Trigonometric Identities Sum Help

Introduction to trigonometric identities sum help:

Trigonometry is arrived from the Greek word, trigonon = triangle and metron = measure. The father of trigonometry is Hipparchus. He designed the first trigonometric table. Trigonometry has wide range of applications in many fields like science, technology, astronomy etc. Identity is defined as an equation that is true for all probable values of its variables. Online help is one of the comfortable method of getting help from anywhere around the globe. Through online study, students can get about trigonometric identities sum. In this topic, we are going to see about, trigonometric identities sum help. I like to share this Pythagorean Trigonometric Identities with you all through my article.

Trigonometric Identities Sum Help - Trigonometric Identities:

The list of trigonometric identities sum are shown below,

Sum or difference of two angles:

sin (a ± b ) = sin a cos b ± cos a sin b

cos(a ± b) = cos a cos b ± sin a sin b

tan(a ± b) = `(tan a +- tan b)/ (1 +- tan a tan b)`

Sum and product formulas:

sin a + sin b = `2sin((a+b)/2)cos((a-b)/2)`

sin a - sin b = `2cos((a+b)/2) sin((a-b)/2)`

cos a + cos b = `2cos((a+b)/2) cos((a-b)/2)`

cos a – cos b = `-2sin((a+b)/2) sin((a-b)/2)`

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Trigonometric Identities Sum Help: - Examples

Example 1:

Evaluate Sin 153

Solution:

Sin 153 = Sin (90+63)

= sin 90 cos 63 + cos 90 sin 63

= 1(0.454) + 0(0.891)

= 0.454 + 0

= 0.454

The answer is 0.454

Example 2:

Evaluate Cos 128

Solution:

Cos 128 = Cos (90 + 38)

= cos 90 cos 38 – sin 90 sin 38

= 0(0.788) – 1(0.616)

= 0 – 0.616

= -0.616

The answer is -0.616

Example 3:

Evaluate tan 38

Solution:

Tan 38 = Tan (45 - 7)

= `(tan 45 - tan 7)/(1+tan 45*tan 7)`

= `(1-0.123)/(1+(1*0.123))`

= `0.877/(1+0.123)`

=` 0.877/ 1.123`

= 0.781

The answer is 0.781

Example 4

Evaluate Cos 124

Solution:

Cos 124 = cos (90 + 34)

= cos 90 cos 34 - sin 90 sin 34

= 0(0.829) - 1(0.559)

= 0 – 0-0.559

= -0.559

The answer is -0.559

Example 5:

Evaluate, sin 50 - sin 40

Solution:

sin a - sin b = `2cos((a+b)/2)sin((a-b)/2)`

sin 50 - sin 40 = `2 cos((50+40)/2)sin((50-40)/2)`

= `2 cos (90/2) sin(10)/2`

= 2 cos 45 sin 5

= 2 (0.707)(0.087)

= 2 * 0.062

= 0.124

The answer is 0.124

Graphs and Histograms

Graphs are pictorial representation of data.Histogram is graphical representation of data in statistics.

Introduction to graphs and histograms:

A histogram graph is representation of a frequency distribution as a graph . The graph  consists of rectangles constructed with class intervals as bases and heights proportional to corresponding frequencies such that there is no gap between any two successive rectangles.

A histogram deals with continuous type of data.

Different types of histograms are:

Histogram of continuous grouped frequency distribution with equal class intervals.

Histogram of a continuous grouped frequency distribution with unequal class intervals.

Histogram when mid-points are given

Histogram for grouped frequency with inclusive classes(discontinuous class-intervals)

Steps for Construction of Histogram:

Choose a suitable scale on the x-axis and represent the class-limits on it

Choose a suitable scale on the y-axis and represent the corresponding frequencies on it

Draw rectangles with class intervals as bases and the respective frequencies as heights

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Example Histogram with Equal Class Interval:

Ex : 1 Constuct a histogram to represent the following data:

Marks
Number of

students

0-10    4
10-20    7
20-30    12
30-40    20
40-50    9
50-60    2
Sol:

Choose a scale

Step 1:Along x-axis 1 cm = 10 marks

Step 2:Along y-axis 1 cm = 5 students

Step3:Starting from 0, mark 10,20,30,40,50,60,70 on x-axis and 4,7,12,20,9,2 on the y-axis

Step4:Then we draw the rectangles with class intervals as bases and corresponding frequencies as heights.

