Trigonometry and its applications

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Trigonometry And its Applications S.Midun Class 10 Section–B P.S.G. PUBLIC SCHOOLS

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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. Trigonometry

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 . :

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 .

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6/2/2012 4 In the 13th century , the Germans fathered modern trigonometry by defining trigonometry functions as ratios rather than lengths of lines. After the discovery of logarithms by the Swedish astronomer, the history of trigonometry took another bold step with Isaac Newton. He founded differential and integral calculus. Euler used complex numbers to explain trigonometry functions and this is seen in the formation of the Euler's formula. The history of trigonometry came about mainly due to the purposes of time keeping and astronomy. Modern Trigonometry

Right Triangle:

5 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 Trigonometry deals with Right Triangles

Trigonometric ratios :

6/2/2012 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 q Name of the ratio Abbreviation Sine of q Sin q Cosine of q Cos q Tangent of q Tan q Cotangent of q Cot q Secant of q Sec q Cosecant of q Csc q

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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. 6/2/2012

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6/2/2012 8 RELATION WITH RATIOS 1)Relation with cosine sin 2 A + cos 2 A = 1 sin 2 A = 1 - cos 2 A Hence sin A = (1 - cos 2 A) 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 sin 2 A (1 + tan 2 A) = tan 2 A sin 2 A (1 + 1/cot 2 A) = 1/cot 2 A sin 2 A (cot 2 A + 1)/cot 2 A = 1/cot 2 A sin A = (cot 2 A + 1) -1/ 2 2) Relation with secant sin 2 A = 1 - cos 2 A = 1 - 1/sec 2 A     ( secant is the reciprocal of cosine) = (sec 2 A - 1)/sec 2 A 4) Relation with Tangent sin 2 A = cos 2 A tan 2 A = (1-sin 2 A)*tan 2 A sin 2 A + sin 2 A*tan 2 A = tan 2 A sin 2 A (1 + tan 2 A) = tan 2 A sin A = {tan 2 A/(1 + tan 2 A) } 1/2

Standard Identities :

6/2/2012 9 sin 2 A + cos 2 A = 1 1 + tan 2 A = sec 2 A 1 + cot 2 A = cosec 2 A 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=cos 2 A - sin 2 A tan2A=2tanA/(1-tan 2 A) 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:

Trigonometric ratios of complementary angles 10 Trigonometrical ratio of angle Trigonometrical ratio of complementary angle Formulas

Calculator:

6/2/2012 11 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.

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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.

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cos = hypotenuse adjacent side

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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 .

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Engineers of various types use the fundamentals of trigonometry to build structures/systems, design bridges and solve scientific problems. Angle of elevation 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

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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) Music

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Digital imaging- 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.

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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 LAW Sine i Sine r = Refractive index

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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. Mirage Looming

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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

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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

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