Statistics, Basic Concept and Key Terms

The next series of days will focus on Statistics.

Statistics is the science and art of collecting, analyzing, and making conclusions about data. All data is collected from a set population. Subsets of the population can be called members, or units. If every subset of the population is collected, then we have a census, however, often just a small portion of the population is sampled. The sample is then thought to represent the whole population.

Care must be taken when choosing any given sample. For example, if we want to get an idea of how a population will vote on a new cigarette tax it does no good to sample non-voters like children. Also, the sample must be random in order to assure accurate results. If we only ask voters leaving a cigarette store, we are not likely to get good results and arrive to correct conclusions.

Proof of Herons Formula to find the area of a non-right triangle

Proof of Herons Formula to find the area of a non-right triangle

Consider the following triangle:


There are no right angles in the triangle, so therefore we must use Heron's forumla which first defines a variable s

s=1/2(a+b+c) where a,b,c each represent the length of a side.
then the area of the triangle is equal to
sqrt(s(s-a)(s-b)(s-c))

For the example we see that s=21
and the area is equal to
sqrt(21(21-10)(21-15)(21-17))
=
sqrt(21*11*6*4)=sqrt(5,544)=~74.46 which is the area of the triangle.

How do we prove Herons formula? Consider the triangle:


Here we have taken a triangle with no right angles, and cut a line so that we create two right triangles, and two new lengths at the base: x, and C-x. Now we need to solve for two unknown values x and h. We will solve for x first.

We can use pythagorean theorem to create equations for our new values, we see we have:

x² + h² = B²
and
(c-x)² + h² = A²
=
C² - 2cx + x²+ h² = A²

We know that B= x²+h² so we substitute that in to the equation above and get

C² - 2cx + B² = A²

Now we can solve for x!

x= (A² - B² - C²) / - 2c

Now we have x we can plug it back in to the equation

x² + h² = B²

thus

h = sqrt(B² - (A² - B² - C²) / - 2c)²)

Now, we know our base is equal to C and our height is equal to h, if we substitute these two values into the formula for the area of a triangle we get
(1/2)*b*h = 1/2*C²*sqrt(B² - (A² - B² - C²) / - 2c)²)

That is a complicated formula, so Heron found that you could define a term
s = (1/2)*( A + B + C) and then found the area formula could become a simpler

sqrt(s(s-a)(s-b)(s-c))

If you plug s in to the area formula above you will find the result we saw above:
1/2*C²*sqrt(B² - (A² - B² - C²) / - 2c)²)

which is the area of the triangle.

Prove the area of a right triangle is equal to half the base*height

Prove that the area of a triangle is equal to 1/2*base*height.

The essence of this proof is to show that any right triangle can be made into a rectangle of the same base and height.

Consider the following two triangles:


Both these triangles can be made into rectangles with the same base and height as shown below:


The area of a rectangle is length times width, or in this case base times height.
The hypotenuse of the triangle cuts the rectangle directly in half, and thus the area of a triangle is 1/2 of a rectangle, or 1/2 base times height.

The Law of Cosines

When we know the length of two sides of a triangle, and the angle between them, we can use the law of cosines to find the length of the remaining side.

The law of cosines states:
C^2 = A^2 + B^2 - 2AB*cos(theta)

Here is an example of the law of cosines:


Proof of the Law of Cosines
The law of cosines looks similar to the pythagorean theorem (C^2 = A^2 + B^2) and indeed the two are similar. What we have to do to prove the law of cosines is to create a right triangle and define coordinates for that right triangle so we can find our remaining side.

Consider the diagram below:


Imagine the triangle on a coordinate plane. We define the origin, the point (0,0) at the end of side B. Thus the coordinate to the left is (-B,0). The coordinate at the top of the triangle (A*cos(theta)-B,A*sin(theta)) is derived from the right triangle sketch in on the right of our triangle. Although the coordinates appear complex, keep in mind they represent two numbers:
(A*cos(theta)-B,A*sin(theta)) is equal to some (x,y) on the coordinate plane.

Now that we have defined some coordinates we can draw a line down from the top angle to some point on side B, this creates a right triangle as shown below.


Using the coordinates which we defined we can define the length of the sides of this triangle.
The bottom side has length: |A*cos(theta)-B|.
A*cos(theta) comes from the dotted triangle we sketched in the previous image and represents the length that we have chopped off of side B. This the new length of the triangle is A*cos(theta)-B , it could also be B-A*cos(theta). Because we don't know which way to subtract, we take the absolute value so that both equations give us the same length, and write the distance as |A*cos(theta)-B|. The length of the vertical side also comes from the previous dotted triangle, and is simply A*sin(theta).

With these lengths now defined we can find C with the pythagorean theorem.
C²= (A*sin(theta)² + (A*cos(theta) - B)²
Mutliply this out and you get
C²=A²*sin²(theta) +A²cos²(theta) - 2AB*cos(theta) + B²
Factoring out the A² we get
C²=A²(sin²(theta)+cos²(theta))-2AB*cos(theta)+B²

Knowing that sin²(theta)+cos²(theta)=1 we get

C²=A² - 2AB*cos(theta) + B²

whic is the Law of Cosines.
C² = A² + B² - 2AB*cos(theta)

The Law of Sines

When we know the measure of the angles of a triangle, and the measure of one of its sides, we can use the law of sines to find the length of the other two sides.

The law sines states for any triangle with sides A,B, or C that
sin(a)/A = sin(b)/B = sin(c)/C
or equivalently
A/sin(a) = B/sin(b) = C/sin(c)

So if we have a triangle as such, with side x unknown we can find it by using the law of sines:


Proof of the Law of Sines
Consider the following diagram:


In the first step of the proof we divide the triangle into two right triangles by drawing a line of length h1.

Now we have two right triangles we can say that
sin(b)=h/A or h=A*sin(b)
similarly
sin(a)=h/B so h=B*sin(a)

now we can see that
B*sin(a)=h=A*sin(b)

divide both sides by AB and we get
sin(a)/A = sin(b)/B which is the first half of the law of sines.

For the next part, we draw a right triangle out from side B, creating a new length h2.

From this new triangle we see that
sin(b)=h/C and h=sin(b)*C
we also see that
sin(180-c) = h/B or h=sin(180-c)*B

but from the unit circle below we see that sin(180-c)=sin(c)


so we can write sin(c)*B=h and from this conclude
sin(b)*C=h=sin(c)*B
again we divide both sides by BC and get
sin(b)/B = sin(c)/C

so we have sin(a)/A=sin(b)/B=sin(c)/C which is the law of sines.

An Example of Vector Physics

Assume you try to pull a box across the room. You exert 150 pounds of pressure (which you could measure with a spring) on the box, how much force will be used to drag the box and how much to lift it? This problem can be solved with trigonometry.



Thus from the sketch we see we can calculate horizontal force as
150 * cos(40) = 115 lbs of force
We get this equation knowing that the cosine of an angle is adjacent over hypotenuse, thus
adjacent = hypotenuse*cosine(angle)
or
115 = 150 * cos(40)

By the same method we can find the opposite side which will give us the force of lift.
150 * sin(40) = 96 pounds

Whether or not this is enough force to move the box depends on the friction and weight of the box, as well as several other variables.

Vectors

Vectors in math are lines which represent both distance, and direction. They can also be used to represent magnitude or force.

A Vector: