## Trigonometry Identities

There a number of “identities” in trigonometry that can be found from the basic ideas of sin, cos and tan as explained in my earlier post. These identities can help in solving equations involving trig functions, especially when there are 2 or more different functions as the often allow you to write the equation in terms of one function, eg sin, that you can then solve. One of the identities is: **sin ^{2} + cos^{2} = 1. ** To prove this consider a right angled triangle with side a,b and c as shown below

From this we can use Pythagoras theorem to say: a^{2}+b^{2}=c^{2 now we know sin t = b/c so b = csin t cos t = a/c so a = ccos t} substituting these values in the above equation we get (csint)^{2} +(ccost)^{2} = c^{2} canceling the c^{2} we get **sint ^{2} + cost^{2} = 1**

There are trig functions that are equal to 1 over sin, cos and tan called cosec = 1/sin, sec = 1/cos and cot = 1/tan. These can be remembered using the third letter rule as the third letter of each of these corresponds to the the function it is one over.

Using these a cos

^{2}+ sin

^{2}= 1 we can calculate other identities

**tan**We can obtain this by dividing through by cos

^{2}t + 1 = sec^{2}t^{2}as we know sin/cos = tan, cos/cos = 1 and 1/cos = sec. Other similar identities can be obtained for cosec and cot.

## Proof of Pythagorases Theroem

Please visit my main site at www.breakingwave.co.nr

Pythagoras theorem, that the square of the longest side of a right angled triangle is equal to the sum of the squares of the other 2 sides.

This can be proved quite easily by drawing a square into which fit 4 of the same right angled triangle as shown below

As you can see the area of the whole square is equal to the the sum of the 2 shorter sides squared or (a+b)^{2}. The area of the green square left is the square of the longest side c^{2}. We also know that the area of each of the triangles is 1/2 x base x height = ab/2

From these 3 areas we can prove the theorem. The know that the total area of the square is equal to the area of the green square plus 4 of the triangles ie)

**(a+b) ^{2}=c^{2}+ 4ab/2**

a^{2} + b^{2} + 2ab = c^{2}+ 2ab

The 2ab ‘s cancel and we are left with Pythagoras theorem

**a ^{2} + b^{2} = c^{2}**

View the rest of my posts on this blog or at my website *www.breakingwave.co.nr* by David Woodford.

## Proof of Cosine Rule

Below is the proof by Pythagoras’s theorem of the cosine rule, a^{2}=b^{2}+c^{2}– 2bccosA.

This assumes you understand Pythagoras’s theorem (visit pythagoras’s theorm to view my lesson on it), how to use basic trigonometry(basic trigonometry lesson). If you want to learn how to use the cosine and sine rule, opposed to just learning the proof) visit by sine and cosine rule page.

The proof is done using the letters of the following triangle

and we are trying to prove the cosine rule:

**a ^{2}=b^{2}+c^{2}– 2bccosA**

**In triangle CBL**

a^{2} = (c-x)^{2} + h^{2}

a^{2} = c^{2} – 2cx + x^{2} + h^{2}

h^{2} = a^{2} -c ^{2}– x^{2} + 2cx *<<EQN1*

**in triangle CLA**

b^{2} = h^{2} + x^{2}

h^{2} = b^{2} – x^{2 }* <<EQN2*

*eqn1 – eqn2 ::* 0 = a^{2} – c^{2} – b^{2} +2cx

a^{2} = c ^{2}+ b^{2} – 2cx *<<EQN3*

** in CLA**

cosA = x/b

x = bcosA

**in eqn3**

**a ^{2} = c^{2} + b^{2} – 2bccosA**

So there is the proof for the cosine rule using pythagorases therom. If you found that usefull try looking at my other maths lessons

## Pythagorases Theorm

Hello, this is my TREVOR Pythagoras blog about the GAME, however people keep looking at it because of the mathematician Pythagoras so ive decided to do a post on his triangle theorem

## The Theorem

“The square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides in a right angled triangle”

or mathematically

c^{2} = b^{2} + a^{2}

This is very useful for allot of applications.

## Applications of the theorem

There are many applications of Pythagoras besides simple triangles

eg)The difference between colors. Yes using pythag we can measure the “difference” between to colors as a number. This is how

- take you first and second color as a rgb number eg) red = 256 ,0,0 and a light shade of blue is 30, 100, 256.
- square the difference between the values for red, green and blue
- then find the square root of the sum of these values

Why does this work, because the hypotenuse of one triangle can be used as the base of another, and this triangle can be tilted by 90 degrees into the z plane the theorem can work in 3D :). We can then change the x,y and z axis to values for r,g,b. As pythag works in 3D we can calculate the distance between to colors(which are points in the rgb axis)

## So there you go – Pythagoras can be used for measuring colors ðŸ™‚

## IM BACK

helo, i havnt posted many blogs recentyl because ive too busy actually creating trevor and my anti virus has run out. It seems my trevor download link isnt working so theres not much pointme uploading any new updates – however they do exist!!!!!!!!!!!!!!!!!!!!!!.

1)I have finally created a level 1 which you can play and includes all the objects that i have created so far ðŸ™‚

2)Ive made a sheild for trevor so that he can get 10seconds protection from evil tihngs

On my site I have uploaded alot of new pages in the computing section (dwebs) these include a java tutorial and a download page for the games:) theres also a download for my html editor which does work thankfully.

when i get my anitviruses sorted out ill carry on my diablo and juggling posts. Cya soon

## Merry Christmas

Merry christmas evreyone. Dont expect many posts over christmas because i wont be going online. However I hope to finish the basic trevor system by the end of the holidays and to create a fully working test level. ðŸ™‚

**Plans for trevor**

On the system Ive still got to implement animations on the characters, mainly Trevor, so that the game runs more smoothly (so Im looking for some decent pictures). Ive also got to get the screen scrolling working properly so that you can create large levels without everything being crushed up.

Im also creating a level reader so that complex levels can be designed in another program(notepad at the moment) and then saved in a text file. The level reader will then read these into the game so that they can be played. This is much easier than coding out all the levels using co-ordinates and also means a single complied version of the game can run any level you want. However the most complex levels using many custom objects will use a mixture of the level reader with a specially complied class for the level (this will also keep out the cheats!!!!!!!!!!)

Im also planning on converting it to an applet so that i can embed it in a web page(though the download times will be LONG). This means you should be able to play it online asap :).

**Juggling**

I am going to be uploading some more juggling videos to youtube soon and on to my site. I will also be adding a juggling section to this blog where each week i will post a new trick with an explination and video ðŸ™‚ so keep reading.

I am also looking to learn some new tricks over Christmas so expect a wider range of pages on my site (www.dwebs.bravehost.com)

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