduction to Symmetry:  

Symmetry is happens not only in geometry, it also happens in all other branches of math’s. It is really similar as invariance.

• Lines of symmetry
• Rotational symmetry
• Planes of symmetry
• Axes of symmetry
• Congruent shapes
• Congruent triangles

 

Measure of Quadrilaterals angles :-

 

The interior angles of quadrilateral always add up to 360 degrees. Quadrilateral also have rotational and lines of symmetry.

Perimeter  of Quadrilaterals

The perimeter of quadrilateral means add up the outside of a shape.

 

Types of Symmetry

 

Line Symmetry:

A line of symmetry is a line which divides the shape into half and they form a mirror image of the other, when it folded one side over the top of another along the line of symmetry they are equal.

Rotational symmetry:

Rotational symmetry is in which the point is moved about the shape and fit on itself more than once in 360 degrees. The shape is moved around the point is called centre of rotation.  The rotational symmetry is not one it has 2, 3 or more the order of rotational symmetry explains about the shape can fit itself within 360 degrees.

Planes of symmetry

In two dimensional shapes we have looked flat and seen lines of symmetry and rotational symmetry. In Three dimensional shapes we look at planes of symmetry and axis of symmetry.

Congruent shapes

The congruence or congruent means exactly equal. If any two shapes to be congruent they have the equal size and shapes.

Congruent triangles

To show triangles are congruence they are four ways.

1. If all three sides of triangle are same then they are congruent. (SSS)
2. If two sides of triangle and the angle connecting them are the same ( SAS)
3. Two angles and a corresponding side (ASA or AAS)
4. Right angled triangles only the hypotenuse and one side are the equal (RHS)

 

example for symmetry in quadrilaterals :

 

• Rectangle is an example for quadrilateral is symmetry.   The opposite sides of length and breadth are equal.

• A kite has only one line of symmetry, across a cross section.

Power Calculator Online

We can easy to get  the power of a  number using online calculator ,Consider  the unknown number `x^y` , Here x is base and y is the power  of x . The online calculator performs the  operation of  x^y     

           Let us consider the  number 2³

Here 2 is the base(x)  and 3 is the power of 2(y). We calculate 2³= 2 x 2 x 2 =8

Definition of Power :

It  shows how many times  to use the given number (x) in the multiplication.’    Power is also said to be ” exponential “

 

How to use Power online calculator:

 

To calculate x^y

Enter y as an interger ,it may positive or negative number and x is as real number .

INPUT	OUTPUT
22	4
23	8
32	9
33	27

 

Let us discuss about calculation of power by manual, consider the number 10²

Here , 10 is base  and 2 is power .

So, Mulitply the base number into power times.

10 x 10 =100

Therefore , 10²   = 100

I am planning to write more post on Pythagorean Theorem Formula with example, Acute Angles. Keep checking my blog.

 

Examples of power using Positive integers:

 

Example 1:

5 ²

Solution:

2 times 5 =  5 x 5   =25

Example 2:

25²

Solution:

2 times 25 = 25 x 25   =625

Example 3:

17³

Solution:

3 times 17 = 17 x17 x17  = 4913

Example 4:

22⁴

Solution:

4 times 22 = 22 x22 x22 x22 =234256

Example 5:

25³

Solution:o

3 times 25 = 25 x25  x25  =15625

Example  problems  of power using Negative integers:

Note:  (negative)   x (negative)  =  Positive

 

1)`(-2)^2`   = -2 x -2  =  4

 

2)`(-5)^4`   =  (-5 ) x (-5) x (-5 ) x (-5)

= 25      x     25

=625

3)`(-10)^2`   = (-10 )   x   (-10)

= 100

4)`(-7)^2`   = (-7)   x   (-7)

=49

5)`(-11)^3`   = (-11)  x (-11)  x (-11)

= 121  x  (-11)

= -1331

 

Example problems of power using radical numbers:

 

