Number of Angles in a Triangle

Introduction to number of angles in triangle:
                              The three line segments which enclosed to form the triangle .In a triangle there are three sides and three vertices which forms three angles. The angles in a triangle can be represented by the capital letters and the sides of triangle are by small letter. There are number of angles in a triangle according to their measurements. Now we are going to see about the number of angles in triangle.



Number of Angles in a Triangle:

Acute angle:

                     An acute angle is nothing but whose measure is less than 90 degree. Some of the examples for acute angles are 40°, 45°, 50° etc.

                                                                                  
Obtuse angle:

                      The obtuse angle is nothing but an angle whose measure will be greater than 90° degree but less than 180° degree is called as an obtuse angle.

                                                                 
Right angle:

                      In a right angle triangle the measurement of one angle will be 90° degree.

                                                               

Problems for Number of Angles in a Triangle:

Ex1:

                 Whether the following be the measuring angles of a triangle which are given below

45°, 80°, 55°

Sol:

                 The sum of the measure of the three angles is

               45 + 80 + 55 = 180

               Therefore 45°, 80°, 55° can be the measure of the angles of a triangle

Ex 2:

                 The angles of triangle are given as 40°, 70°, 90° . Determine the type of angle in triangle?

Sol:

                               The sum of the three angles = 40 + 70 + 65

                                                                     = 175.

          The sum of the three angles which gives 175 degrees. The angle in the given triangle is an obtuse angle.

Ex 3:

               Find the height of the stick where the length of one side is 600m and the angle is 60°.

Sol:

            The adjacent side which is present next to the angle and the greater side of the triangle is the hypotenuse.

         In this triangle the cosine formula is used to find its height are,

         Cos 60° = adjacent/hypotenuse

         Cos 60° = h/600

         Cos 60° = ½

         h/600 = ½

              h = 300.

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Volume of Hollow Cylinder

Introduction to volume of hollow cylinder:

          A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since deep antiquity.

The above figure represents the hollow cylinder. Formula for finding volume of hollow cylinder is

Formula:

`pi` R2h - `pi` r2h = `pi` h (R2 – r2)

Here R represents the radius of the outer surface

Similarly r represents the radius of the inner surface

Volume of Hollow Cylinder - Example Problems

Example: 1

Find the volume of the hollow cylinder with radius of the inner surface is 6 meter; radius of the outer surface is 8 meter and height is 9 meter.

Solution:

Already we know that volume of the hollow cylinder is `pi` h (R2 – r2)

Here R = 8 meter, r = 6 meter, h = 9 meter

Substitute this value into the formula we get

Volume = 3.14 * 9 (82 – 62)

Simplify the above equation we get

= 28.26 (64 – 36)

= 28.26 *28

= 791.28 meter cube

Therefore the volume of the hollow cylinder is 791.28 meter 3

Example: 2

Find the volume of the hollow cylinder with diameter of the inner surface is 22 meter; radius of the outer surface is 13 meter and height is 8 meter.

Solution:

We know the formula `pi` h (R2 – r2)

Here given is diameter of the inner surface we have to the radius

Radius = `(diameter ) / 2`

Radius = `22/2` = 11 meter

Now r = 11, R = 13, h=8 `pi` = 3.14

Substitute this value into the formula we get

Volume = 3.14 * 8 (132 – 112)

= 25.12 (169 – 121)

= 25.12 * 48

= 1205.76 meter cube

Therefore the volume of the hollow cylinder is 1205.76 meter3

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Volume of Hollow Cylinder - Practice Problems:

Find the volume of the hollow cylinder with radius of the inner surface is 9 meter; radius of the outer surface is 12 meter and height is 11 meter.
Answer = 2177.12 meter 3

Find the volume of the hollow cylinder with diameter of the inner surface is 24 meter; diameter of the outer surface is 30 meter and height is 12 meter.
Answer = 3053.63 meter3

Introduction to minimum deviation of a prism

Introduction to minimum deviation of a prism:

A prism is a transparent medium bounded by two plane surfaces inclined to each other at a suitable angle. The angle between two surfaces is known as refracting angle or angle of prism. In a prism, a ray of light suffers two refraction and the result is deviation. In other words, we say that after passing through prism the ray of light deviates through a certain angle from its original path.

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Minimum Deviation of a Prism:

Look at the given diagram, here ABC is the principal section of the prism and the angle of prism is A.

 

A ray of light KL is incident on face AB of the prism at Ði₁. It bends towards the normal NO and refracted along LM at Ðr₁. The refracted ray LM is incident at Ðr₂ on face AC of the prism. It bends away from the normal PO and emerges MS at angle Ði₂. In passing through the prism, the ray KL suffers two refraction and finally turned through an ÐQTM = d, this is the angle of deviation.

