Algebra Coefficient Variable

In mathematics, a coefficient is defined as the number in front of the variable. For example 2x is the given expression here the coefficient is 2. The coefficient is usually in numeral with the variable. The expression contains variable and coefficient of the variable. Here in this topic we are going to see about coefficient of variables.

Example problem for the coefficient:

Example 1:

Find the coefficient of the variables in the given algebraic expression:

x + 2y

Solution:

Given that x + 2y

Here x and y is the expression with the coefficient

The coefficient of x is 1

The coefficient of y is 2.

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Example 2:

Find the coefficient of variables in the given algebraic expression:

y2 + 2y + 3xy +1

Solution:

Given that y2 + 2y + 3xy +1

Here y2 , 2y,  3xy, is the expression with the coefficient and 1 is without variable

The 3xy having two variables that two variable consider as a one variable

1 is the constant term of the given algebraic expression

The coefficient of y2 is 1

The coefficient of y is 2

The coefficient of xy is 3

Example 3:

Find the coefficient of variables in the given algebraic expression:

z5 + z + y +6y

Solution:

Given that z5 + z + y

Here z5, z, y, 6y is the expression with the coefficient

The coefficient of z5 is 1

The coefficient of z   is 1

The coefficient of   y is 1

Example 4:

Find the coefficient of  variables in the given algebraic expression:

x3 + y 5 +2xy + 5yx

Solution:

Given that x3 + y 5 +2xy + 5yx

Here x3, y 5, 2xy, 5yx the expression with the coefficient

The coefficient of x3 is 1

The coefficient of y5 is 1

The coefficient of xy is 2

The coefficient of yx is 5.

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Example 5:

Find the coefficient of variables in the given algebraic expression:

100x2 + 2z3 + 102z

Solution:

Given that 100x2 + 2z3 + 102z

Here 100x2, 2z3, 102z the expression with the coefficient

The coefficient of x2 is 100

The coefficient of z3 is 2

The coefficient of z is 102.

Law of Cosines Calculator

Law of cosines calculator is a tool to calculate calculation applying law of cosines easily. First let's understand the concept of law of cosines. In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) is a statement about a general triangle that relates the lengths of its sides to the cosine of one of its angles. The law of cosines states that,

c2 = a2 + b2 – 2ab cos γ,

The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle γ is a right angle (of measure 90° or `pi/2` radians), then cos(γ) = 0, and thus the law of cosines reduces to,

c2 = a2 + b2

Formula for Law of Cosine:

The law of cosines is used to solve the third side of a triangle, when other two sides and the angle are known.

By changing the sides of the triangle, one can find the following two formulas also to solve for the law of cosines,

a2 = b2 + c2 – 2bc cos α,

b2 = a2 + c2 – 2ac cos β.

Law of Cosines Using Distance Formula:

To solve for the law of cosines, consider a triangle with a side length of a, b, c, and θ is the measurement of the angle opposite to the side length c.

A = (bcosθ, bsinθ), B = (a, 0), and C = (0, 0).

By using distance formula, we have,

c = `sqrt((a - bcostheta)^2 + (0 - bsin theta)^2)` ,

Now, squaring on both sides, we get,

c2 = (a – bcos θ)2 + (–bsin θ)2

c2 = a2 – 2ab cos θ + b2cos2θ + b2sin2θ,

c2 = a2 – 2ab cos θ + b2(cos2θ + sin2θ)

c2 = a2 + b2 – 2ab cos θ.               [where, cos2θ + sin2θ = 1].

Law of Cosines Using Trigonometry:

To solve for the law of cosines, draw the perpendicular to the side c as shown in the figure,

c = a cosβ + b cosα.

Multiply by c, we get,

c2 = ac cosβ + bc cosα                                                 (1)

By considering the other two perpendiculars, we get,

a2 = ac cosβ + ab cosγ                                                (2)

b2 = bc cosα + ab cosγ                                                (3)

Adding equations (2), and (3), we get,

a2 + b2 = ac cosβ + ab cosγ + bc cosα + ab cosγ

a2 + b2 = ac cosβ + 2ab cosγ + bc cosα                       (4)

Subtracting equation (1) from the equation (4), we get,

a2 + b2 – c2 = (ac cosβ + 2ab cosγ + bc cosα) – (ac cosβ + bc cosα)

a2 + b2 – c2 = 2ab cosγ,

a2 + b2 – 2ab cosγ = c2,

c2 = a2 + b2 – 2ab cosγ.

