Trigonometry

GREEN House X-c

Submitted to: Math Department

Presented By: Bijaya Bharati

ONE CANNOT SUCCEED ALONE NO MATTER HOW GREAT ONE’S ABILITIES ARE, WITHOUT THE COOPERATION OF OTHERS. THIS PROJECT, TOO, IS A RESULT OF EFFORTS OF MANY. I WOULD LIKE TO THANK ALL THOSE WHO HELPED ME IN MAKING THIS PROJECT A SUCCESS.I WOULD LIKE TO EXPRESS MY DEEP SENSE OF GRATITUDE TO MY MATHS TEACHER, MR. JANAK SINGH SAUD WHO WAS TAKING KEEN INTEREST IN OUR LAB ACTIVITIES AND DISCUSSED VARIOUS METHODS WHICH COULD BE EMPLOYED TOWARDS THIS EFFECT, AND I REALLY APPRECIATE AND ACKNOWLEDGE HER PAIN TAKING EFFORTS IN THIS ENDEAVOUR.

SUPERVISED BY – MR. JANAK SINGH SAUDCREATED BY – BIJAYA BHARATI

WHAT IS TRIGONOMETRY ?

Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.

Trigonometry is most simply associated with planar right-angle triangles (each of which is a two-dimensional triangle with one angle equal to 90 degrees). The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, in elliptic geometry (a fundamental part of astronomy and navigation). Trigonometry on surfaces of negative curvature is part of hyperbolic geometry.

Cont…………..

Ancient History• Sumerian astronomers studied angle measure,

using a division of circles into 360 degrees.

• The ancient Nubians used a similar method. In the 3rd century BC, Hellenistic mathematicians such as Euclid (from Alexandria, Egypt) and Archimedes (from Syracuse, Sicily) studied the properties of chords and inscribed angles in circles, and they proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically.

• In 140 BC, Hipparchus (from Iznik, Turkey) gave the first tables of chords, analogous to modern tables of sine values, and used them to solve problems in trigonometry and spherical trigonometry.

• In the 2nd century AD, the Greco-Egyptian astronomer Ptolemy (from Alexandria, Egypt) printed detailed trigonometric tables (Ptolemy's table of chords) in Book 1, chapter 11 of his Almagest.

• Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years.

Modern History• By the 10th century, Islamic mathematicians were using all six

trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry.

• Knowledge of trigonometric functions and methods reached Western Europe via Latin translations of Ptolemy's Greek Almagest as well as the works of Persian and Arabic astronomers.

• One of the earliest works on trigonometry by a northern European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus.

• Driven by the demands of navigation and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics.

• Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.

• Gemma Frisius described for the first time the method of triangulation still used today in surveying.

• It was Leonhard Euler who fully incorporated complex numbers into trigonometry.

It is used in navigation to

find the distance of the shore

from a point in the sea.

It is used in oceanography in calculating the

height of tides in oceans

Uses Of

Trigonometry

It is used in finding the

distance between

celestial bodies

The sine and cosine functions are

fundamental to the theory of periodic

functions such as those that describe sound

and light waves.

Architects use trigonometry to

calculate structural load, roof slopes,

ground surfaces and many other aspects,

including sun shading and light angles

ANGLE

If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the above figure

quick fact (in a right angled triangle trigonometric ratios of angles are defined as)

sin= cos= tan=

cosec= sec= cot=Reciprocal of sin Reciprocal of cos Reciprocal of tan

Hypotenuse=h (greatest side)Perpendicular=p (side opposite to reference angle)Base=b (remaining side)

sin 90⁰=1

Basics ofTheoretical proofSin = opposite over hypotenuse.

