DA PUBLIC SCHOOL (O&A LEVELS) Trigonometry and Applications BY : MUSHTAQ-UR-REHMAN HEAD OF MATHEMATICS DEPARTMENT D.A.P.S. O & A LEVELS Allah says in the Holy Quran “Indeed we have created everything in a proper measure.” (Surah Al-Qamr) Trigonometry and Applications Fields of discussion What is Mathematics? Prince of Mathematicians What is Trigonometry? History and the meaning of the word sine and cosine. Trigonometric functions, Circular functions or cyclometric functions Fields of Trigonometry Ancient Egypt and the Mediterranean world Applications of Trigonometry Angle measurement Properties of sines and cosines The Law(Rule) of sines,cosines Trigonometric Equations Applications of Trigonometric Equations What is Mathematics? Etymology The word “Mathematics" comes from the Greek word (máthēma), which means learning, study, science, and additionally more technical meaning “Mathematical study", Mathematics (Definition) A group of related subjects , including ALGEBRA, GEOMETRY, TRIGONOMETRY and CALCULUS, concerned with the study of number ,quantity, structure, shape and space. Applications Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Prince of Mathematicians Carl Friedrich Gauss Himself known as the "prince of mathematicians“, referred to Mathematics as "the Queen of the Sciences". This German Mathematician contributed to many areas of Mathematics, including probability theory, algebra, and geometry. He proved that every polynomial has at least one root, or solution; this theory is known as the fundamental theory of algebra. Gauss also applied his mathematical work to theories of electricity and magnetism. What is Trigonometry? Etymology The word Trigonometry is derived from three Greek words ‘tries’(three), ‘goni’(angle) and ‘metron’(measurement). So literally, this word means “measurement of the triangle”. Trigonometry (Definition) The branch of Mathematics concerned with the properties of trigonometric functions and their application to the determination of the sides and angles of triangles. Trigonometry has now a wide application in higher Mathematics in fact, any attempt to study Higher Mathematics would be an utter failure without a working knowledge of trigonometry. It has applications in both pure mathematics and applied mathematics, where it is essential in many branches of science and technology. History and the meaning of the word sine and cosine Interesting word history for "sine” The Hindu mathematician Aryabhata (about 475–550 A.D.) used the Sanskrit word “jya” or “jiva” for the half-chord which was sometimes shortened to jiva. This was brought into Arabic as jiba, and written in Arabic simply with two consonants jb, vowels not being written. Later, Latin translators selected the word sinus to translate jb thinking that the word was an arabic word jaib, which meant bosom, fold, or bay, The Latin word for bosom, bay, or curve is “sinus”. In English, sinus was imported as "sine". This word history for "sine" is interesting because it follows the path of trigonometry from India, through the Arabic language from Baghdad through Spain, into western Europe in the Latin language, and then to modern languages such as English. Trigonometric functions, Circular functions or cyclometric functions Any of a group of functions expressible in terms of the ratios of the sides of right-angled triangle. Sine Ratio The sine of an angle in a right triangle equals the opposite side divided by the hypotenuse: sin =opp/hyp Cosine Ratio. Cosines are just sines of the complementary angle. Thus, the name "cosine" ("co" being the first two letters of "complement"). The complementary angle equals the given angle subtracted from a right angle, 90°. For instance, if the angle is 30°, then its complement is 60°. Generally, for any angle x, cos =adj/hyp cos x = sin (90° – x). Or cos 50ᵒ = sin (90 – 50) = sin40ᵒ Tangent Ratio tanx = sinx/cosx tanx = opp/adj Trigonometric functions, Circular functions or cyclometric functions Secant: sec q = 1/cos q Cosecant: csc q = 1/sin q Cotangent: cot q = 1/ tan q cot q = cos q/sin q tan q = sin q/cos q Trigonometric Ratios Trigonometric Identities The following formulas, called identities, which show the relationships between the trigonometric functions, hold for all values of the angle θ, or of two angles, θ and φ, for which the functions involved are Fields of Trigonometry Plane Trigonometry In many applications of trigonometry the essential problem is the solution of triangles. If enough sides and angles are known, the remaining sides and angles as well as the area can be calculated, and the triangle is then said to be solved. Triangles can be solved by the law of sines and the law of cosines. Surveyors apply the principles of