86-Area Volume Calculation Tool

Special Note: The Following link will help you to get all the calculations that are based on two values given by you.

The most amazing part of this blog is that you will get all calculated values such as perimeters, area volume, curved surface area, and total surface area, using the software of which the link is provided here.

Click on the following Button for Area Volume Calculator:

85-Magic Square-9 (magic square of some particular date Part-3)

Click here for the previous part of the magic square.

The most amazing part of this blog is that you can prepare your own two Magic Squares using the software for which the link is provided here.

You can prepare magic squares simply by clicking the following images.

A) Magic Square Maker for any Date:

B) Magic Square Maker for Any Numbers:

In the previous blog, we observed, the addition of all the numbers in other patterns 1, 2, and 3.
Today we will discuss some new concepts of the magic square.

7) Addition of all the numbers in other pattern-04: 

See the following diagrams in which each row is highlighted.

Here, two more patterns have the sum of the numbers as the same. For the first pattern, the middle numbers are to be taken from the 1st and second rows, and for the second pattern, we need to take the middle numbers from the 3rd and the 4th row. Please see the following diagram to understand more about this concept.

Fig (27)                           Fig (28)

04 + 19 + 67 + 31  =  121        21 + 02 + 29 + 69  =  121

In the second case,  

see the following diagram.

Fig (29)                       Fig (30)

69 + 18 + 02 + 32  =  121      01 + 33 + 70 + 17  =  121 

Click here for the next part of the magic square.

ANIL SATPUTE

84-Magic Square-8 (magic square of some particular date Part-2)

Click here for the previous part of the magic square.

In the previous blog, we observed, the addition of all the numbers in each row, column, diagonal and broken diagonal. Today we will discuss some new patterns of the same magic square.

4) Addition of all the numbers in other pattern-01: 

83-Magic Square-7 (magic square of some particular date Part-1)

Click here for the next blog on the magic square.

Today we will discuss something new about the Magic square prepared for some specific date.

See the following examples for the date 30/04/1968.

Here the 1st row of both the squares is 30/04/1968.

The addition of all the numbers in each row, column, and diagonal is the same. Like this all together we 26 kind of groups in which addition of all these numbers is same. We will discuss all these groups one by one. Let us observe all these groups in the following diagrams.

1) Addition of all the numbers in each row: 

82-5 Colors and 5 Cubes - Fun with mathematics- graph theory

Today, we will see something new, other than our regular study.

We will take 5 colors and 5 cubes. We will paint them in such a way that in some particular position, no color is repeated on any cube to that side. If we look at all the 5 cubes, from the front, we will see that no color is repeated. In the same way, the same situation is there from all the remaining 5 sides.  Now we will see the following diagram.

Diagram-01

81-Basics of Trigonometry - 09 Important key points

Click here for the previous basics of trigonometry.

Basics of trigonometry

Now we will study the next part of trigonometry. 

1) Fundamental Identities:

      a) sin² Θ + cos² Θ = 1

      b) sec² Θ - tan² Θ = 1

      c) csc² Θ - cot² Θ = 1

Proof:

      a) sin² Θ + cos² Θ = 1 

Here, 

       1)  x² + y²  = r²  -----------------(1)

            Dividing equation 1 by r² , we get,  

       2)  (x² / r² + y² / r² = r² / r²

       3)  (x / r) ² + (y / r) ² = 1 -------(2)

       4) We know that, 

          y/r = sin Θ

          x/r  =  cos Θ, ----- from equation 2, we have,

      5)  (cos Θ) ² + (sin Θ) ² = 1 ---(3)         

      6)  (sin Θ) ² + (cos Θ) ² = 1

      b) sec² Θ - tan² Θ = 1

Here, 

       1)  x² + y²  = r²  -------------------------(1)

            Dividing equation 1 by x²,we get,  

       2)  (x² / x² + y² / x² = r² / x²

       3)  (x / x) ² + (y / x) ² = (r / x) ² -------(2)

       4) We know that, 

           y/x =  tan Θ

           r/x =  sec Θ, ----- from equation 2, we have,

      5)  (1) ² + (tan Θ) ² =  (sec Θ) ² ---(3)         

      6)  (sec Θ) ² - (tan Θ) ² = 1

      c) csc² Θ - cot² Θ = 1

Here, 

       1)  x² + y²  = r²  -------------------------(1)

            Dividing equation 1 by y²,we get,  

       2)  (x² / y² + y² / y² = r² / y²

       3)  (x / y) ² + (y / y) ² = (r / y) ² -------(2)

       4) We know that, 

           x/y =  cot Θ

           r/y =  csc Θ, ----- from equation 2, we have,

      5)  (cot Θ) ² + (1) ² =  (csc Θ) ² ---(3)         

      6)  (csc Θ) ² - (cot Θ) ² = 1

Click here for the next basics of trigonometry.