If your angle is 90 degrees then the side opposite your angle is the hypotenuse so

sin= and anything divided by itself is 1

OtherRatios

CanBe

Found In

SimilarWay

ByFirst

Calcula-tingCosAnd

taking ratios

OfSin

And Cos

sin 0⁰=0

Basics of Theoretical proofAssume angle BCA to be x.sin x=now assume the value of x to decrease towards zero keeping BC constant. As x decreases AB would decrease. When x is very less say 0.001 AB would be very short in length . So as x tends to 0 x also tends to 0.So sin 0=0.

OtherRatios

CanBe

Found In

SimilarWay

ByFirst

Calcula-tingCosAnd

taking ratios

OfSin

And Cos

In game development, there are a lot of situations where you need to use the trigonometric functions. When programming a game, you'll often need to do things like find the distance between two points or make an object move. Here are a few examples:• Rotating a spaceship or other vehicle• Properly handling the trajectory of projectiles shot from a rotated

weapon• Calculating a new trajectory after a collision between two objects

such as billiard balls or heads• Determining if a collision between two objects is happening• Finding the angle of trajectory (given the speed of an object in the x

direction and y direction)

Use of trigonometry in video games

The first step in measuring the distance between the Earth and the Sun is to find the relative distances between Earth and other planets. (For instance, what is the ratio of the Jupiter-Sun distance to the Earth-Sun distance?) So, let us say that the distance between Earth and the Sun is "a". Now, consider the orbit of Venus. To a first approximation, the orbits of Earth and Venus are perfect circles around the Sun, and the orbits are in the same plane.Take a look at the diagram below (not to scale). From the representation of the orbit of Venus, it is clear that there are two places where the Sun-Venus-Earth angle is 90 degrees. At these points, the line joining Earth and Venus will be a tangent to the orbit of Venus. These two points indicate the greatest elongation of Venus and are the farthest from the Sun that Venus can appear in the sky. (More formally, these are the two points at which the angular separation between Venus and the Sun, as seen from Earth, reaches its maximum possible value.)

Another way to understand this is to look at the motion of Venus in the sky relative to the Sun: as Venus orbits the Sun, it gets further away from the Sun in the sky, reaches a maximum apparent separation from the Sun (corresponding to the greatest elongation), and then starts going towards the Sun again. This, by the way, is the reason why Venus is never visible in the evening sky for more than about three hours after sunset or in the morning sky more than three hours before sunrise.

Now, by making a series of observations of Venus in the sky, one can determine the point of greatest elongation. One can also measure the angle between the Sun and Venus in the sky at the point of greatest elongation. In the diagram, this angle will be the Sun-Earth-Venus angle marked as "e" in the right angled triangle. Now, using trigonometry, one can determine the distance between Earth and Venus in terms of the Earth-Sun distance:(distance between Earth and Venus) = a × cos(e)

Similarly, with a little more trigonometry:

(distance between Venus and the Sun) = a × sin(e)

how to measure distance between celestial objects

sine wave or sinusoid is a mathematical curve that describes a smooth repetitive oscillation. It is named after the function sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields.This wave pattern occurs often in nature, including ocean waves, sound waves, and light waves.A cosine wave is said to be "sinusoidal", because cos ( x ) = sin ( x + ) , which is also a sine wave with a phase-shift of radians. Because of this "head start", it is often said that the cosine function leads the sine function or the sine lags the cosine.The human ear can recognize single sine waves as sounding clear because sine waves are representations of a single frequency with no harmonics.To the human ear, a sound that is made of more than one sine wave will have perceptible harmonics; addition of different sine waves results in a different waveform and thus changes the timbre of the sound. Presence of higher harmonics in addition to the fundamental causes variation in the timbre, which is the reason why the same musical note (the same frequency) played on different instruments sounds different. On the other hand, if the sound contains aperiodic waves along with sine waves (which are periodic), then the sound will be perceived "noisy" as noise is characterized as being aperiodic or having a non-repetitive pattern.

Sine and cosine can be used to describe sound waves

ConclusionIt is pretty clear that there is

great importance of trigonometry in our daily life so

I conclude the presentation without making the conclusion

too long and deliberately wasting your time.

The End

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