Steps to Draw Histogram of Unequal Class Interval:
Choose a suitable scale on the x-axis and represent the class-limits on it
Determine a class interval which has the minimum class size. Let the minimum class size be h
Find the adjusted frequency of each class by using the formula :
Adjusted frequency of the class = `h/("Class - size of the class)` x frequency of the class

Choose suitable scale on the y-axis and represent the corresponding adjusted frequencies on it
Draw the rectangles , the width of rectangles will be according to class limit

Direction Vector of a Line

Introduction to direction vector of a line:

The direction vector is a vector point of direction and it indicates the direction of line. The direction vector of a line is based on real values of line segment equation. In math, the vector points are used in Euclidean space. Now we are going to see about direction vector of a line.

Explanation for Direction Vector of a Line
Direction vector:

The line is represented as `vecAB` and the arrow mark symbol is indicating the direction of line. The direction vectors are determined from line equation.

In Euclidean space, the direction vector of line D is determined by using line equation form real numbers that is the line equation form is ax + by + c = 0. Here a, b, and c are real numbers and the direction vector of line D is (-b, a). We can also consider the multiples of (-b, a) as direction vectors. Understanding empirical probability is always challenging for me but thanks to all math help websites to help me out.

More about Direction of Vector of Line

Example problems for direction vector of a line:

Problem 1: Find out the direction vector of a line segment D from line form.

4x + 3y + 1 = 0.

Solution:

The given line equation form is 4x + 3y + 1 = 0.

The line segment D has two end points AB.

The direction vector `vecAB` is determined from line form.

`vecAB` = (-3, 4).

Therefore, the direction vector of line is (-3, 4), (9, 16)…

Problem 2: Find out the direction vector of a line segment D from line form.

x - 2y + 4 = 0.

Solution:

The given line equation form is x - 2y + 4 = 0.

The line segment D has two end points AB.

The direction vector `vecAB` is determined from line form.

`vecAB` = (2, 1).

Therefore, the direction vector of line is (2, 1), (4,1)…

Exercise problems for direction vector of line:

1. Find out the direction vector of line D from 3x + 4y + 2 = 0.

Solution: The direction vector `vecAB` is (-4, 3).

2. Find out the direction vector of line D from 6x - 3y - 5 = 0.

Solution: The direction vector `vecAB` is (3, 6).

Number Theory Problems and Solutions

Introduction to number theory  problems and solutions

Number theory is the branch of pure mathematics that concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. Number theory may be subdivided into the several fields, according to the methods used and the type of questions investigated and let we about number theory problems and solutions. (Source – Wikipedia).

Number Theory Problems and Solutions
Example 1:

Find the solutions of consecutive numbers and the product of the number is 143
Solution:-

Let as assume the two consecutive numbers be x, x +1.

Where the product is 143 so,

X * (x + 1) = 143

x2 + x = 143

x2 + x – 143 = 0

x2 + 13x – 11x – 143 = 0

(x + 13) (x - 11) = 0

(x + 13) = 0 (or) (x - 11) = 0

x = -13 (or) x = 11

-13 is not possible to get 143

Therefore we take x = 11

So, x + 1 = 11 + 1 = 12

The consecutive numbers is 11 and 12.

Example 2:

Verify the given sequence described by an = 12n2 + 1 and A.P.?

Solution:

an = 12n2 + 1

a1 = 12(1)2 + 1 = 13,

a2 = 12(2)2 + 1 = 49

a3 = 12(3)2 + 1 = 109,

a4 = 12(4)2 + 1 = 193

The  number theory problems solutions in sequence is 13, 49, 109, 193...

Here, 49 – 13 = 36

Example 3 :

Find the missed term in the given numbers 31, 29, ____, 25.
The above stated number is in descending order.

