trigonometry and application
Download as PPTX, PDF5 likes5,381 views
Trigonometry is derived from Greek words meaning "three angles" and "measure". It deals with relationships between sides and angles of triangles, especially right triangles. The document discusses the history of trigonometry dating back to ancient Egypt and Babylon, and how it advanced through the works of Greek astronomer Hipparchus and Ptolemy. It also discusses the six trigonometric ratios and their formulas, various trigonometric identities, and applications of trigonometry in fields like architecture, engineering, astronomy, music, optics, and more.
More Related Content
Similar to trigonometry and application (20)
![Atoms and molecules [autosaved]](https://cdn.slidesharecdn.com/ss_thumbnails/atomsandmoleculesautosaved-160615024046-thumbnail.jpg?width=560&fit=bounds)
trigonometry and application
- 2. 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 degrees Triangles 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.
- 3. Trigonometry a beginning The 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 .
- 4. Right Triangle A 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
- 5. Values of trigonometric function of Angle A sin = a/c cos = b/c tan = a/b cosec = c/a sec = c/b cot = b/a
- 6. Trigonometric ratios The 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
- 7. 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.
- 8. RELATION WITH RATIOS 1)Relation with cosine sin2 A + cos2A = 1 sin2 A = 1 - cos2A Hence sin A = (1 - cos2A)1/2 Also sin A = cos (90-A) 3) Relation with cosecant sin A = 1/ cosec A (cosecant is the reciprocal of sine) 5) Relation with cotangent sin2A (1 + tan2A) = tan2A sin2A (1 + 1/cot2A) = 1/cot2A sin2A (cot2A + 1)/cot2A = 1/cot2A sin A = (cot2A + 1)-1/ 2 2) Relation with secant sin2A = 1 - cos2A = 1 - 1/sec2A ( secant is the reciprocal of cosine) = (sec2A - 1)/sec2A 4) Relation with Tangent sin2A = cos2A tan2A = (1-sin2A)*tan2A sin2A + sin2A*tan2A = tan2A sin2A (1 + tan2A) = tan2A sin A = {tan2A/(1 + tan2A) }1/2
- 9. • 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
- 10. Trigonometric ratios of complementary angles Trigonometrical ratio of angle Trigonometrical ratio of complementary angle Formulas
- 11. 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
- 12. Calculator This 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.
- 13. 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.
- 14. A few examples of this include Soccer ball begin kicked A baseball begin thrown An athlete long jumping Fireworks and Water fountains Projectile motion refers to the motion of an object projected into the air at an angle.
- 15. cos = hypotenuse adjacent side
- 16. Applications of Trigonometry in Astronomy The 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.
- 17. 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.
- 18. 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 H opposite side D Some structures that were built using trigonometry are: Leaning tower of Pisa Eiffel Tower Qutub Minar
- 19. 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
- 20. 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)
- 21. 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-
- 22. Whenlight moves from a dense toa less dense medium, suchas from water to air, At this point, light is reflected in the incident medium, knownas internal reflection. Beforethe ray totally internally reflects, thelight refractsat 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 LAW Sine i Sine r = Refractive index
- 23. 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.
- 24. 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
- 25. 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 Cartography Chemistry Civil Engineering Computer graphics Geophysics Crystallography Economics Electrical engineering Electronics Land surveying Geodesy Physical sciences Mechanical engineering, machining Medical imaging (CAT scans and ultrasound) Meteorology Music theory Number theory (and hence cryptography), Oceanography Optics Pharmacology Phonetics Probability theory Psychology Seismology Statistics Visual perception So Where Ever You Go Trig Will Follow You
- 26. 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