Trigonometry And its Applications

Trigonometry is derived from Greek words trigonon (three angles) and matron ( measure).Trigonometry is the branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degreesTriangles on a sphere are also studied, in spherical trigonometry. Trigonometry specifically deals with the relationships between the sides and the angles of triangles, that is, on the trigonometric functions, and with calculations base on these functions.

Trigonometry a beginningThe history of trigonometry dates back to the early ages of Egypt and Babylon . Angles were then measured in degrees. History of trigonometry was then advanced by the Greek astronomer Hipparchus who compiled a trigonometry table that measured the length of the chord subtending the various angles in a circle of a fixed radius. This was done in increasing degrees of 71.In the 5th century, Ptolemy took this further by creating the table of chords with increasing 1 degree. This was known as Menelaus theorem which formed the foundation of trigonometry studies for the next 3 centuries. Around the same period, Indian mathematicians created the trigonometry system based on the sine function instead of the chords. Note that this was not seen to be ratio but rather the opposite of the angle in a right angle of fixed hypotenuse. The history of trigonometry also included Muslim astronomers who compiled both the studies of the Greeks and Indians .

Right TriangleA triangle in which one angle

is equal to 90 is called right triangle.

The side opposite to the right angle is known as hypotenuse.AB is the hypotenuse

The other two sides are known as legs.

AC and BC are the legs

Values of trigonometric function of Angle Asin = a/c

cos = b/ctan = a/bcosec = c/asec = c/bcot = b/a

Trigonometric ratiosThe relationships between the angles and the sides of a right triangle are expressed in terms of TRIGONOMETRIC RATIOS. The six trigonometric ratios for the angle

Name of the ratio Abbreviation

Sine of Sin

Cosine of Cos

Tangent of Tan

Cotangent of Cot

Secant of Sec

Cosecant of Csc

Name of Ratio Abbreviation Explicit Formula Formula Memory Aid

Sin Of q. Sin q. SOH

Cosine Of q. Cos q. CAH

Tangent Of q. Tan q. TOA

Cotangent of q. Cot q.

Secant of q. Sec q.

Cosecant of q. Cos q.

RELATION WITH RATIOS1)Relation with cosinesin2 A + cos2A = 1sin2 A = 1 - cos2AHence sin A = (1 - cos2A)1/2

Alsosin A = cos (90-A) 3) Relation with cosecantsin A = 1/ cosec A (cosecant is the reciprocal of sine) 5) Relation with cotangentsin2A (1 + tan2A) = tan2Asin2A (1 + 1/cot2A) = 1/cot2Asin2A (cot2A + 1)/cot2A = 1/cot2Asin A = (cot2A + 1)-1/ 2

2) Relation with secantsin2A = 1 - cos2A = 1 - 1/sec2A ( secant is the reciprocal of cosine) = (sec2A - 1)/sec2A

4) Relation with Tangentsin2A = cos2A tan2A = (1-sin2A)*tan2A sin2A + sin2A*tan2A = tan2Asin2A (1 + tan2A) = tan2Asin A = {tan2A/(1 + tan2A) }1/2

• sin2A + cos2A = 1• 1 + tan2A = sec2A• 1 + cot2A = cosec2A• sin(A+B) = sinAcosB + cosAsin B• cos(A+B) = cosAcosB – sinAsinB• tan(A+B) = (tanA+tanB)/(1 – tanAtan B)• sin(A-B) = sinAcosB – cosAsinB• cos(A-B)=cosAcosB+sinAsinB• tan(A-B)=(tanA-tanB)(1+tanAtanB)• sin2A =2sinAcosA• cos2A=cos2A - sin2A• tan2A=2tanA/(1-tan2A)• sin(A/2) = ±{(1-cosA)/2}• Cos(A/2)= ±{(1+cosA)/2}• Tan(A/2)= ±{(1-cosA)/(1+cosA)}

Standard Identities

Trigonometric ratios of complementary angles

Trigonometricalratio of angle

Trigonometricalratio of complementary

angle

Formulas

Values of Trigonometric function

0 30 45 60 90

Sine 0 0.5 1/2 3/2 1

Cosine 1 3/2 1/2 0.5 0

Tangent 0 1/ 3 1 3 Not defined

Cosecant Not defined 2 2 2/ 3 1

Secant 1 2/ 3 2 2 Not defined

Cotangent Not defined 3 1 1/ 3 0

CalculatorThis Calculates the values of trigonometric

functions of different angles.First Enter whether you want to enter the angle

in radians or in degrees. Radian gives a bit more accurate value than Degree.