The known difference between the number 29 – 31 = 2.

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More Number Theory Problems and Solutions

Example 1:

Identify the type of numbers -7, 3 ÷ 4, `sqrt(8)`

Solution:

-7 - Integer

3 ÷ 4 - Fraction

`sqrt(8)` - Irrational Number

Example 2:

Mention the values of the consecutive numbers where the sum of the two numbers is 91.

Solution: Let we mention the two consecutive numbers be x, x+1.

Where the sum is 91 so,

x + x + 1 = 91

2x + 1 = 91

2x = 90

x = 45

Therefore, x + 1 = 45 + 1 = 46

So the  number theory problems solutions of an  consecutive numbers is 45 and 46.

The above problems are stated in number theory  problems and solutions

Tenth Grade Math Problems

Introduction to tenth grade math problems:

Now in this article we are going to see about tenth grade math problems. Tenth grade math topics are linear algebra, calculus, integral, slope equation, logarithmic, line equation, geometry, trigonometry function, trigonometry identities, word problems, and quadratic equations. Students know the basic differentiation and integration problems in this grade before entering into their high school. We solve more problems in tenth grade mathematics. In this article, we solve some problems used in tenth grade mathematics.

Example Problems for Tenth Grade Math

Tenth grade math example problem 1:

Solve the first order linear equation y = x + 7 and 6x + y = 70

Solution:

Given first order linear equation is y = x + 7 and 6x + y = 70

Here,

y = x + 7 -------- (1)

6x + y = 70 ------------ (2)

Substitute the equation 1 in equation 2, we get

x + 6x + 7 = 70

After simplification, we get

7x + 7 = 70

Subtract 7 on both the sides, we get

7x = 63

Divide by 7 on both the sides, we get

x = 9

Substitute the value of x in equation 1, we get

y = (9) + 7

y = 16

The answer is x = 9 and y = 16

Answer:

The final answer is x = 9 and y = 16.

Tenth grade math example problem 2:

Find the second order value of the function y = 4x3 + 2x2 + 98x

Solution:

The given function is y = 4x3 + 2x2 + 98x

First find the first order of derivative value of the function y, we get

`(dy / dx)` = (4 * 3)x2 + (2 * 2)x + (98 * 1)

= 12x2 + 4x + 98

Second order derivative of the function y is given as,

`((d^2y) / (dx^2))` = (12 * 2)x + (4 * 1)

= 24x + 4

Answer:

The final answer is 24x + 4


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Tenth Grade Math Example Problem 3:

A train travelling in a speed of 23km/h. In 3 hours how long train travel from the starting point.

Solution:

From the given,

Distance = ?

Time = 3 hours

Speed = `23((km)/(h))`

We know the formula,

Speed = `((D)/(t))`

Here, D = Distance and t = Time

Rearrange the above formula, we get

Distance = Speed * Time

Substitute the given values, we get

= `23((km)/(h))` * 3hours

= 69km

Therefore, speed of the train is 69km

Answer:

The final answer is 69km

Factors and Product in Math

Introduction to factors in math:
Divisor, an integer which evenly divides a number without leaving a remainder
Factorization, the decomposition of an object into a product of other objects
Integer factorization, the process of breaking down a composite number into smaller non-trivial divisors
A coefficient, a multiplicative factor in an expression, usually a number
A von Neumann algebra with a trivial center
Factor (graph theory), a spanning sub graph
(Source: Wikipedia)

Example Problem for Factors and Product in Math

Factors and product in math example problem 1:

Find the factors of the given quadratic equation  x2 - 11x - 180

Solution:

Given quadratic equation is x2 - 11x - 180

First factorize the given equation, we get

(x2 - 11x - 180) = (x2 - 20x + 9x - 180)

Grouping the first two terms and second two terms, we get

= (x2 - 20x) + (9x - 180)

= x (x - 20) + 9 (x - 20)

= (x - 20) (x + 9)


The factors of the given quadratic equation is (x - 20) and (x + 9)

Answer:

The final answer is (x - 20) and (x + 9)

Factors and product in math example problem 2:

Find the factors of the given quadratic equation  x2 + 13x + 42

Solution:

Given quadratic equation is x2 + 13x + 42

First factorize the given equation, we get

(x2 + 13x + 42) = (x2 + 7x + 6x + 42)

Grouping the first two terms and second two terms, we get

= (x2 + 7x) + (6x + 42)

= x (x + 7) + 6 (x + 7)

= (x + 6) (x + 7)


The factors of the given quadratic equation is (x + 6) and (x + 7)

Answer:

The final answer is (x + 6) and (x + 7)

Factors and product in math example problem 3:

Find the product of the given two factors (x - 2) and (x + 3)

Solution:

Given two factors are (x - 2) and (x + 3)

Multiply the each term of the two factors together, we get

(x - 2) * (x + 3) = (x * (x + 3)) - (2 * (x + 3))

= x2 + 3x - 2x - 6

= x2 + x - 6

The product of the two factors are x2 + x - 6

Answer:

The final answer is x2 + x - 6

Factors and product in math example problem 4:

Find the product of the given two factors (x - 4) and (x + 6)

Solution:

Given two factors are (x - 4) and (x + 6)

Multiply the each term of the two factors together, we get

(x - 4) * (x + 6) = (x * (x + 6)) - (4 * (x + 6))

= x2 + 6x - 4x - 24

= x2 + 2x - 24

The product of the two factors are x2 + 2x - 24

Answer:

The final answer is x2 + 2x - 24

Practice Problems for Factors and Product in Math

Factors and product in math practice problem 1:

Find the factors of the given quadratic equation  x2 - 31x - 360

Answer:

The final answer is (x - 40) (x + 9)

Factors and product in math practice problem 2:

Find the product of the given two factors (x + 4) and (x + 3)

Answer:

The product of the two factors are x2 + 7x + 12

Determining Polynomial Functions

Introduction about polynomial:

In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminate) and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x2 - 4x + 7 is a polynomial, but x2 - 4/x + 7x3/2 is not, because its second term involves division by the variable x and because its third term contains an exponent that is not a whole number. (Source: Wikipedia)

Example Problems for Determining Polynomial Functions :

Determining polynomial functions - Example problem 1:

Simplify f(x) = (21x + 15y) + ( 9x – 21y ) – (11x – 17y)

Solution:

f(x) = (21x + 15y) + ( 9x – 21y ) – (11x – 17y) = 21x + 15y + 9x – 21y – 11x + 17y

= 21x + 9x - 11x + 15y – 21y + 17y

= 19x +11y

The answer for f(x) = (21x + 15y) + ( 9x – 21y ) – (11x – 17y) = 19x + 11y

Determining polynomial functions - Example problem 2:

Simplify f(x) = (15x^2 – 11x – 21) + (7x^2 – 16x – 19) - (–7x^2 + 5x + 21)

Solution:

(15x^2 – 11x – 21) + (7x^2 – 16x – 19) - (–7x^2 + 5x + 21)

= 15x^2 – 11x – 21 + 7x^2 – 16x – 19 + 7x^2 - 5x - 21

= 15x^2 + 7x^2 + 7x^2 – 11x – 16x - 5x – 21 – 19 - 21

= 29x^2– 36x – 61

The answer for f(x) = (15x^2 – 11x – 21) + (7x^2 – 16x – 19) - (–7x^2 + 5x + 21) = 29x^2– 36x – 61

Determining polynomial functions - Example problem 3:

Simplify f(x) = (14x3 + 8x^2 + 21x – 16) – (18x3 – 15x^2 – 7x + 17) - (8x^2 – 11x – 21)

Solution:

(14x3 + 8x^2 + 21x – 16) – (18x3 – 15x^2 – 7x + 17) - (8x^2 – 11x – 21)

= (14x3 + 8x^2 + 21x – 16) – 1(18x3 – 15x^2 – 7x + 17) – 1(8x^2 – 11x – 21)

= (14x3 + 8x^2 + 21x – 16)– 1(18x3) – 1 (–15x^2) – 1(–7x) – 1(17) - 1(8x^2) -1(-11x) -1(-21)

