Mathematical Proof of Cosine Identity (cosine) Theorem
Nov/112
Last summer I was randomly surfing on internet at home and saw cosine theory on wiki and wanted to proof the identities by myself. However, despite having a good trigonimetry and geometry background, I couldn’t remember the proof. Honestly I got a bit sad as I realized that it is being more than ten years since high school. I searched for its proof and I found a nice geometric proof. This time I got angry as I wasn’t able to find this geometric proof. It was time to get my hands dirty.
Here is a perpendicular triangle.
Mathematical proof of cosinus idendity theorem – step 1 putting the known parts together
Step-1
Let’s say A is divided by a line segment
|AD| = t,
then |AB| = t*cos(A), BD = t*sin(A)
Now let’s draw a hight from D to |AC| named H so we have similarity between the big triangle and the small one; (ABC) ~ (DHC)
|HD| = t * sin(B),
|DC| = t * sin(B) / cos(A+B),
|HC| = t * sin(B) * sin(A+B) / cos(A+B)
|HA| = t * cos(B)
Mathematical proof of cosinus idendity theorem - step 2
Step-2
Looking at the big triangle (ABC), its hypotenus is equal to |AH| + |HC| which also is also equal to |AB| / cos(A+B)
Mathematical proof of cosinus idendity theorem - step 3
Step-3
Multiply both sides with cos(A+B), get rid of t and organize the equation
Mathematical proof of cosinus idendity theorem - step 4
Mathematical proof of cosinus idendity theorem - step 5
Mathematical proof of cosinus idendity theorem - step 6
Mathematical proof of cosinus idendity theorem - step 7
Mathematical proof of cosinus idendity cosinus theorem - step 8
step-4, 5, 6, 7, 8
Organizing the equation
Mathematical proof of cosinus idendity cosinus theorem - step 9
step-9
Here we come the the most interesting part, which we have a parabolic equation which means we are close to the solution
Mathematical proof of cosinus idendity cosinus theorem - step 10
Mathematical proof of cosinus idendity cosinus theorem - step 11
Steps 10,11
are the discriminant solution for the parabolic equation
Mathematical proof of cosinus idendity cosinus theorem - step 12
Step 12 is the simplification of inside the root
Mathematical proof of cosinus idendity cosinus theorem - step 13
And finaly step 13,
As Lois warned, as sin(A)*sin(B) comes out of the root it has one minus root
cos(A+B) = cos(A)*cos(B) – sin(A)*sin(B)
Hacking Captcha with non-artificial intelligence
Nov/110
Nowadays I am looking through twitter api (twitter4j) and trying to learn twitter services as I want to find out how many times status containing the words ’steve jobs died’ or ‘want buy mazda 3′ tweeted. As you just noticed using tweets as a source of information, it is possible to find out which car is popular nowadays… examples can be extended to far different points …
however as the company i am working is restricting access to twitter, I couldn’t use twitter api. But hold on! I can also do that by using google searches. if you type site:twitter.com inurl:”/status” steve jobs died in the google search box you will get lots of tweets apprising death of steve jobs to people. What’s more is if you have a programming background you can get search result in your program to analyse. By doing that, for instance, you may find out peoples’ impression; if they are happy or unhappy… I know it was a silly example but you can deal with it 
At this point another problem emerges. After a few sequence of requests, google gets suspicious if you are a robot and asks you a captch question. for those who doesn’t know captcha may just skip reading this post 
or check wiki.
Ok now we know that captcha is a common problem for robots trying to hack web sites or collecting data but how to deal with it? up to now many methods has emerged mostly using artificial intelligence (specifically anns) but I have another idea (well I have no idea if people has already tought or are presently using), solving this problem not with ai but with non-artificial intelligences called human beings (well I am not much religious as u just spot) . why don’t we ask captchas to other people and give rewards to those answering correctly? Assume we have an online game and if a user answers a captch correctly, give him/her additional chips/resources or whatever.
Here is the diagram of the entities represented with capical letter prefixes and actions represented with number prefixes.
hacking captch
At the first step our robot sends a search request to google, google either sends the result or asks for a captcha. if it is a captcha our robot sends the request to our web site and asks to real users. if any of the users answer, our site sends the request to the robot back and redirects it to the google with the answer. if the answer is true, system gives a present to the user.
that’s all for now. if you have any questions or ideas please don’t hesitate to give me a feedback.
I hope you didn’t find it stupid and enjoyed