We know that (√ x )² = x

1)`sqrt3^2` = `sqrt3`   x `sqrt 3 `   =  3

2)`sqrt(2)^3`   =   `sqrt2`    x `sqrt2`    x   `sqrt 2`    =  `2sqrt2`

3)`sqrt(5)^4`   =`sqrt5` x`sqrt5`x`sqrt5`x`sqrt5`

= 5   x    5

=  25

4)`sqrt10 ^6`   =`sqrt10`   * `sqrt10`  * `sqrt10`  * `sqrt10` * `sqrt (10)` *`sqrt10`

=          10          *        10            *          10

=     1000

 

Practice problems of power using Online calculator:

 

Find Power of positive Integers using online calculator:

1)23²   

Answer :529

2)12⁴

Answer :20736

3)8⁵

Answer :32768

Find power of negative integers using Online calculator:

1)(-9)²

Answer :81

2)(-11)³

Answer:-1331

3)14⁴

Answer:38416

Find power of radical numbers using online calculator:

1)`sqrt 7^3`

Answer :7`sqrt7`

2)`sqrt9^5`

Answer :81`sqrt9`

3)`sqrt(12)^4`

`Answer:144`

`<br>`

Inverse Tangent Tutor

Introduction :

            The inverse tangent is the multivalued function tan^-1z , also referred as arctan z or arctg z, that is the inverse function of the tangent. Sometimes the variants Arctan z and Tan ^-1 z are used to refer to explicit principal values of the inverse cotangent.

The inverse tangent function  tan^-1x is plotted above along the real axis.

 

Inverse Tangent Tutor:

 

Sometimes  the notation arctan z is used for the principal value, with Arctan z being used for the multivalued function. In the notation tan^-1z, tan z refers the tangent and -1 the inverse function, not the multiplicative inverse.

The inverse tangent is a multivalued function and it requires a branch cut in the complex plane, which Mathematica‘s convention places at (-i∞, -i] and [i, i∞). The inverse tangent can be caculated as,

tan^-1 z = ½ i[In(1 – iz) – In(1 + iz)]

                                      (or)

                  tan^-1 (z)  = (i/2 )log(i +z/i-z)

 

Examples by Inverse Tangent Tutor:

 

Example 1:Find the angle when z=35.

Solution:

Angle = arctan z

Angle = arctan 35

Angle = 1.5422326689561 radian (or) 88.363422958383  degree.

Example 2:Find the angle when z=60.

Solution:

Angle = arctan z

Angle = arctan 60

Angle = 1.554131203081 radian (or) 89.045158746128  degree.

Example 3:Find the angle when z=90.

Solution:

Angle = arctan z

Angle = arctan 90

Angle = 1.5596856728973  radian (or) 89.363406424037  degree.

Example 4:Find the angle when z=300.

Solution:

Angle = arctan z

Angle = arctan 300

Angle = 1.5674630058072  radian (or) 89.80901477564  degree.

Example 5:Find the angle when z=120.

Solution:

Angle = arctan z

Angle = arctan 120

Angle = 1.5624631863548radian (or) 89.52254622269  degree.

Example 6:Find the angle when z=222.

Solution:

Angle = arctan z

Angle = arctan 222

Angle = 1.5662918527563  radian (or) 89.741912648664  degree.

 

Practices Problems by Inverse Tangent Tutor:

 

Problem 1:Find the angle when z=250

Solution: Angle= 1.566796348128  radian (or) 89.770818104246  degree.

Problem 2:Find the angle when z=24.50

Solution: Angle= 1.530002643927  radian (or) 87.662694140876  degree.

Problem 3:Find the angle when z=40.75

Solution: Angle= 1.5462613737322  radian (or) 88.594250738954  degree.

Problem 4:Find the angle when z=270.45

Solution: Angle= 1.5670988025103  radian (or) 89.788147463846  degree.

The above examples are describes the concept of inverse tangent tutor.

Formula For Foci Of a Hyperbola

• A hyperbola is a conic in which strangeness is greater than unit. Normally hyperbola is a curve which moves; therefore the ratio of the distance from a fixes point to its distance from a fixed straight line is always greater than 1. The fixed point is represented as focus or foci and the fixed straight line is said to be directrix and the constant ration is said to be eccentricity of the hyperbola.

 

Description about formula foci of a hyperbola:

 

The diagram will given below the structure of fcoi of the hyperbola:

Formula: The origin of the hyperbola center is (0,0), is the graph of

`x^2/a^2`-`y^2/b^2` =1             or          ` y^2/a^2` -`x^2/b^2` =1

The following properties are used in the above graph: x intercepts at ± a , no y intercepts, foci at (-c , 0) and (c , 0), asymptotes with equations y = ± x (`b/a` ). The following properties are used in the right side part:  y intercepts at ± a , no x intercepts, foci at (0 , -c) and (0 , c), asymptotes with equations y = ± x (`a/b` ) .

a, b and c are related by  c² = a² + b².

The transverse axis  of the length is 2a, and the length of the conjugate axis is 2b.

 

Practice problems for formula for the foci of a hyperbola:

 

Example Problem for foci of  a hyoerbola:  Given the following equation

4m² – 9n² = 36

a) Solve the m and n intercepts, if possible,  of the graph of the equation.

b) Solve the coordinates of the foci.

c) Draw the graph for the equation.

I am planning to write more post on Slope of a Vertical Line with example, math equation solver free. Keep checking my blog.

Solution to Problem

a) Here we need to  write the given equation in standard form. We can get the exact form by dividing by 144 on both sides of the equation.

`(“4m^2”) / (“36”)`  –  `((9n)^2)/ (36)` =  1

`(m^2)/(9)-n^2/4`   =  1

`m^2/3^2-n^2/2^2`   =  1

form the above step we can get the value for a and b, comparing the above equation with the standard form we can write the value for a =4 and b=3.

Make  y = 0 in the equation and here we can find the x values.

`x^2/3^2`   = 1

find for x.                      x²  = 3²

x= ± 3

put x = 0 in the equation and find the y intercepts.

`- y^2/2^2`  = 1

NO y intercepts, and in that there is no real solutions.

b) here we can find the c value.

c² = a² + b²

a and b were found in part a).

c² = 3² + 2²

c² = 16

Solve for c.

c = ± 4

The foci are    F₁ (4 , 0) and  F₂ (-4 , 0)

c) 1 – Find the asymptotes y = – (`b/a` ) x and y = (`b/a` ) x and plot them.

y = -(`2/3` ) x  and y = (`2/3` ) x

2 – plot the x intercepts

3 – Find extra points (if necessary)

set x = 6 and find y                4(6)² – 9y² = 36

– 9y² = 36 – 144

-9y² = -108

y² = `108/9`

y =+- 12.

Solve for y                               y = 12 and            y = -12

so the points (6,  12x) and    (6,  -12x) are on the graph of the hyperbola.

Also because of the symmetry of the graph of the hyperbola, the points (-6, 12x) and

(-6, -12x) are also on the graph of the hyperbola.

Recursive Series

Introduction:

Recursive sequence {f(n)}n , also recognized as a reappearance sequence, is a series of numbers f(n) indexed by an integer n and generated by solving a recurrence equation. The conditions of a recursive sequences is denoted by symbolically in a numeral of dissimilar notations, such as fn, f (n). Here f is a symbol instead of the series. For a series a₁, a₂, a₃, . . . , a_n, . . .

 

Recursive series formula

 

Recursive formula is used to make a decision of subsequently expression of a series using one or additional of the preceding conditions.

For Example:-

Recursive method for the chain 5, 25, 125, 625 ….is a_n = 5 a_n-1.

Plan of sequence in subsequently conditions are deduced from former ones, which is unwritten in the standard of mathematical induction, dates to antiquity.