= 14x3 + 8x^2 + 21x – 16 – 18x3 + 15x^2 + 7x – 17 - 8x^2 + 11x + 21

= 14x3 – 18x3 + 8x^2 + 15x^2 -8x^2 + 21x + 7x + 11x – 16– 17 + 21

= -4x3 + 15x^2 + 39x –20

The answer for f(x) = (14x3 + 8x^2 + 21x – 16) – (18x3 – 15x^2 – 7x + 17) - (8x^2 – 11x – 21) = -4x3 + 31x^2 + 39x –20

Practice Problems for Determining Polynomial Functions :

1. Simplify f(x)= (15x + 8y) + ( 7x – 15y ) – (8x – 9y)

The answer for f(x)= (15x + 8y) + ( 7x – 15y ) – (8x – 9y) = 14x + 2y

2. Simplify f(x)= (7x^2 – 9x – 8) + (3x^2 – 8x – 11) - (–6x^2 + 9x + 15)

The answer for f(x)=(7x^2 – 9x – 8) + (3x^2 – 8x – 11) - (–6x^2 + 9x + 15)= 16x^2– 26x – 34

Radius of a Circle Calculator

Introduction to radius of a circle calculator:

In mathematics, radius of a circle calculator is the calculator used to find the radius of the circle. In the radius of a circle calculator, we can find the radius value from any of the two values either by area or circumference. In the radius of a circle calculator, we can get the value of radius by just giving the value of circumference or area. The steps used in radius of a circle calculator are given below.

Radius of a Circle Calculator - Steps:

The diagram for the circle is given below:



Steps used in radius of a circle calculator are given below:

Step 1: Enter the value of circumference or area in the box given in the calculator.

Step 2: Then press "calculate" button.

Step 3: After some more second the process complete and shows the value of radius in the calculator.

Some of the examples based on this calculator is given below.

Radius of a Circle Calculator - Examples:

Example 1: Find the radius of the circle given that the area of circle is 153.86?

Solution:

Steps used in radius of a circle calculator  are given below:

Step 1: Enter the value of area as 153.86 in the box given in the calculator.

Step 2: Then press "calculate" button.

Step 3: After some more second the process complete and shows the value of radius as 7 in the calculator.I like to share this 6th grade math problems online with you all through my article.

Example 2: Find the radius of the circle given that the circumference of circle is 56.52?

Solution:

Steps used in radius of a circle calculator  are given below:

Step 1: Enter the value of circumference as 56.52 in the box given in the calculator.

Step 2: Then press "calculate" button.

Step 3: After some more second the process complete and shows the value of radius as 9 in the calculator.

Radius of a circle calculator – Practice problem:

Problem 1: Find the radius of the circle given that the area of circle is 379.94?

Answer is given below:

The output for that calculator is given below:



Problem 2:  Find the radius of the circle given that the circumference of circle is 81.64?

Answer is given below:

The output for that calculator is given below:

Solving Variable Expressions

Introduction to solving variable expressions:

In mathematics, an expression is a finite combination of symbols that are well-formed according to the rules applicable in the context at hand. Symbols can designate values (constants), variables, operations, relations, or can constitute punctuation or other syntactic entities. The use of expressions can range from simple arithmetic operations like 3+ 5 x ((-2)^7 – 3/2) . (Source: Wikipedia)

Types of Math Solving Variable Expressions:-

In solving variable expressions to study the algebra expressions in following types are used in algebra expressions.

Variable expressions Variable expressions using word

Variable expressions  

In variable expressions means to form a number and word in expression like as add, plus, greater, less than, increase, decrease etc.

For Example,

24 increased by x?

24 + x

Variable expressions using word

In variable expressions using word means the expression numbers and word are shows the sentence formation like as add, plus , greater ,less than ,decrease etc

For Example,

Haley earned 31 bonus points. Marisol earned b more bonus points than Haley. Choose the expressions that show how many bonus points Marisol earned.

