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Foundations of Trigonometry Pre-calculus chapter 10 A little history of trigonometry: While the work of the Alexandrians Hipparchus (c. 150 BC), Menelaus (c. AD 100), and Ptolemy (c. AD 150) in astronomy laid the foundations of trigonometry, further progress was piecemeal and spasmodic. From about the time of Aryabhata I (c. AD 500), the character of the subject changed, and it began to resemble its modern form. Subsequently it was transmitted to the Middle Easterners, who introduced further refinements. From the Middle East, the knowledge spread to Europe, where a detailed account of existing trigonometric knowledge first appeared under the title De triangulis omni modis, written in 1464 by Regiomontanus. Pre-calculus chapter 10 A little history of trigonometry: On account of their shapes, the arc of a circle, the arc ACB, was known as the "bow" (capa) and its full chord, the line segment AMB, as the "bow string" (samastajya). In their study of trigonometric functions, Indian mathematicians more often used the half chord, the segment AM or MB. The half chord was known as ardhajya orjyardha, later abbreviated tojya to become the Indian sine. Three functions were developed, whose modern equivalents are defined here with reference to the figure: jya a = AM = r sin a, kojya a = 0M = r cos a, utkramajya a = MC = OC = r — r cos a = r(l — cos a) = r versin a. Pre-calculus chapter 10 A little history of trigonometry: As with so many other areas of mathematics, the Middle Eastern scholars (the author calls them Islamic, but I think that the name is misleading, for example we don’t call European scholars Christian!) selected Hellenistic and Indian concepts of trigonometry and combined them into a distinctive discipline that bore little resemblance to its precursors. It then became an essential component of modern mathematics. 1. The introduction of six basic trigonometric functions, namely sine and cosine, tangent and cotangent, secant, and cosecant. 2. The derivation of the sine rule and establishment of other trigonometric identities. 3. The construction of highly detailed trigonometric tables with the aid of various interpolation procedures. Pre-calculus chapter 10 A little history of trigonometry: There were two types of trigonometry: one based on the geometry of chords and best exemplified in Ptolemy's Almagest, and the other based on the geometry of semi-chords, which was an Indian invention. Pre-calculus chapter 10 A little history of trigonometry: From the tenth century onward, starting with the work of Abu Nasr Mansur (c. 960 —1036), Islamic mathematicians brought the sine function closer to its modern form with a few defining it for the first time in terms of a circle of unit radius, although it remained defined for an arc of a circle rather than the angle subtended at the center. The etymology of the word "sine" is instructive, for it shows what can happen as a result of imperfect linguistic and cultural filtering. The Sanskrit term for sine in an astronomical context wasjya-ardha (half chord), which was later abbreviated tojya. From this came the phonetically derived Arabic wordjiba, which, following the usual practice of omitting vowels in Semitic languages, was written asjyb. Early Latin translators, coming across this word, mistook it for another word, jaib, which had among its meanings the opening of a woman's garment at the neck, or bosom; jaib was translated as sinus, which in Latin had a number of meanings, including a cavity in facial bones (whence sinusitis), bay, bosom, and, indeed, curve. And hence the Pre-calculus chapter 10 present word "sine." A little history of trigonometry: During the ninth century the Islamic astronomer Habash al-Hasib examined the length of the shadow of a rod of unit length horizontally mounted on a wall when the sun was at a given angle to the horizontal. It is easily shown, see the figure, that the length s of the shadow on the wall can be calculated as sin s tan cos where alpha is the angle of elevation of the sun above the horizon. The length t of the shadow cast by a vertical rod, see the figure, is cos t cot sin Pre-calculus chapter 10 A little history of trigonometry: During the ninth century the Islamic astronomer Habash al-Hasib examined the length of the shadow of a rod of unit length horizontally mounted on a wall when the sun was at a given angle to the horizontal. It is easily shown, see the figure, that the length s of the shadow on the wall can be calculated as sin s tan cos where alpha is the angle of elevation of the sun above the horizon. The length t of the shadow cast by a vertical rod, see the figure, is cos t cot sin Pre-calculus chapter 10 Angles and their Measure A ray is usually described as a ‘half-line’ and can be thought of as a line segment in which one of the two endpoints is pushed off infinitely distant from the other, as pictured below. The point from which the ray originates is called the initial point of the ray. When two rays share a common initial point they form an angle and the common initial point is called the vertex of the angle. Pre-calculus chapter 10 Angles and their Measure The measure of an angle is a number which indicates the amount of rotation that separates the rays of the angle. There is one immediate problem with this, as pictured below. One commonly used system to measure angles is degree measure. Quantities measured in degrees are denoted by