Proofs

First, note that sin(a+b)=sin(a)cos(b)+sin(b)cos(a)

[For a proof, see: www.mathsroom.co.uk/downloads/Compound_Angle_Proof.ppt

For the case of b=a, we have:

sin (a+a)=sin(a)cos(a)+sin(a)cos(a)

sin (2a)=2*sin(a)cos(a)

A computer can only distinguish between two states: on and off. So it can only use two numbers.
Because the first computer consisted of rows of switches which were either on - or off. The binary number system uses a 1 and 0 in place of the on and off functions.
Hello,

The computer uses the binary system because the information in a computer is actually electrical pulses (on/off) And because of that we use 0 (off) and 1 (on) to make it easier for us to calculate it :)

Actually, we don't really have a choice in using the binary system. Computers are electrical machines, so we are pretty much forced to use it as the lowest level of communication. Then past that there's hexadecimal, then plain text.
the computer uses binary number system because the computer can understanding the binary language that is (0,1),and data which are entered by the input device the computer can convert each word into binary number that is (0,1)and then it process the data and again convert into machine language(binary language) to high level language and then we can saw and understand the output result.
Binary is used in all computers because at the hardware level everything is simply on or off, 1 or 0. With only two possible symbols to physically represent, binary systems are extremely simple to implement, whether through mechanical means or through electro-chemical means: a switch is either on or off; a capacitor either holds enough electrical charge or it does not; polarised material is either positively charged or it is negatively charged; a punch card either has a hole at a specific location or it does not. Any medium that can easily differentiate between two possible states can therefore be used to store and retrieve digital data. The simpler the data representation is to implement, the faster the machine will operate. And there simply is no numbering system any simpler than binary.
You can represent 0 and 1 with a simple change of state--on or off. And they can be used to convey the most complex data.

Here's an example. Imagine you have a little box with 4 buttons on it. These are the buttons:

1

2

4

8

If you press a button, it equals that number. If you don't press the button, it equals 0. If you press more than one button, they're added together.

Now look what happens if you press the buttons in different combinations:

no presses = 0

press 1 only, total = 1

press 2 only, total = 2

press 1 + 2, total = 3

press 4 only, total = 4

press 1 + 4, total = 5

press 2 + 4, total = 6

press 1 + 2 + 4, total = 7

press 8, total = 8

press 1 + 8, total = 9

And that is exactly how one "byte" (which contains 8 "bits," each of which can just be on or off) expresses the digits from 0 to 9. And so you can make any number, any number at all, because all numbers consist of the digits from 0 to 9. And you can add, subtract, multiply, divide, and more by turning those buttons or switches on and off.

It gets a little more complicated with letters and special characters (all the symbols you can type that aren't letters and numbers), but the principle is the same.

Along with the data, though, you have to make rules for interpreting it. For instance, you have to make a rule that says the first bit is worth 1, the second 2, and so on. And there are very elaborate rules for how the computer receives, translates, and returns all kinds of data. Those rules are definitions and programs. It even has to have rules for making definitions and programs, and those rules add up to what we call a "language."

You see what we can do with computers. If you're old enough to have watched the possibilities grow over the decades, you are probably more amazed than if you have just grown up with it. And we are really only just beginning. What we have now is still very primitive.

Those 0's and 1's (which we call "binary" and which we can use to mean on/off or yes/no) are incredibly powerful when you combine them with logical reasoning.

There is no largest number. One simple way to consider this is to look at the function:

f(x) = x + 1

Pick any number, and plug it into that function. No matter what number you plug in, the function will always give you a number that is one larger than the previous one. It doesn't matter how big "x" is, there will always be an "x + 1" value, which will always be larger than x. This is a concept we refer to as infinity, which is usually represented with the symbol ∞. It is important to note that infinity is not in itself a number, but the concept of a set of numbers that has no limit.

There is however a large finite named number. That would be a googolplexian, which is a one followed by a googolplex of zeroes. A googolplex itself is a one followed by a googol of zeroes, and a googol is a one followed by a hundred zeroes. In other words:

1 googol = 1 × 10100

1 googolplex = 1 × 10googol or 1 x 10^(10^100)

1 googolplexian = 1 × 10googolplex or 1 x 10^(10^(10^100))

There are also large numbers that can only be expressed as complex exponentials, such as Graham's Number. Once again though, none of these are the largest number that exists, as there is no such thing. You need merely to add another value on top to reach a larger number.

That's only true if the "legs" are indeed legs, i.e. the triangle is a right triangle, and the legs

include a 90-degree angle.

This cannot be proven, because it is not generally true. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. And conversely, the diagonals of any parallelogram bisect each other. However not every parallelogram is a rhombus.

However, if the diagonals are perpendicular bisectors, then we have a rhombus.

Consider quadrilateral ABCD, with diagonals intersecting at X, where

AC and BD are perpendicular;

AX=XC;

BX=XD.

Then angles AXB, BXC, CXD, DXA are all right angles and are congruent.

By the ASA theorem, triangles AXB, BXC, CXD and DXA are all congruent.

This means that AB=BC=CD=DA.

Since the sides of the quadrilateral ABCD are congruent, it is a rhombus.