Examples based on Law of Cosine:

Ex 1: Evaluate the length of side A using law of cosine formula.

Sol:

law of cosine example 1

Step 1:   A² = B² + C² −2(B)(C)cos (`-<`1) ( law of cosine formula)

Step 2:  Plug in the values of B and C

A² = 20² + 13² −2(20)(13)cos(66)

A² = 400 + 169 −520 cos(66)

Step 3: Add and subtract

A² = 569 −211

A² = 358

Step 4:  Take square root

A= √358 = 18.9

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Ex 2:  Evaluate x using law of cosine formula.

Sol:

law of cosine example 2

Step 1:   A² = B² + C² −2(B)(C)cos ( 1)

25² = 32² + 37² −2(32)(37)cos(x)

Step 2:  Solve the equation

625 = 2393 − 2,368 cos(x)

- 1760 = -2, 368 cos (x)

. 7432 = cos x

Cos-1 (.7432) = 42.0 º

Least Common Multiples

In arithmetic and number theory, the least common multiple or lowest common multiple (LCM) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both of a and of b. Since it is a multiple, it can be divided by a and b without a remainder. A few word problems for least common multiples is given below.

(Source: Wikipedia)

Example of word problems for least common multiples:

Word problem 1:

Find the largest number of four digits which when divided by 5, 10, 15, it leaves a remainder 4 in each case.

Solution:

Step 1: Given numbers

5, 10, 15

Step 2: Find least common multiple of 5, 10, 15

Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40....

Multiples of 10 = 10, 20, 30, 40, 50, 60, 70, 80...

Multiples of 15 = 15, 30, 45, 60, 75, 90, 105....

From the list of multiples of 5, 10 and 15 the smallest common number in each list 30.

Therefore, the least common multiples of 5, 10 and 15 are 30.

Step 3: Find multiple of 30 which should be slightly less than five digit.

30 * 333 = 9990

So, 9990 is the largest number of four digit which is divisible by 5, 10, 15 and leaves a remainder 0.

Step 4: Find number which leaves a remainder 4.

To get remainder 4, we should add 4 to the obtained number.

Therefore, the required number is 9990 + 4 = 9994

Word problem 2:

Three children John, Nick and Shane run on a round  track. John takes 50 seconds, Nick takes 55 seconds and Shane takes 60 seconds to run a round. If all three of them start together at a point, when do they meet again?

Solution:

Step 1: Find least common multiple of 50, 55, 60

10 |       50       55       60

-----------------------------------------

5  |      5          55       6

-----------------------------------------

1           11       6

Least common multiple = 10 * 5 * 1 * 11 * 6 = 3300

Step 2: Solution

Therefore, they meet after 3300 seconds = 55 minutes.

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Homework of word problems for least common multiples:

1) Find the largest number of three digits which when divided by 7, 14, 28 it leaves a remainder 2 in each case.

2) Two children Joseph and Fleming run on a round track. Joseph takes 75 seconds and Fleming takes 80 seconds to run a round. If both of them start together at a point, when do they meet again?

Solutions:

1) 982

2) 1200 seconds

How to Find Number of Factors

In mathematics, factorization (also factorization in British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2) (x + 2). In all cases, a product of simpler objects is obtained.

How to find number of factor:example problems

Example 1:

Find all number of factors 50.
Solution:

50 = 1x50
= 2x25
= 5x10

So, the factors of 50 are 1, 2, 5, 10, 25 and 50.In the above example for how to find number of factors.

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Example 2:

Find all numbers of factors 80

Solution:

80 = 1x80
= 2x40
= 4x20
= 5x16
= 8x10

So, the factors of 80 are 1, 2,4,5,8,10,16,20,40 and 80.In the above example for how to find number of factors.

More explanation of how to find number of factors:-

Every number greater than 1 have atleast two factors: 1 and itself.

Example,

2 = 1 * 2

3 = 1 * 3

4 = 1 * 4

In two numbers are factors of another number are multiplied.