Calculation for the Minimum Deviation of a Prism:

In  TLM, `delta` = TLM + ÐTML

 `delta` = (i₁ – r₁) + (i₂ – r₂)

 `delta` = (i₁ + i₂) - (r₁ + r₂)                       ….(1)

In D OLM, ÐO + r₁ + r₂ = 180°           ….(2)

In quadrilateral ALOM,

ÐL + ÐM = 180°

ÐA + ÐO = 180°

Put this value in equation (2), we get

ÐO + r₁ + r₂ = ÐA + ÐO

r₁ + r₂ = ÐA                                                     ….(3)

Put this value in equation (1)

 `delta` = (i₁ + i₂) – A                                                ….(4)

Let n be the refractive index of the medium of prism with respect to air

n = Sin i₁ / Sin r₁ = i₁ /r₁                        (when angles are small)

i₁ = n r₁ , Similarly i₂ = n r₂

Put this value in equation (4), we get

d = n(r₁ + r₂) – A

d = n A – A

 `delta` = (n – 1)A

In case of minimum deviation of a prism, i₁ = i₂ = i so that r₁ = r₂ = r, so put these values in equation (3) and in equation (4), we get

r + r = A

r = A/2

 `delta`_m = (i + i) – A = 2i – A   where d_m is the minimum deviation angle

i = ( `delta`_m + A)/2

n =   

Introduction to prime numbers and divisibility

Introduction to prime numbers and divisibility:

         In mathematics, a prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself. The first twenty-five prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

          Infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC, although the density of prime numbers within natural numbers is 0.                                                                                                                                                                                                                

Introduction to Prime Numbers and Divisibility

        In the mathematics, divisibility has many rules. This divisibility rules are discussed by the following methods. These divisibility rules are very useful to understand the divisibility of the number, which is even number, odd number, and decimal number etc., Here we discuss the rules of dividing the numbers 2 to 10.

Example on Prime Numbers and Divisibility

Example for finding Prime factor:       

       Following numbers 3,5, and 7 are prime numbers and  8 is composite number. Here one prime devisor is common for all  positive integers m,  therefore, it gives the own prime devisor if m is prime. The prime factor completely reduced by the m. Here m is composite.

8= 2*2*2, 24=3*2*2*2, 56=8*7=2*2*2*7.

Rules of divisibility and its examples:

Rule 1: (Dividing by 2)

All even numbers can be divide by number two (2).

Example: Any number ending with these numbers 0,2,4,6 or 8can be divide by number two (2).

Rule 2 :( Dividing by 3)

To adding, the all digits in the given number and determine its total, if the total can be divide by 3, therefore the rule is suitable for that number.

     Example: 32121 (3+2+1+2+1=9) 9 is divisible by3, therefore 32121 is divisible by 3

Rule 3: (Dividing by 4)

To find the last two digits of the given number can be divisible by 4. So, the number is suitable for this rule.

For example: 434712 ends in 12, which is divisible by 4, thus so is 434712.

Rule 4 :( Dividing by 5)

In this rule, the given numbers ended with a 5 or a 0 can be divisible by 5.

Rule 5: (Dividing by 6)

When the given number can be divisible by two and three, the number can be divisible by 6 also.

Introduction about complex fraction calculator

Fraction:

Fraction is defined as an element of quotient field. Fraction can be represented as " x / y " here fraction variable 'x' denotes the value called as numerator and fraction variable 'y' denotes the value called as denominator and the denominator 'y' is not equal to zero.

Complex Fractions:

If a fraction of numerator and denominator contains a fraction, it is called complex fraction.

Simplify Using Complex Fraction Calculator:

The complex fraction is also called as a rational expression because it has a numerator or denominator as a fraction. Otherwise, the overall fraction includes at least one fraction.

For ex:

1) (1/3)/2

Here 1/3 is numerator and 2 is denominator.

2) ((2/5)+7)/(9/11)

Here the numerator has (2/5) +7 and the denominator has 9/11.

General Rules:

To simplify the complex fraction calculator following steps are involved,

Rule1: Simplify  the numerator and denominator in the form of single fractions by adding or subtracting.

Rule2: For dividing fractions, multiply the numerator by the reciprocal of denominator.

Rule3: Simplify the whole fraction.

Solve Example of Complex Fractions Using Calculator:

Ex 1: Simplify  8/ (1/4)

Sol: 8/ (1/4)

indicates

               = 8 / 1/4

Numerator can be written as fraction as follows,

8 as         = 8/1

That is     = 8/1 ÷ 1/4

From rule 2 it can be written as

                = 8/1 * 4/1

Finally we get

                = 32/1

Ex 2: Simplify ((1/2)-(1/3))/ (1/12)

Sol:

Step1: Subtracting the numerator

We get,

                  = (1/2) – (1/3)

From LCD 6 we can rewrite as,

                  = (3/6) - (2/6)

                  = 1/6
There is no operation in denominator.

So substitute the above value back into complex fraction.

                  = (1/6)/ (1/12)

Step2: Multiply the numerator by reciprocal of denominator

           1/12 = 12/1

So              = (1/6) * (12/1)

                  = (12/6)

Step3: Finally simplify the fraction

         (12/6) = 2

So answer is 2

Ex 3: Calculate the following complex fractions  (4/7)/(2/21)*(2/3)/6

Sol:

Change the numerator as a simple fraction,

That is  4/7 ÷ 2/21 = 4/7 * 21/2

                          = 6

Change the denominator as a simple fraction,

                2/3 / 6 = 2/3 * 1/6

                          = 1/9

Finally multiply the numerator and denominator,

      ((4/7)/ (2/21))*((2/3)/6) = 6 * (1/9)

                                      = 6/9

Reduce the final fraction, `2/3` is the answer.