Geometry of circle

A circle is a conic section. When a cone is cut by a plane that is exactly perpendicular to the axis of the cone, the cross section we get is a circle. All points on the circle are equidistant from a fixed point in the circle. This fixed point is called the centre.

Radius of a circle:

The distance between the centre of a circle and any point on the circle is called the radius of that circle. The radius is half the diameter of the circle. So if we denote the radius by r and the diameter by d then, r = d/2

Circle formula:

We can measure the circumference of a circular object by winding a piece of fine string around the curved surface of the object exactly once and then measuring the length of the string with a meter scale. On measuring the circumference of a number of objects we find that the value of the ratio Circumference/diameter in each case is almost the same. This would always be some number between 3.1 and 3.2. This constant ratio is named by the Greek letter pi (pronounced as pi). Therefore we can write the formula for circumference of a circle as : C = pid, where d = diameter of the circle.

Area of a circle is given by the formula: A = pi*r^2 = pi * (d/2)^2 = (pi/4)d^2

Area of a semicircle:

We know that a semicircle is formed when a diameter divides a circle into two equal halves. So obviously the area of a semicircle is exactly half the area of the circle of the same diameter. Mathematically it is written like this,
A.S. = (pi/2)*r^2, where A.S. = area of semi circle, r = radius of the semi circle.

Area of a quadrant of a circle = (1/4)* pi*r^2

Circle Geometry:

Consider a unit circle centered at the origin(O) of a co-ordinate axis. A point Q on the circle is such that the line segment OQ makes an angle of h with the positive x axis. Then the co-ordinates of the point Q would be (cos(h), sin(h)). If the radius of the circle is r, then the co-ordinates of the point Q would be (r*cos(h), r*sin(h)).

From the above figure we see that triangle OQS is a right triangle. The radius of the circle is 1, so OQ = 1. Therefore adjacent side to angle h = cos(h) and opposite side to angle h = sin(h).

When and How to Use Law of Sines and Cosines

What type of triangles use Law of Cosines and Sines 
In a right triangle, we can find the unknown sides or angles using the Pythagorean Theorem. But if the given triangle is not right triangle and is an oblique triangle then how do we go about? In such triangles we use the Law of Cosines and Sines to solve triangles. Sine and Cosine Laws are given as follows: Law of Sines: a/Sin(A)=b/Sin(B)=c/Sin(C)
Law of Cosines: a^2=b^2+c^2-2bc Cos A; b^2=c^2+a^2-2ca Cos B ; c^2=a^2+b^2-2ab Cos C

In a given oblique triangle, when do we use  Law of Sines and Law of Cosines
The Law of Sines are used when we know two sides and one opposite side or when we know two angles and one opposite side of an oblique triangle.  The Law of Cosines are used when we know two sides and the included angle or given the three sides of an oblique triangle.

 Law of Sines and Cosines
The Law of Sines help to establish a relationship between the side lengths and angles of a triangle ABC. There are three Sines and hence the relationship explains the plural ‘s’ of Law of Sines.
 The Law of Sines are, a/Sin[A] = b/Sin[B] =c/Sin[C] or we can even write them as Sin[A]/a=Sin[B]/b=Sin[C]/c ;  where a,b,c are the side lengths and A, B and C are the opposite angles of the respective sides a,b,c in the  oblique triangle ABC
The Law of Cosines is used most widely than the Law of Sines. When we know two sides of a triangle and their included angle, then Law of Cosines enables us to find the third side. The plural‘s’ in the law of Cosines is used as there are three cosines and hence by rotation similar formulas are valid for other angles. Law of Cosines are, a^2=b^2+c^2-2bc Cos A; b^2=c^2+a^2-2ca Cos B ; c^2=a^2+b^2-2ab Cos C.  The Law of Sines and Cosines, are also known as the Sine Rule and the Cosine Rule.

Solving  Law of Sines and Cosines Word Problems
Let us solve some Law of Sines and Cosines Problems with the given side lengths and angles
Example: Peter wants to measure the height of a tree. He walks 100ft from the base of the tree and looks up. The angle of elevation found is 33 degrees. This particular tree grows at an angle of 83 degrees with respect to the ground rather than vertically. Calculate the height of the tree.
Solution:
Angle B=83 degrees , Angle A= 33 degrees, c=100ft [two angles and included side]
Angle C= 180-[angle A+angleB] = 180-116= 64 degrees
Using Sine Rule,  a/SinA = c/SinC
              a = c . SinA/SinC = 100. Sin(33)/Sin(64) = 100. (0.606) = 60.6
So, the height of the tree calculated by Peter is 60.6 ft

Know more about the Math Homework Help,online Math help. This article gives basic information about Standard deviation. Next article will cover more statistics concept and its advantages,problems and many more. Please share your comments.

Negetive exponents

Exponents:

When a number or variable is multiplied with itself a number of times, it gives rise to the exponential value of that number or variable. That is to say, if we have a * a * a * a, we write that as a^4, where 4 is the exponent and a is the base. We use 4 here because a is multiplied to itself 4 times. In general if a is multiplied to itself n times like this: a*a*a*a*a….. n times, then that would be same as writing a^n. Where n = exponent and a = base. So, 2^3 = 2*2*2 = *8; or 3^2 = 3*3 = 9 etc.