Note:- In 4 has some other factors besides 1 and 4:

4 = 1 * 4

4 = 2 * 2

A number 36 is a factors 4,

36 =  1 * 36

=  2 * 18

=  3 * 12

=  4 *  9

=  6 *  6

In those factors 1 and 36:

36 = 1 * 36

Divide by the next highest number after 1 and 2 is goes to 36. 18 times, 2 and 18 are a pair of factors:

36 =  1 * 36

=  2 * 18

=  3 * 12

=  4 *  9

Note:- 5 doesn't work, so leave that, and go on to 6:

36 =  1 * 36

=  2 * 18

=  3 * 12

=  4 *  9

=  6 *  6

Again, 7 doesn't work, and neither does 8. But 9 works:

36 =  1 * 36

=  2 * 18

=  3 * 12

=  4 *  9

=  6 *  6

=  9 *  4

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but it would be add on the small  table, because it says the same thing as an earlier entry:

36 = 1 * 36

= 2 * 18

= 3 * 12

= 4 *  9  <-- p="">
= 6 *  6     |  the same factors

= 9 *  4  <-- p="">
Any numbers larger than 6 remaining are smaller than 6. The factors of 36 are 1, 36, 2, 18, 3, 12, 4, 9, and 6.

Numerical Integrals

The process of finding an integral; it may be definite integral or an indefinite integral is referred as integration. Numerical integration is mainly used for finding the numerical value of a definite integral. Numerical integration is also used to finding the numerical solution of differential equations. Numerical integration is also referred to as numerical quadrature.

I like to share this Solving Indefinite Integrals with you all through my article.

Numerical integrals:- types of integrals

1. Indefinite integrations:

An indefinite integration is the family of functions that have a given function as a common derivative. The indefinite integral of f(x) is written ∫ f(x) dx.

2. Definite integrations:

If F(x) is the integral of function f(x) over the interval [a, b] ,i.e., ∫ f(x) dx = F(x) then the definite integral of function  f(x) over the interval [a, b] is denoted by int_a^bf(x)dx and is defined as int_a^bf(x)dx  = F(b) – F(a).

Where 'a' is called the lower limit and b is called the upper limit of integration and the interval [a, b] is called of integration.

Methods of integration:

It is not possible to integrate each integral with help of the following methods but a large number of varieties of the problems can be solved by these methods so, we have the following methods of integration:

1. Integration by substitution

2. Integration by parts.

3. Partial fractions

Example Problems on indefinite and definite integration:

Numerical Integrals Problem:

Example 1:

find ∫ 1 / sin2x cos2x dx.

Solution:

We have ∫ 1/ sin2x cos2x dx

= ∫sin2x+cos2x / sin2x+cos2x dx

=∫ (1/cos2x+1/ sin2x )dx

=∫sec2 x d x+∫cosec 2 x dx

=tan x -cot x+C.

Example 2:

∫ sin x sin (cos x) dx.

Solution:

Let cos x = t

dt = -sin x dx

Therefore we have

∫sin x sin (cos x) dx = - ∫sint dt = cost + C=cos (cos x) + C.

where

t=cos x

Example 3:  int_0^1dx/(1+x2)

Solution:

=[tan-1 x]10

=tan-1 1 - tan-1 0

= /4 -0 = / 4

π/2

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Example 4: Find the value int sin⁷xdx

-π/2

Solution:

Let f (x) = dx.

Then f(-x) = -sin7 x=-f (x)

so, f (x) is odd function.

π/2

π/2

Therefore, int f(x)dx=0 => int sin7x dx = 0.

-π/2                  -π/2

Laws of Exponents for Real Numbers

Let ‘a’ be a positive number. Exponential notation is ax , where x is an integer. When x is a rational number, say p/q with p, an integer and q, a positive integer, we define ax by ax  = root(q)(a) p

Example:

53/8 = root(3)(5)8

laws of exponents for real numbers: Exponents Notation

When x is an irrational number, ax can be defined to represent a real number.

For any a > 0, ax can be defined and that it represents an unique real number u and write u = ax.

The real number u is written in the exponential form or in the exponential notation.

Here the positive number 'a' is called the base and x, the index or the power or the exponent.

