TRIGNOMETRY IN A NUT-SHELL

Draw a right angled triangle of angle 30 degree as shown in the figure. Let the length of the hypotenuse be 1 unit.
We know sine 30 = opposite/hypotenuse = opposite side/1 unit
Hence sin30 is the length of the opposite side which is 0.5 unit. Hence sin30=0.5
similarly
cosine30=cos30=adjacent /hypotenuse =.87 unit
This ratios sin30 and cos30 holds good for any smaller or larger triangles. We can find any side of the triangle using these ratios.

We can find the sine and cos values for angles from 0 to 90 degree in this way. Also they are available in ready table form. This 'sin and cos' concept is also extended to the angles 91 to 360 degree, Here, we have to incorporate positive and negative signs suitably (refer).

Now,
tan x = sin x/cos x = opp/hyp/ adj/hyp = opp/adj
When 'sin' is divided by 'cos' , the ratio of opposite side/ adjacent side is obtained. We call it 'tan'. (tangent)

Say, you are standing 40 m from a tall building and you are viewing the top of the building. Say, the viewing angle is 45 degree from the horizontal.
Here the line of sight is the 'hypotenuse', building is the 'opposite side' and the distance between yourself and the building is adjacent. The tan45 is 1. That means, adjacent and opposite must be equal. The building height must be 40 m. This is just one example for the use of trigonometry.

Sun, Earth and moon sometimes form a right angled triangle. Atoms may lie in a triangle. Trigonometry can be applied in both the cases. That is real beauty.

From sin and cos, hundreds of trignometric functions and formulas can be derived.
for example;
sin^2x + cos^2 x = 1
in our case (.5)^2 + (.87)^2
for 30 degree .25 +.75 =1 verified.
The above equation is the trigonometric form of Pythagoras theorem.

Light ray practically travels in straight line. Hence, in geometrical optics, trigonometry is widely used.
The value of sine function for 0 to 360 degree oscillates between 0 -1- 0- (-1). Hence a wave form is created. Sine wave is applied everywhere in physics.
The trigonometry connects, distance, angle, circle, waves using compact functions like sin x.

Comments

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

sciencecapsules

Search This Blog