how to generate sine, cosine and tangent curves; interactive exercise  
Definition  Draw a circle with cenre (0,0) and radius 1. P is a point on the circle which makes an angle x with the positive xaxis. Idenitify the lengths sinx, cosx, tanx, cotx.  
Trig graphs 
the 4th applet 

the 3rd applet  
y=Asinx, y=sinx+d, y=sin(xc), y=sinbx  
Revision  
the 3rd applet "recognise functions 3"  
the 4th applet "recognise graphs 3"  
Application  
Polynomials




Useful
websites




A brainteaser about the sum of an A.P.
Part I
In a kingdom far, far away, there once lived a King and 10 wealthy jewelers, very wellknown in their trade. Every year the 10 jewelers paid tax to the King, each gave the King a stack of 10 gold coins. The real coins weighed exactly 1 oz each. This year the King received report that one and only one stack contained 10"light" coins, each having exactly 0.1 oz of gold shaved off the edge. The King now ordered his personal adviser, Mr. Fischer, to IDENTIFY the crooked jeweller and the stack of light coins with JUST ONE SINGLE weighing on a scale. 
It took Mr. Fischer, intelligent as he was, the whole afternoon to think of a perfect solution. Even then, he wouldn't reveal much to his aides. All he asked them to do was label the stacks of the coins 1, 2, 3, ..., 10. Take one coin from stack 1, two coins from stack 2, three coins from stack 3, four coins from stack 4, and so on up to stack 10. "Weigh the coins you just collected from the stacks," he instructed his aides. "54.3 oz was the reading, sir!" The aides said after the ONE and ONLY ONE weighing allowed by the King. "Very well, the crooked jeweler was the who gave stack number ...!" Mr. Fischer whispered to himself. "What did you say, sir?" The aides were eager to know. "See how many coins you just weighed and you will know which stack was faulty!" Mr. Fischer wanted to test his aides.

Now
which stack was the one from the crooked jeweller?

Part II

The King was so impressed with Mr. Fischer's many achievements (finding the crooked jeweller was just one) that he told Mr. Fischer,"Ask me for anything you want. Whatever you ask I will give you, up to half my kingdom!" 
Mr. Fischer immediately took the King to the royal court whose marble floor was exactly an 8x8 chessboard. "Your majesty, if you so wish, please ask one of your servants to put one grain on the first square, two grains on the second square, four grains on the third square, eight grains on the fourth square and so on, doubling the amount of the grains when it comes to the next square until the last square of the chessboard is filled, then let your humble servant have all the grains placed on the chessboard!"

The King was again very pleased with Mr. Fischer. "How modest this Mr. Fischer is! How considerate! He could have asked a lot more!" The King was about to grant Mr. Fischer what he asked for. Nonetheless, reason got the better of him. As a mere formality, the King summoned his bookkeeper, Mr. Anderson, to calculate how much all this would cost him. When Mr. Anderson came back and showed the King his calculations, the King simply couldn't believe his eyes. Why? 
Can you calculate how many grains there are on the chessboard? (At that time, annual grain production of the whole world is roughly 10 ^{16}grains.)

What should the King do with this Mr. Fischer? Should he honour his word and let Mr. Fischer be his biggest creditor or should he take back his word and ...? 
Part III

Using similar calculations, which of the following is the best deal for Tom's pocket money for the next 2 weeks? 
(a) $ 7 every day 
(b) $ 1 for the first day, $ 2 for the second day, $ 3 for the third day, ... 
(c) 1 c for the first day, 2 c for the second day, 4 c for the third day, 8 c for the fourth day, ... 
Afterthought 
(1) How different are (a)(b)(c) if the period of 2 weeks is extended to a month? 
(2)
How will (a)(b)(c) look in a graph of total amount of pocket money vs
time? Each of them represent a standard type of growth, can you name all
of them?

LINEAR PROGRAMMING
three interesting problems; Birthday problem, Monty Hall problem and one involving infinite series
The Birthday problem Surprise  Surprise ! A challenge to your intutive sense but quite simple in a mathematical sense! (A) Let's start with an easy one first. At least how many people must be gathered so that we can be 100% certain that some of them share the same birthday? (B) Now comes the serious part ... How many people should there be in a class so that there is at least a 50% chance that some of them share the same birthday? Should it be half of what we have in (A) above? How about one third? say 180 people? 120 people? Think about your own class ( around 40 in size ), do you know any 2 people having the same birthday? Ask another class, is it common to have 2 or more people having the same birthday? Discussion (2) How to approach the problem mathematically? (Assume there are only 365 different birthdays.)
(3) A full explanation Does the answer agree with your intuition? 

A probability problem involving an infinite series 2 monkeys throwing coconuts at each other ! 

The
famous Monty Hall Problem
*
Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors? If you were the contestant, which of the following would have a better chance to win the big prize? Strategy 1 (stick): Stick with the original door Strategy 2 (switch): Switch to the other door or it doesn't matter since the two strategies have equal chance of winning the big prize *This problem was named Monty Hall in honor of the long time host of the American television game show "Let's Make a Deal." During the show, contestants are shown three closed doors. One of the doors has a big prize behind it, and the other two have junk behind them. The contestants are asked to pick a door, which remains closed to them. Then the game show host, Monty, opens one of the other two doors and reveals the contents to the contestant. Monty always chooses a door with a gag gift behind it. The contestants are then given the option to stick with their original choice or to switch to the other unopened door. Discussion



DATA


Normal Distribution (Gaussian distribution)