The laws of indices which we have stated for integer exponents can be obtained for all real exponents.


laws of Exponents for Real Numbers

We state them here and call them, the laws of exponents for real number:

root(x)(root(y)(a))  = root(xy)(a)


root(x)(a / b)  =  root(x)(a) / root(x)(b)


(root(x)(a) )x = a


axxx ay = ax+y


ax// ay = ax-y


(ax)y = axy


a-x  = 1/ ax


axxx bx = abx


a0 = 1  for a != 0


(a/b)x = ax/bx


a1/x =  root(x)(a)


(root(x)(a) )y/x = (root(x)(a) )


root(x)(a) root(x)(b) = root(x)(ab)



Example sums for Exponents for Real Numbers

Example 1:

Find the exponential form of (25)1/17

Solution of example 1:

Given = (25)1/17

Exponential form  =  root(17)(25)

Example 2:

Simplify 25225 / 53253

Solution of example 2:

Given 25225 / 53253

Now we use the exponents formula for real numbers

By using

(i)axxx ay = ax+y

(ii)axxx bx = abx

We get

= 253 / (5(25))3

= 253 / 1253

= 253 / (5)3(25)3

Cancel the same variables

Now, we get

= 1 / 53

= 1 / 125

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Example 3:

Solve ( root(3)(root(2)(64))92xx 62) / ((43)2+(5/25)2)

Solution of example 3:

Given = ( root(3)(root(2)(64))92xx 62) / ((43)2+(5/25)2)

Now we use the exponents formula for real numbers

By using below laws of exponents formula we can simplify the given data.

(i) root(x)(root(y)(a)) = root(xy)(a)

(ii)axxx bx = abx

(iii)(ax)y = axy

(iv)(a/b)x = ax/bx

= ( root(3)(root(2)(64))92xx 62) / ((43)2+(5/25)2)

Use the (i) formula

We get

= root(6)(49))92xx 62) / ((43)2+(5/25)2)

Use the (ii) formula

We get

= ((root(6)(64))542) / ((43)2+(5/25)2)

Use the (iii) formula

We get

= ((root(6)(64))542) / ((46)+(5/25)2)

Use the (iii) formula

We get

= ((root(6)(64))542) / ((46)+(52/252))

= (2 (542) / ((256) + (25 / 252))

Cancel the same elements

We get

= 2(542) / ((256) + (1 / 25))

= 2(542) / ((6400 + 1) / 25 )

= 2(2916) / (6401/25)

= 2(2916)(25) / (6401)

= 50(2916) / 6401

= 145800 / 6401

= 22.77

What are Perpendicular lines

Two intersecting lines will have four angles formed at the intersection points. If all the four angles are equal, then the two lines are said to be perpendicular to each other. We already know by linear postulate theorem that the two vertically opposite angles are equal. Hence if these two lines are perpendicular, then all four angles are 90 degrees.

Examples of perpendicular lines:

In the graph paper, The X-axis and Y-axis are perpendicular.
In an ellipse two axes, minor axis, and major axis are perpendicular.
For a line segment, any shortest line from a point outside the circle is perpendicular.
Tangent and normal to any curve are perpendicular lines.

Slopes of two perpendicular lines:  In coordinate Geometry, when two lines are perpendicular, the product of the slopes of the lines is -1.  This property has a lot of applications in finding the equation of perpendicular lines, length of perpendicular segment from a point to a given line, etc.

For any curve in a graph with equation y = f(x), the slope of the tangent is defined as the rate of change of y with respect to x at that point. The normal to this curve at this point is perpendicular to the tangent line.

Example:  In a circle, with centre at the origin and radius 3, the equation will be of the form (x)²+(y)² = 3². Take any point say (0,3). To find the tangent, we have to find dy/dx.

Differentiating, 2x+2y  =0

Hence, the slope of the normal is perpendicular to x axis or parallel to y axis.

Example for Perpendicular lines from a point to a line

Let AB be a line with coordinates (1,2) and (3,4).  Measure the length of perpendicular line from (-1,1) to this line segment.

We know that the perpendicular line from (-1,1) has a slope  of -1/slope of AB.

Equation of AB is (x-1)/(3-1) = (y-2)/(4-2)  Or x-1 = y-2 Or y = x+1

Slope of AB passing through (1,2) and (3,4) is 4-2/3-1 =1.

Slope of perpendicular line to AB is -1.

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Since the perpendicular line passes through (-1,1) equation of the perpendicular is y-1 = -1(x+1)  or y =-x -1 +1 or y = -x.

To get the foot of the perpendicular line on AB, we solve the two equations by substitution method.

y = x+1 = -x   This on simplification gives 2x=-1 or x=-1/2.

Since y = -x , we have y = +1/2,

So, foot of the altitude from the point (-1,1) is (-1/2,1/2).

The length of the perpendicular segment is between (-1,1) and (-1/2,1/2) is

√[ (-1/2+1)²+(1/2-1)²] = √(1/4+1/4)   =  √(1/2) = 1/1.414 =0.707 approximately.