geometry and trigonometry in determining the shapes, measurements and position of features on or beneath the surface of the Earth. Such topographic surveys are useful in the design of roads, tunnels, dams, and other structures. Fields of Trigonometry Spherical Trigonometry Spherical Trigonometry involves the study of spherical triangles, which are formed by the intersection of three great circle arcs on the surface of a sphere. Great Circle A great circle is a theoretical circle, such as the equator, formed by the intersection of the earth’s surface and an imaginary plane that passes through the center of the earth and divides it into two equal parts. Navigators use great circles to find the shortest distance between any Two points on the earth’s surface. Fields of Trigonometry Analytic Trigonometry Analytic Trigonometry combines the use of a coordinate system, such as the Cartesian coordinate system used in analytic geometry, with algebraic manipulation of the various trigonometry functions to obtain formulas useful for scientific and engineering applications. Ancient Egypt and the Mediterranean world Several ancient civilizations—in particular, the Egyptian, Babylonian, Hindu, and Chinese—possessed a considerable knowledge of practical geometry, including some concepts of trigonometry. A close analysis of the text, with its accompanying figures, reveals that this word means the slope of an incline, essential knowledge for huge construction projects such as the pyramids. It shows that the Egyptians had at least some knowledge of the numerical relations in a triangle, a kind of “prototrigonometry.” Ancient Egypt and the Mediterranean world Applications of Trigonometry . Fields that use trigonometry or trigonometric functions include Astronomy (especially for locating apparent positions of celestial objects(star or planet), in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves. Music theory, acoustics(study of sound ), optics, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory cryptology(coding), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy(cartography), architecture, phonetics (sounds of human speech), economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, crystallography and game development. Applications of Trigonometry Marine sextants like this are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer(timer), the position of the ship can then be determined from several such measurements. Applications of Trigonometry Wave Mathematics Waves are familiar to us from the ocean, the study of sound, earthquakes, and other natural phenomenon. Ocean waves come in very different sizes to fully understand waves, we need to understand measurements associated with these waves, such as how often they repeat (their frequency), and how long they are (their wavelength), and their vertical size (amplitude). The importance of the sine and cosine functions is in describing periodic phenomena—the vibrations of a violin string, the oscillations of a clock pendulum, or the propagation of electromagnetic waves, sound and light waves. Applications of Trigonometry Sine waves in nature i)Sound waves are sine waves whenever we listen to music , we are actually listening to sound waves. ii) light waves are also sine waves. iii)Radio waves are sine waves. iv)Simple harmonic motion of a spring when pulled and released is a sine wave. v) Alternating current (AC) is a sine wave. vi) Pendulum clock oscillations are sinusoidal in nature vii) Waves of ocean are sinusoidal . viii) The vibrations of guitar strings when played are sinusoidal in nature. Applications of Trigonometry Graph of Trigonometric Functions Graph of sine function f(x) = a sin ( bx + c ) Graph of sine function f(x) = a cos ( bx + c ) Graph of tangent function f(x) = tanx Applications of Trigonometry The 17th and 18th centuries saw the invention of numerous mechanical devices. A notable application was the science of artillery—and in the 18th century it was a science. Galileo Galilei (1564–1642) discovered that any motion—such as that of a projectile under the force of gravity—can be resolved into two components, one horizontal and the other vertical, This discovery led scientists to the formula for the range of a cannonball when its muzzle velocity v0 (the speed at which it leaves the cannon) and the angle of elevation A of the cannon. Applications of Trigonometry Applications of Trigonometry Fourier series An infinite trigonometric series of terms consisting of constants multiplied by sines or cosines, used in the approximation of periodic functions. The trigonometric or Fourier series have found numerous applications in almost every branch of science, from optics and acoustics to radio transmission