31+b

Example Problems for Solving Variable Expressions:-

Problem 1:-

Solving variable expressions for 370 added to v.

Solution:-

Adding the variable v to number 370 and form an expression as V+370

Answer:- V+370 

Problem 2:-

Solving variable expression for 28 minus w

Solution:-

Subtracting the variable w to number 28 and form an expression as 28-w. I like to share this Algebra 2 problem solver with you all through my article.

Answer: 28 –w

Problem 3:-

Solving variable expression for 703 increased by z

Solution:-

Increased the variable z to number 703 and form an expression as 703+z

Answer: 703+z

Problem 4-

Solving variable expression for 45 decreased by p

Solution:-

Decreased the variable p to number 45 and form an expression as 45-p

Answer: 45-p

Problem 5-

Talia earned 64 bonus points. Jones earned d more bonus points than Talia. Choose the expression that shows how many bonus points Jones earned.

Solution:-

Adding the jones variable d to talia earned 64 bonus points and form an expression as 64+d

Answer: 4+d

Vector Component Calculator

Introduction to vector component calculator:

The collection of ordered components are called as vector and the types of components are x component and y component. The calculator is a device that gives the output of all math operations. The vector component is also determined by calculator. Both vector components are representing the direction by sign. Now we are going to see about vectors component calculator.

Explanation for Vector Component Calculator

Vector component:

In math, the vector component is used in algebra. The x component is representing the horizontal component that is x-axis value and the y component is representing the vertical component that is y-axis value. If two vectors are present means find out the components by addition operation.

Vector component calculator:

The vector calculator is calculating the components by addition method. The x component of first vector is added with x component of second vector. Similarly the y component of first vector and second vector is added.I like to share this math homework help answers with you all through my article.

The graph representation is used in vector calculator. The calculator contains graph with in it. We can see the calculator operations stepwise.

More about Vector Component Calculator

Following steps are operations of vector calculator:

Step 1: First plotting the first vector in graph.



The first vector is plotted as above and this vector is represented with x components 5 and y components 4.

Step 2: Plotting the second vector in graph.



The second vector is plotted as above diagram.The second vector is represented with x component 6 and y component -4.

Step 3: The addition of two vectors.



The given two vectors are added by calculator. The result of vector addition in calculator is shown above.

The calculator represents the x component value is 11 that is 5 + 6 and the y component is 0 that is 4 - 4. Another application of vector calculator is magnitude and direction determination.

Array Math Definitions

Introduction to array

Let us discuss about the array math definition. The definition of array is number or objects arranged in rows and columns. The array is the main tools in the mathematics. The basic operation of the array is multiplication and division. The array is declaring the square bracket. The example of the array is [8]. Next we see the definition of an array.

Definition of the Array in Math

The definition of the diagonal array is square matrix.  A diagonal matrix having the element only in the diagonal position. The remaining position elements are zero.

`[[9,0,0],[0,5,0],[0,0,7]]`

The array of number is 3 rows and 3 columns. That is defining the B [3] [3].

Definition of row matrix

The definition of row matrix is called as the only one row. It is represents the 1 x n. the n is declare the number of columns.

Examples of row matrix: [4   5   9]

Math definition of column matrix

The column matrix is defined as the matrix contains only one column. It is declaring the up to down format. It is represents the n x 1. The n is called as the number of rows.

Examples of column matrix:

`[[8],[11],[14]]`

Math Definition of even Array

The definition of the even array is the number of rows and column is a even number. the even number is normally divisible by two. The example of the even number is 2, 4, 6 etc. the example of the even array is,

`[[5,8],[7,6]]`

This array is the 2 x 2 array.

The definition of the 2 x 2 array is declare only 2 rows and 2 column should be followed.

2 x 2 = 2 + 2

= 4.    

Sample E Learning

Introduction to sample e learning:-

Sample e learning is important for students. Student’s does learning the sample e learning and also solve the e learning problems. Here e means `e^x` . In math exponential function means ex, where e is the significance of ex the same value again consequent.
For example,
`17e^x` this is a way to write an exponential function.
e = 2.718 is the value of e.