Common form of recursive series

Location of expressions the involvement is obvious in the multipliers in every line.

3^rd =a₁+a₂

4^th=a₁+a₂+a₃+a₄

5^th=a₁+2a₂+2a₃+2a₄+2a₅

6^th=a1+3a2+4a3+4a4+3a5+t6

a₁ is forever multiplied by 1

a₂ in all container increments by 1. [0 1 2 3 4 5 6 ….]

a₃ in all container increments used by prior addtion intervals [0 1 2 3 4 5 6 ….] to provide [1 1 2 4 7 11 …].

I am planning to write more post on Acute Angles with example, Convert to Scientific Notation. Keep checking my blog.

 

Example problems for recursive series

 

Problem 1

What is the 10th term of the sequence defined by a_n = (n – 2) (3 – n) (4 + n)?

Solution

Putting n = 10, we obtain

a₁₀ = (10 – 2) (3 – 10) (4 + 10)

= 8 × (– 7) × (14) = – 784

 

Problem 2

Let the sequence a_n be defined as follows:

a₁ = 2, a_n = a_n – 1 + 2 for n ≥ 2.

Calculate five primary conditions and write corresponding series.  

Solution  

We have

    a₁ = 2, a₂ = a₁ + 2 = 2 + 2 = 4, a₃ = a₂ + 2 = 4 + 2 = 6,

   a₄ = a₃ + 2 = 6 + 2 = 8, a₅ = a₄ + 2 = 8 + 2 = 10.

Hence,

Conditions of the sequence are 1,3,5,7 and 9. The matching series is 2 + 4 + 6 + 8 + 10 +…

Problem 3

What is the 5th term of the sequence defined by a_n = (n – 1) (2 – n) (3 + n)?

Solution

Putting n = 5, we obtain

a₅ = (5 – 1) (2 – 5) (3 + 5)

= 4 × (– 3) × (8) = – 96

Napier’s Analogy

Trigonometry:                                                                                                                                                                   Trigonometry is that branch of mathematics which deals with the measurement of sides and angles of triangles.

Angles

An angle is an amount of rotation of a half-line (or ray) in a plane about its end point from an initial position to a terminal position. The important terms are: Measurement of angle, Positive and Negative angles, Lines at right angles, Quadrants, Angle in standard position.

Trigonometric Functions:

The circle whose radius is 1 unit whose centre is the origin of a rectangular co-ordinate system is called the unit circle.

1. cosq = x.
2. sinq = y.
3. tanq = y/x.
4. secq = 1/x.
5. cosecq = 1/y.
6. cotq = x/y.
The six functions of q defined by the above equation are called trigonometric functions of q or circular functions of q.

 

Proof of Napier`s Analogy:

 

Proof of  Napier`s Analogy:

In any triangle ABC, with sides a = BC, b = CA and c = BA, then prove that

 

Napier`s analogy proofs:

 

Theorem:

In any triangle ABC, with sides a = BC, b = CA and c = BA, then prove that

Proof:

Method I

Similarly, (ii) and (iii) can be proved.

Method II

I am planning to write more post on Obtuse Angles with example, Definition of Rectangle. Keep checking my blog.

 

napier’s analogy : Solution of triangle:

 

• From geometry, we know that when any three elements are given of which necessarily a side is given, the triangle is completely determined i.e, remaining three elements can be determined.
• The process of determining the unknown elements knowing the known elements is known as the solution of a triangle.
• Napier analogy can be used .
•  A triangle has six parts or six elements viz three sides and three angles.
• From geometry, we know that when any three elements are given of which necessarily a side is given, the triangle is completely determined i.e, remaining three elements can be determined.
• The process of determining the unknown elements knowing the known elements is known as the solution of a triangle

 

Napier’s analogy : Example:

 

Example:

If b = 251, C = 147, A = 47^o, find the remaining angles.

Answer:

= 0.268 x 2.2998