ANGLES, DEGREE AND RADIAN

The earth circles the sun in 365 days. So the ancient people devided the circle into 360 degree. And 360 is also the base of 60. Hence they felt comfortable. We use this concept even today. But there is more convenient unit called Radian. Imagine the circle of unit radius.
Here
circumference = c= 2*pi*r=2*pi.*1 =2*pi
c= 2*3.14 = 6.28
Hence, we can say, the total angle of the circle 360 degree(full circle) is proportional to 2pi. or 180 degree(half circle) is proportional to pi. 90 degree(quarter circle) is proportional to pi/2.

Hence we can measure the angle in terms of pi. That is, we can measure the angle in terms of arc length in unit circle.This unit is called radian.

360 degree in radian is 2pi=2*3.14 =6.28....
hence 1 degree in radian is 0.0175 radian.

There are some advantages in this unit.

1.Suppose you want to know the length of arc held by the angle T radian in the circle of radius R, you simply find R*T, you get the arc.
Logic: T is the arc in unit circle for the angle T radian. If you find R*T, the arc is scaled to the circle of radius R.

2. Suppose you want to find the speed of a car. The car's wheel makes 10.5 rotations per second. Or it makes 10*2pi radians (angle) in one second. Multiply 20*pi by the radius of the wheel, you get the length traveled by the wheel in one second or the velocity of the car.

3. If the angle X measured in radian is very small then,
sin(x) =x
Radian makes the calculation easy.

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LISSAJOUS FIGURES

Definition: "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures". We know sine wave motion and circular motion is basically same. Hence we draw two circles A and B perpendicular to each other. The circle B rotates twice faster than circle A. That is, frequency of circle B is two times than that of A. A particle at the intersection of two circles is subjected to two sine wave motion A and B at 90 degree simultaneously. The particle will describe figures depending on the frequency and phase of A and B . In our case, the ratio of frequency is 1:2 and the two waves are in phase. To draw lissajous figures : A moving point in both the circles are chosen. Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two. Hence the points are marked on the circles according to their speed. Then straight lines

THE PARABOLA

A jet of water shooting from a hose pipe will follow a parabolic path. What is the so special about parabola. Y= x^2 Draw a graph for the above equation. It will result in a parabola. This parabola is also called unit parabola. Any equation involving square will yield a parabola. Example: Y = 2x^2 +3x+3 (also called quadratic equation) X= 2 and -2, both satisfies the equation 4 = X^2. Parabolic equations always have two solutions. Any motion taking place freely under gravity follows parabolic path. Examples: An object dropped from a moving train, A bomb dropped from flying plane, A ball kicked upwards. If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point. This fact is made use of in Dish Antenna, Telescope mirrors, etc. Inverted parabola shape is used in the construction of buildings and bridges. Because the shape is able to bear more weight. A plane

CASINO'S GAME

Let us find out how the casino survives with mathematics. Say, your friend invite you for a game of dice. You must bet (wager) 2 dollars. If you roll 'six' you will get back 8 dollars. The game will go on for 30 rounds. All sounds good. The probability of rolling 'six' is 1/6. Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5. That is, you are expected to win 5 rounds out of 30. Hence your gain will be 5 * 8 =40 dollars. ok. This also implies that you will loose 25 rounds. Hence your loss will be 25*2 =50 dollars. Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars. There will be a loss of -1/3 or 0.33 dollars per round. It is not a fair game. Let us make a simple formula to calculate 'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect

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