Example:
A sample of bacteria doubles every hour. If the initial number is 100, how many bacteria would be there after 8 hours?
Solution: The above problem pertains to exponential growth. So after one hour, the number of bacteria would be = 100 * 2. After another one hour it would be = 100 * 2 * 2 = 100 * 2^2. After the third hour the number would be = 100 * 2^2 * 2 = 100 * 2^3 and so on. So after 8 hours the number would be = 100 * 2^8 = 100 * 256 = 25600.

Negative exponents:

Exponents can also be negative. A negative exponent would mean the positive exponent of the reciprocal of the base. Therefore a^(-n) = (1/a)^n (or 1/a^n ). Therefore, 5^(-2) = (1/5)^2 = 1^2/5^2 = 1/25, or 4^(-3) = (1/4)^3 = 1^3/4^3 = 1/64.

Multiplying exponents:

Instead of multiplying a*a*a*a…. n times, if we are asked to multiply (a^m) * (a^m) * (a^m) * (a^m) …… n times, then the answer would be (a^m)^n = (a*a*a*a…. m times)^n = (a*a*a*a…. m times) * (a*a*a*a…. m times) * (a*a*a*a…. m times) * (a*a*a*a…. m times) ….. n times = a*a*a*a….. (m*n) times. Therefore (a^m)^n = a^(mn). For example, (11^2)^3 = 11^(2*3) = 11^6, because 11^2 = 11*11, and (11^2)^3 = (11*11)^3 = (11*11) * (11*11) * (11*11) = 11^6

Extending the above concept further, if we have ((a^m)^n)^p = a^(mnp). For example, ((5^2)^2)^2 = 5^(2*2*2) = 5^8.

Example: A corporate deposit is such that $500 invested doubles every 5 years. What would be the amount due after 25 years?

Solution: Amount after the 1st 5 years = 500*2 = 1000
Amount after the 2nd 5 years = 500*2 * 2 = 2000
Therefore the amount after 5th years = 500 * 2^5 = 500 * 32 = 16000.

Prime Number

Let us define prime number. A Prime number is a whole number which is divided by number 1 and itself so this is the prime numbers definition. Prime numbers has following properties:-

• They are always positive.
• They are non-zeros numbers.
• They are always greater than 1.
• They always have two factors.

Therefore a prime number can be defined as the whole number greater than 1 which exactly has two divisors that is number 1 and the number itself.

Zero and one are not considered as prime numbers. 2 is the smallest and only even prime number. All other even numbers are divided by 2 so they are not prime numbers.

The prime numbers up to 100 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 .The list of prime numbers to 100 is given below:-

The table above shows the list of all prime numbers to 100.The numbers that are multiplied to get a product are called the factors of the product. Prime numbers are numbers which do not have any factors other than 1 and number itself. To find out if the number is a prime number, we start dividing it by 2, if it is divisible then it is not a prime number and if it is not then we continue dividing it by 3, 5, 7, and 11 and so on till we get any factors. If we get any factors then it is a non-prime number or a composite number.

Examples:-

• 2 have factors 1 and 2 so 2 is a prime number.
• 3 have factors 1 and 3 so 3 is a prime number.
• 4 have factors 1, 2 and 4 so it is not a prime number.
• 5 have factors 1 and 5 so it is a prime number.
• 6 have factors 1, 2, 3 and 6 so it is not a prime number.

Help in finding the slope of a line

Introduction to Finding the slope of a line
Define Slope :
Slope of a line is referred to as the values of the angle that a straight line makes with the positive direction of x-axis in the anticlockwise sense.

Generally , for any straight line equations of the form y = mx + c , m is referred to as slope.This could also help us on periodic function. It is the coefficients of the x term from the equation given.

Keep reading may be in the next session let me help you on algebra formula.

Note on angles of a triangle

Introduction - study about angles of a triangle:
Three-sided shapes that lie in single level surface are triangle. Triangles are a kind of polygons. The abstract of every single one the angles in any triangle is 180º. Triangles can be prepared according to the measurement of its angles. The angle is the diagram is framed by the sharing common end points. Let us study the detailed about the triangle and angle.

Angle - Study about Triangle Angles:
In general angle is the shape in the geometry. They are dissimilar's kinds of the angle is available in the mathematics.

We study detailed about different type of the angle is given below: This could also help us on formula for surface area

Right angle: Exact of the 90 degree is the right angle.
Acute angle: Minor than 90 degree is the acute angle
Obtuse angle: Upper than 90 degree and less than 180 degrees
Straight angle: The 180 degree of the angle dimension is the straight line
Reflex angle: The 360 degree of the angle dimension is the reflex angle
Complete angle: The angle dimension is the 90 degree

Keep reading may be in the next session let me help you on long division problems

Help with Statistics Variance

Introduction to Statistics Variance calculator:
Calculator is a device, in which the given inputs are processed and the result is manipulated and gives the output for the input given. In the statistics variance calculators, when the given data set is put in the required input field, the calculator keeps processing the given input and gives the required output. Here some of the examples for the statistics variance calculators and steps for calculating the variance manually.

Definition Statistics Variance Calculator:
Definition of variance:
Variance is the measures of the summations of the total squared mean deviation value which is divided by the total number of values subtracted by one. For measuring the statistics variances we have to manipulate the mean value initially.This could also help us on graphing equations
Mean is the average of the given total numbers, total of all the given data set values to that divide it by the total given values in the data set.

Keep reading may be in the next session let me help you with exponents and radicals