Basic Properties of Sample E Learning:-

In the following basic properties of  sample e learning:

`e^x e^y = e^(x+y)`
`(e^x)^p = e^(px)`
`(de^x)/(dx) = e^x`
`(de^ax)/(dx) = ae^(ax)`
`(d^n e^ax)/(dx^n) = a^n e^(ax)`
`e^x/ e^y = e^(x-y)`
`root(p)(e^x) = e^(x/p)`
`inte^x dx = e^x `

Example Problems to Sample E Learning:-

Problem 1:-

Solve the exponential function equation `e^x = 47`

Solution:-

Here the natural log is the inverses of exponential function, so use ln to get fast solve this problem.

ln `e^x` = ln 47

x = ln 47 (take natural log of 47)

= 3.850

So the answer is 3.850


Problem 2:-

Solving add the exponential equation `e^(18x) +e^(11x)`

Solution:

Given: `e^(18x) +e^(11x)`

We know the property `e^x e^y = e^(x+y)`

Take the common term e.

= `e^(18x+11x)`

= `e^(29x)`

Adding the both values and get 29x.

Finally we get an answer as `e^(29x)`


Problem 3:-

Solving whether the point (0, 1) lies on the graph of the function y = 18(4)x.

Solution:-

Substitute x = 0 in the function y = 18(4)x.

We know the property `e^x`

y = 18(4)0

= 18(1)

= 18

The y–coordinate of the point is 1, which does not match with the obtained value y = 18.

So, the graph of the function y = 18(4)x does not contain the point (0, 1).

Trigonometry Radian Measure

Introduction to trigonometry radian measure:

An angle is determined by rotating a ray about its endpoint. One way to measure angles is in radians. To signify that a given angle is in radians, a superscript c, or the abbreviation rad might be used. If no unit is given on an angle measure, the angle is assumed to be in radians. `(3pi^c)/2-=(3pi)/2 rad.-=(3pi)/2`                                                                                                                            (Source: Wikipedia)

Trigonometry Radian Measure

Middle angles of a circle contain an angle measure of 1° if it subtends an arc to be 1/360 of the boundary of the circle. These appearances of angle determine is reasonably general. Another appearance of angle determine namely in utilize is radian measure. If a middle angle subtends an arc i.e. identical toward the radius of the circle after that the middle angle contain a computing of one radian.

If a middle angle ? of a circle through radius r subtends an arc of length s, subsequently their radians establish is describing since  `theta= s/r`

Given, radius is 4 cm, and length of arc is 60 cm.

We know that the formula for radian measure of `theta=s/r` .

As a result,`theta = 60/4`

?=15

The angle of the arc is 15 radians.
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Examples for Trigonometry Radian Measure

Example 1 for trigonometry radian measure

Calculate the angle of the arc, if the radius is 6 cm, and length of arc is 120 cm

Solution:

Given, radius is 6 cm, and length of arc is 120 cm.

We know that the formula for radian measure of `theta=s/r` .

As a result,`theta = 120/6`

?=20

The angle of the arc is 20 radians.

Example 2 for trigonometry radian measure

Calculate the angle of the arc, if the radius is 8 cm, and length of arc is 135 cm

Solution:

Given, radius is 8 cm, and length of arc is 135 cm.

We know that the formula for radian measure of `theta=s/r` .

As a result,`theta = 135/8`

?=17

The angle of the arc is 17 radians.

Example 3 for trigonometry radian measure

Convert 1650 into radians

Solution:

We know that,

`(radians)/pi=(degrees)/180^0`

`radians=degrees pi/(180^0)`

`Given radians = `1650

`radians=165 pi/(180)`

Therefore,`radians=(11pi)/(12)`

Percent Discount Formula

Introduction to percent discount formula:

The word percent means out of hundred or per hundred. Percentage is a fraction with its denominator 100. The numerator of such a fraction is called rate percent.

Ex: (i) `3 / 5` = `3 / 5` `xx` 100% = 60 %

(ii) 0.25 = 0.25 `xx` 100% = 25%

(iii) 15% of 80 = `15 / 100` `xx` 80 = 12.