the familiar ‘◦’ symbol. One complete revolution as shown below is 360◦, and parts of a revolution are measured proportionately. Pre-calculus chapter 10 Angles and their Measure The measure of an angle is a number which indicates the amount of rotation that separates the rays of the angle. There is one immediate problem with this, as pictured below. One commonly used system to measure angles is degree measure. Quantities measured in degrees are denoted by the familiar ‘◦’ symbol. One complete revolution as shown below is 360◦, and parts of a revolution are measured proportionately. Pre-calculus chapter 10 Angles and their Measure Using our denition of degree measure, we have that 1 represents the measure of an angle which 1 constitutes of a revolution. Even though it may be hard to draw, it is nonetheless not dificult 360 to imagine an angle with measure smaller than 1 . The choice of `360' is most often attributed to the Babylonians. Pre-calculus chapter 10 Angles and their Measure Using our denition of degree measure, we have that 1 represents the measure of an angle which 1 constitutes of a revolution. Even though it may be hard to draw, it is nonetheless not dificult 360 to imagine an angle with measure smaller than 1 . The choice of `360' is most often attributed to the Babylonians. There are two ways to subdivide degrees. The first, and most familiar, is decimal degrees. The second way to divide degrees is the Degree - Minute - Second (DMS) system. In this system, one degree is divided equally into sixty minutes, and in turn, each minute is divided equally into sixty seconds. Note : 15 45 117 60 3600 Pre-calculus chapter 10 Angles and their Measure Pre-calculus chapter 10 Angles and their Measure Pre-calculus chapter 10 Angles and their Measure We need to extend our notion of `angle' from merely measuring an extent of rotation to quantities which can be associated with real numbers. To that end, we introduce the concept of an oriented angle. As its name suggests, in an oriented angle, the direction of the rotation is important. We imagine the angle being swept out starting from an initial side and ending at a terminal side, as shown below. When the rotation is counter-clockwise from initial side to terminal side, we say that the angle is positive; when the rotation is clockwise, we say that the angle is negative. Pre-calculus chapter 10 Angles and their Measure To further connect angles with the Algebra which has come before, we shall often overlay an angle diagram on the coordinate plane. An angle is said to be in standard position if its vertex is the origin and its initial side coincides with the positive x-axis. Angles in standard position are classified according to where their terminal side lies. For instance, an angle in standard position whose terminal side lies in Quadrant I is called a `Quadrant I angle'. If the terminal side of an angle lies on one of the coordinate axes, it is called a quadrantile angle. Pre-calculus chapter 10 Angles and their Measure Two angles in standard position are called conterminal if they share the same terminal side. Pre-calculus chapter 10 Angles and their Measure Graph each of the (oriented) angles below in standard position and classify them according to where their terminal side lies. Find three conterminal angles, at least one of which is positive and one of which is negative. Pre-calculus chapter 10 Radian Measure Let be the central angle subtended by this arc, that is, an angle whose vertex is the center of the circle and whose determining rays pass through the endpoints of the arc. Using proportionality arguments, it stands s to reason that the ratio should also r be a constant among all circles, and it is this ratio which defines the radian measure of an angle. Pre-calculus chapter 10 Radian Measure Pre-calculus chapter 10 Radian Measure Graph each of the (oriented) angles below in standard position and classify them according to where their terminal side lies. Find three conterminal angles, at least one of which is positive and one of which is negative. Pre-calculus chapter 10 Converting degree to radian Pre-calculus chapter 10 The Unit Circle In order to identify real numbers with oriented angles, we make good use of this fact by essentially `wrapping' the real number line around the Unit Circle and associating to each real number t an oriented arc on the Unit Circle with initial point (1, 0). Pre-calculus chapter 10 The Unit Circle Sketch the oriented arc on the Unit Circle corresponding to each of the following real numbers. Pre-calculus chapter 10 Applications of Radian Measure: Circular Motion Suppose an object is moving as pictured below along a circular path of radius r from the point P to the point Q in an amount of time t. Here s represents a displacement so that s > 0 means the object is traveling in a counter-clockwise direction and s < 0 indicates movement in a clockwise direction. Average velocity displacement s v time t Average angular velocity angle measure in radian time t Velocity for Circular Motion: For an object moving on a circular path of radius r with constant angular velocity , the (linear) velocity of the object is given by v r. Pre-calculus chapter 10 Applications of Radian Measure: Circular Motion Assuming that the surface of the Earth is a sphere, any point on the Earth can be thought of as an object traveling on a circle which completes one revolution in (approximately) 24 hours. The path traced out by the point during this 24 hour period is the Latitude of that point. San Jacinto Community College is at 29.6911° N latitude, and it can be shown that the radius of rotation at this Latitude is approximately 3,439 miles. Find the linear velocity, in miles per hour, of San Jacinto Community College as the world turns. 