Anil Satpute

80-Basics of Trigonometry - 08 Important key points

Click here for the previous basics of trigonometry.

Basics of trigonometry

Now we will study the next part of trigonometry. 

Today we will study the trigonometric ratios of three groups as shown below.

Group-03: 150° (90° + 60°), 240° (180° + 60°), 330° (270° + 60°).

3) Trigonometric ratios of Group-03: 150° (90° + 60°), 240° (180° + 60°), 330° (270° + 60°).

a) An angle q = 150°

Here angle XOP is 150° (Anti-clock-wise-direction).

Here, an inclination of ray OP is 150°, so angle AOP 30° and angle OPA is 30°

We know that the side opposite 30° is 1/2 times the hypotenuse.
So if hypotenuse OP = r, then  AP = r/2 (side opposite of 30°).

As point A is to the negative side of the X-axis, the x-coordinate of point A will be - √3r/2. In the same way, point P is in the 2nd quadrant so the y-coordinate of point P will be r/2. So, the coordinates of point P will be (-√3r/2, r/2).

So, all the trigonometric ratios of q = 150° with

 x = - √3r/2,

 y = r/2,

 r = r.

a) sin 150° = y/r     sin 150° = (r/2)/r        sin 150° = 1/2	b) cos 150° = x/r     cos 150° = (-√3 r/2)/r       cos 150° = - √3/2	c) tan 150° = y/x     tan 150° = (r/2)/(-√3 r/2)         tan 150° = - 1/√3
d) csc 150° = r/y     csc 150° =  r/(r/2)          csc 150° = 2	e) sec 150° = r/x     sec 150° = r/(-√3 r/2)         sec 150° = - 2/√3	f) cot 150° = x/y     cot 150° = (-√3 r/2)/(r/2)        cot 150° = - √3

b) An angle q = 240°

Here angle XOP is 240° (Anti-clock-wise-direction).

Here, an inclination of ray OP is 240°, so angle AOP 60° and angle OPA is 30°

We know that the side opposite 30° is 1/2 times the hypotenuse.

So if hypotenuse OP = r, then  AO = r/2 (side opposite to 30°)

& AP = √3 r/2 (side opposite of 60°).

As point A is to the negative side of the X-axis, the x-coordinate of point A will be - r/2. In the same way, point P is in the 2nd quadrant so the y-coordinate of point P will be √3 r/2. So, the coordinates of point P will be (-r/2, √3 r/2).

So, all the trigonometric ratios of q = 150° with

 x = - r/2,

 y = - √3 r/2,

 r = r.

a) sin 240° = y/r     sin 240° = [(-√3 r)/2]/r        sin 240° = -√3/2	b) cos 240° = x/r     cos 240°  = (-r/2)/r       cos 240° = -1/2	c) tan 240° = y/x     tan 240° = [(-√3 r)/2]/(-r/2)      tan 240° =  √3
d) csc 240° = r/y     csc 240° =  r/[(-√3 r)/2]       csc 240° = -2/√3	e) sec 240° = r/x     sec 240° = r/(-r/2)        sec 240° = -2	f) cot 240° = x/y     cot 240° = (-r/2)/[(-√3 r)/2]     cot 240° = 1/√3

c) An angle q = 330°

Here angle XOP is 330° (Anti-clock-wise-direction).

Here, an inclination of ray OP is 330°, so angle AOP 30° and angle OPA is 60°

We know that the side opposite 30° is half the hypotenuse and the side opposite 60° is √3/2 times the hypotenuse.

So if hypotenuse OP = r, then  AP = r/2 (side opposite of 30°)
and OA = (√3 r)/2 (the side opposite of 60°).

As point A is to the positive side of the X-axis, the x-coordinate of point A will be (√3 r)/2. In the same way, point P is in the 4th quadrant so the y-coordinate of point P will be -r/2. So, the coordinates of point P will be (√3 r/2, -r/2)

So, all the trigonometric ratios of q = 330° with

 x = (√3 r)/2,

 y = - r/2,

 r =  r.

a) sin 330° = y/r     sin 330° = (-r/2)/r        sin 330° = -1/2	b) cos 330° = x/r     cos 330= (√3 r)/2/ r        cos 330° = √3/2	c) tan 330° = y/x     tan 330° = (-r/2)/(√3 r)/2      tan 330° =  -1/√3
d) csc 330° = r/y     csc 330° =  r/(-r/2)     csc 330° = -2	e) sec 330° = r/x     sec 330° = r/(√3 r)/2        sec 330° = 2/√3	f) cot 330° = x/y     cot 330° = (√3 r)/2/(-r/2)     cot 330° = -√3

Click here for the next basics of trigonometry.

Anil Satpute