Now let us see few problems on this topic percent discount formula.

Example Problems on Percent Discount Formula.

Ex 1: What is the single discount which is equivalent to successive discounts of 20%, 15% and 10%?

Soln: Let us consider the price of an article be 100 dollars.

Therefore After the first discount 20%, it will cost = 100 – 20 = 80 dollar.

Second discount = 15%

Therefore 15% of 80 = 15 / 100 * 80 = 12

Therefore the new price = 80 – 12 = 68 dollars.

Third discount = 10%

Therefore 10% of 68 = `10 / 100` `xx` 68 = 6.8

Therefore the new price = 68 – 6.8 = 61.20 dollars

Therefore the single discount is 100 – 61.20 = 38.80%

Here one important thing is, we can get the single discount by adding all the discounts as 20 + 12 + 6.8 = 38.8%.

Ex 2: An article price is 450 dollars. It is sold at a discount of 20%. Find:

(i)                  The discount given

(ii)                The selling prize of the article.

Soln: Given: The price of the article is 450 dollars,

(i) The discount given = 450` xx 20/100` = 90 dollars.

(ii)  The selling price  = 450 – discount

= 450 – 90

= 360 dollars

Ex 3: In a sale, a shopkeeper allows 10% discount on his article. What price must he mark on an article, which costs his 750 dollar, to make a profit of 20%?

Soln: Given: Cost price = 750 dollars.

Therefore to give 10% discount and to gain 20%, his selling price should be as follows:

750 + 20% of 750

= 750 + `20 / 100``xx` 750 = 900 `=>` 90% (x) = 900

`=>` x = 900 `xx` `100 / 90` = 1000

Therefore the marked price = 1000 dollars.

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Practice Problems on Percent Discount Formula.

Find a single discount for 30%, 20% and 10%
[Ans: Single discount = 50.4%]

2. A dealer is selling an article at a discount of 5% on the marked price. What is the selling price if it is marked 140 dollar?

[Ans: Selling price = 133 dollars]

Descriptive Set Theory

Introduction to descriptive set theory:
A set is a well-defined collection of things

The things belonging to a set are called it's members or elements. These elements may be objects, persons, letters, numbers or any items at all.
Well-defined means that given a member, there must be no doubt in deciding whether it belongs to, or does not belong to the given set. Example:-
A set of apples weighting 200 g or over is a well-defined.
A collection of heavy apples is not well-defined and so it not a set.
All elements are separated by a comma.
The elements should be enclosed in braces (”{ }”).
Sets are usually represented by capital letters.
The order in which the members appear is not important.
If one or more elements are repeated, the set remains the same.
A set may consist of a single member like, B = {National bird of India} = {Peacock}. If 'A' is a set and 'a' is an elements of this set, then “a belongs to A”
This article is about descriptive set theory and the topics in that.

Descriptive Set Theory-representation of a Set (notation):

A set is determined by its members or elements. This determination is brought out in the following three ways.

Roster method :-

The method of listing the elements inside the braces is called Roster method. Example :- A set of alphabets of the English language represented by P is written as :
P = {A , B , C , D , E , F , ….}

Description method :-

The method of listing a set by a well-defined statement or description is called description method.
Example :- {Vowels in English alphabet}

Set builder form :-

The method of listing the elements inside the braces by stating the rule or property or formula is known as Set builder form. Example :- If A is a set of a natural numbers less than 80, then it can be represented as, A = {x : x is a natural number and is less than 80} or as

A = {=x : x E N, x<80 br="br">
Types of Sets in Descriptive Set Theory:

Finite set

The elements are limited. Example :-
Natural number less then 60.
A = {1,2,3,4,5,6,7,.....}
Days of a week
W = {Monday, ….. , Sunday}

In finite set

Unlimited numbers of elements Example :-
Set of odd numbers
N = {1 , 3 , 5 , 7....}
Set of even numbers

E = {2 , 4 , 6 , 8....}

Empty set

No elements is represented
Set of odd numbers between 5 and 6
A = { }
30th day of February

D = { }