Then Enter the required trigonometric function in the format given below:

Enter 1 for sin.Enter 2 for cosine.Enter 3 for tangent.Enter 4 for cosecant.Enter 5 for secant.Enter 6 for cotangent.Then enter the magnitude of angle.

Applications of Trigonometry• This field of mathematics can be applied in astronomy, navigation,

music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.

A few examples of this include

Soccer ball begin kicked A baseball begin thrownAn athlete long jumpingFireworks and Water fountains

Projectile motion refers to the motion of an object projected into the air at an angle.

cos = hypotenuseadjacent side

Applications of Trigonometry in

AstronomyThe technique used is triangulation ).

By looking at a star one day and then looking at it again 6 months later, an astronomer can see a difference in the viewing angle. With a little trigonometry, the different angles yield a distance.

Application of Trigonometry in Architecture• Many modern buildings have beautifully curved surfaces.

• Making these curves out of steel, stone, concrete or glass is extremely difficult, if not impossible.

• One way around to address this problem is to piece the surface together out of many flat panels, each sitting at an angle to the one next to it, so that all together they create what looks like a curved surface.

• The more regular these shapes, the easier the building process.

• Regular flat shapes like squares, pentagons and hexagons, can be made out of triangles, and so trigonometry plays an important role in architecture.

Engineers of various types use the fundamentals of trigonometry

to build structures/systems, design bridges and solve scientific problems.

Here the height of the building is determined by using the function

Tan

tan = =adjacent side Hopposite side D

Some structures that were built using trigonometry are:

Leaning tower of Pisa

Eiffel Tower

Qutub Minar

19

• The graphs of the functions sin(x) and cos(x) look like waves. Sound travels in waves, although these are not necessarily as regular as those of the sine and cosine functions.

• However, a few hundred years ago, mathematicians realized that any wave at all is made up of sine and cosine waves. This fact lies at the heart of computer music.

• Since a computer cannot listen to music as we do, the only way to get music into a computer is to represent it mathematically by its constituent sound waves.

• This is why sound engineers, those who research and develop the newest advances in computer music technology, and sometimes even composers have to understand the basic laws of trigonometry.

• Waves move across the oceans, earthquakes produce shock waves and light can be thought of as traveling in waves. This is why trigonometry is also used in oceanography, seismology, optics and many other fields like meteorology and the physical sciences.

Waves

When you think trigonometry you should think triangles -- not just geometric triangles but musical triangles -- because trigonometry is the mathematics of sound and music.

Sound is the variation of air pressure.

The simplest sounds, called pure tones are represented by

f(t) = A sin(2 pi w t)

Computer generation of complex imagery is made possible by the use of geometrical patterns that define the precise location and color of each of the infinite points on the image to be created.

The edges of the triangles that form the image make a wire frame of the object to be created and contribute to a realistic picture.

In medicine it s use in CAT and MRI scans.

Digital imaging-

When light moves from a dense to a less dense medium, such as from water to air,

At this point, light is reflected in the incident medium, known as internal reflection.

Before the ray totally internally reflects, the light refracts at the critical angle

When θ1 > θ crit, no refracted ray appears, and the incident ray undergoes total internal reflection from the interface medium.

SNELL’S LAWSine iSine r

= Refractive index

Once the ray reaches the viewer’s eye, it interprets it as if it traces back along a perfectly straight "line of sight".

This line is however at a tangent to the path the ray takes at the point it reaches the eye.

The result is that an "inferior image" of the sky above appears on the ground.

Cold air is denser than warm air

As light passes from colder air, the light rays bend away from the direction

When light rays pass from hotter to colder, they bend toward the direction of the gradient.

Here, if the angle of elevation between the line of sight and ground is known, the Distance

between can be measured using the Tangent function

Among the scientific fields that make use of trigonometry are these:Acoustics Architecture Astronomy (and hence navigation, on the oceans, in aircraft, and in space)Biology CartographyChemistryCivil Engineering Computer graphicsGeophysics Crystallography Economics Electrical engineering 

ElectronicsLand surveying Geodesy Physical sciences Mechanical engineering, machiningMedical imaging (CAT scans and ultrasound) Meteorology Music theory Number theory (and hence cryptography), OceanographyOptics Pharmacology Phonetics Probability theory Psychology Seismology Statistics Visual perception

So Where Ever You Go Trig Will Follow You

Conclusion

Trigonometry is a branch of Mathematics with several important and useful applications. Hence it attracts more and more research with several theories published year after year

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