and earthquake analysis. Their extension to non periodic functions played a key role in the development of quantum mechanics in the early years of the 20th century. Trigonometry, by and large, matured with Fourier's theorem. Angle measurement The concept of angle is one of the most important concepts in geometry and the subject of trigonometry is based on the measurement of angles. Degree (Angle) There are two commonly used units of measurement for angles. The more familiar unit of measurement is that of degrees. A circle is divided into 360 equal degrees, Degrees may be further divided into minutes and seconds. For instance seven and a half degrees is now usually written 7.5°. Each degree is divided into 60 equal parts called minutes. So seven and a half degrees can be called 7 degrees and 30 minutes, written 7° 30'. Each minute is further divided into 60 equal parts called seconds, and, for instance, 2 degrees 5 minutes 30 seconds is written 2° 5' 30". Angle measurement Radian(Angle) The other common measurement for angles is radians. If the radius of the circle and the length of arc of a sector of the circle are equal then angle is 1 radian. The radian measure of the angle is the ratio of the length of the subtended arc to the radius of the circle. radian measure = arc length/radius (θ = S/r) Degrees Below is a table of common angles in both 90° degree measurement and radian measurement. Radians Π /2 60° Π /3 45° Π /4 30° Π /6 Angle measurement Angle measurement 1. Express the following angles in radians. (a). 12 degrees, 28 minutes, that is, 12° 28'. (b). 36° 12'. 2. Reduce the following numbers of radians to degrees, minutes, and seconds. (a). 0.47623. (b). 0.25412. 3. Given the angle a and the radius r, to find the length of the subtending arc. a = 0° 17' 48", r = 6.2935. 4. Find the length to the nearest inch of a circular arc of 11 degrees 48.3 minutes if the radius is 3200 feet. 5. Given the length of the arc l and the angle a which it subtends at the center, to find the radius. a = 0° 44' 30", l = .032592 Properties of sines and cosines 1.Sine and cosine are periodic functions of period 360° or 2Π ., sin (t + 360°) = sin t, and sin (t + 2 ) = sin t, cos (t + 360°) = cos t. cos (t + 2 ) = cos t. 2 . Sine and cosine are complementary: cos t = sin ( /2 – t) , sin t = cos ( /2 – t) 3 .The Pythagorean identity sin2 t + cos2 t = 1. 4. Sine is an odd function, and cosine is even sin (–t) = –sin t, and cos (–t) = cos t. 5.An obvious property of sines and cosines is that their values lie between –1 and 1. Every point on the unit circle is 1 unit from the origin, so the coordinates of any point are within 1 of 0 as well. The Law(Rule) of sines The Law of Sines is simple and beautiful and easy to derive. It’s useful when you know two angles and any side of a triangle, or sometimes when you know two sides and one angle. Law of Sines — First Form: a / sin A = b / sin B = c / sin C This is very simple and beautiful: for any triangle, if you divide any of the three sides by the sine of the opposite angle, you’ll get the same result. This law is valid for any triangle. Law of Sines— Second Form: sin A / a = sin B / b = sin C / c The Law(Rules) of cosines The Law of Sines is fine when you can relate sides and angles. But suppose you know three sides of the triangle — for instance a = 180, b = 238, c = 340 — and you have to find the three angles. The Law of Sines is no good for that because it relates two sides and their opposite angles. If you don’t know any angles, you have an equation with two unknowns and you can’t solve it. Law of Cosines — First Form: cos A = (b² + c² − a²) / 2bc cos B = (a² + c² − b²) / 2ac cos C = (a² + b² − c²) / 2ab Law of Cosines — Second Form: a² = b² + c² − 2bc cos A b² = a² + c² − 2ac cos B c² = a² + b² − 2ab cos C Trigonometric Equations A formula that asserts that two expressions have the same value ;it is either an identical equation or an identity which is true for any values of the variables or a conditional equation which is only true for certain values of the variables. Example1 Solve the equation sin{1/3(-30)} = 3/2, giving all the roots in the interval 0 360. Example2 Find all the values of in the interval 0 360 for which sin2 = cos36 Example3 Find all the values of in the interval 0 360 for which sin2 - 3 cos2 = 0 Applications of Trigonometric Equations Example The height in meters of the water in a harbor is given by approximately by the formula d=6+3cos30t where t is the time measured in hours from noon. Find the time after noon when the height of the water is 7.5 meters for the second time. THANK YOU Mushtaq ur Rehman H.O.D Mathematics Department D.A.P.S. O&A Levels

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