2 radians the angular velocity of earth 24 hours 12 hours miles linear velocity r 3, 439 miles 900 12 hours hour Pre-calculus chapter 10 Applications of Radian Measure: Circular Motion A r 1 2 2 2A r r r A r 2 r 2 r 2 Pre-calculus chapter 10 The Unit Circle: Cosine and Sine If an object is moving on a circular path of radius r with a fixed angular velocity (frequency) , what is the position of the object at time t? The answer to this question is the very heart of Trigonometry. Consider an angle in standard position and let P denote the point where the terminal side of intersects the Unit Circle. By associating the point P with the angle , we are assigning a position on the Unit Circle to the angle . The x-coordinate of P is called the cosine of , written cos( ), while the y-coordinate of P is called the sine of , written sin( ). Pre-calculus chapter 10 The Unit Circle: Cosine and Sine Find the cosine and sine of the following angles. Pre-calculus chapter 10 The Unit Circle: Cosine and Sine Example) Using the given information about, find the indicated value. 3 9 4 in Q 2 4 2 1) sin( ) cos ( ) 1 cos( ) cos( ) 5 25 5 5 5 5 2 2 in Q 3 2 2) cos( ) sin ( ) 1 sin( ) sin( ) 5 25 5 5 3) sin( ) 1 cos 2 ( ) 1 1 cos( ) 0 Pre-calculus chapter 10 The Unit Circle: Cosine and Sine Reference Angle Theorem. Suppose is the reference angle for . Then cos( ) = cos( ) and sin( ) = sin( ), where the choice of the ( ) depends on the quadrant in which the terminal side of lies. Pre-calculus chapter 10 The Unit Circle: Cosine and Sine Example) Find the cosine and sine of the following angles. is in Q 3 1) 180 225 180 45 cos( ) 0 and sin( ) 0 cos(225) cos(45) 2 2 and sin(225) sin(45) 2 2 11 is in Q 4 cos( ) 0 and sin( ) 0 6 6 11 3 11 1 cos( ) cos( ) and sin( ) sin( ) . 6 6 2 6 6 2 5 is in Q 2 3) cos( ) 0 and sin( ) 0 4 4 5 2 5 2 cos( ) cos( ) and sin( ) sin( ) . 4 4 2 4 4 2 7 is in Q1 4) 2 2 cos( ) 0 and sin( ) 0 3 3 7 1 7 3 cos( ) cos( ) and sin( ) sin( ) . 3 3 2 3 3 2 2) 2 2 Pre-calculus chapter 10 The Unit Circle: Cosine and Sine The Reference Angle Theorem in conjunction with the table of cosine and sine values can be used to generate the following figure, which you should memorized. Pre-calculus chapter 10 The Unit Circle: Cosine and Sine 5 Suppose is an acute angle with cos( ) = . 13 1. Find sin( ) and use this to plot in standard position. 2. Find the sine and cosine of the following angles: (a) (b) 2 (c) 3 ( d ) 2 5 25 12 2 cos( ) = + sin ( ) = 1 sin( ) = 13 169 13 12 isin Q1 sin( ) = 13 5 12 (cos( ),sin( )) ( , ). 13 13 Pre-calculus chapter 10 The Unit Circle: Cosine and Sine 5 Suppose is an acute angle with cos( ) = . 13 1. Find sin( ) and use this to plot in standard position. 2. Find the sine and cosine of the following angles: (a) (b) 2 (c) 3 ( d ) 2 Pre-calculus chapter 10 The Unit Circle: Cosine and Sine Example) Find all of the angles which satisfy the given equation. One solution in Quadrant I is = coterminal with 3 , we find = 3 3 , and since all other Quadrant I solutions must be 2 k for integers k. Similarly in Q4, we find the solution to cos( ) = 1 5 here is 2 3 5 so our answer in this Quadrant is 2 k for integers k. 3 Pre-calculus chapter 10 The Unit Circle: Cosine and Sine Example) Find all of the angles which satisfy the given equation. 7 , and since all other Quadrant III solutions must be 6 7 7 coterminal with , we find = 2 k for integers k. 6 6 11 Similarly in Q4, one solution is 6 11 so our answer in this Quadrant is 2 k for integers k. 6 In Quadrant III one solution is = Pre-calculus chapter 10 The Unit Circle: Cosine and Sine Example) Find all of the angles which satisfy the given equation. One solution is = coterminal with 2 2 , and since all other solutions must be , we find = 2 Similarly for the second angle coterminal to 2 k for integers k. 3 , so our answer for all other angles 2 3 3 is 2 k for integers k. From the figure, both are shortened k . 2 2 2 Pre-calculus chapter 10 Beyond the Unit Circle If Q( x, y ) is the point on the terminal side of an angle , plotted in standard position, which lies on the circle x 2 y 2 r then x r cos( ) and y r sin( ). Moreover, x x y y cos( ) and sin( ) r r x2 y 2 x2 y 2 OPA OQB x r x rx x r cos( ) x 1 y r y ry y r sin( ) y 1 Pre-calculus chapter 10 Beyond the Unit Circle Suppose that the terminal side of an angle , when plotted in standard position, contains the point Q(4, 2). Find sin( ) and cos( ). x 4 and y 2, we have r 42 (2) 2 20 2 5 cos( ) x 4 y 2 5 and sin( ) . r 2 5 r 2 5 5 Pre-calculus chapter 10 Beyond the Unit Circle Assuming that the surface of the Earth is a sphere, any point on the Earth can be thought of as an object traveling on a circle which completes one revolution in (approximately) 24 hours. The path traced out by the point during this 24 hour period is the Latitude of that point. San Jacinto Community College is at 29.6911° N latitude, and it can be shown that the radius of rotation at this Latitude is approximately 3,439 miles. Find the linear velocity, in miles per hour, of San Jacinto Community College as the world turns. In this problem, now we can figure out why the radius of rotation for San Jacinto is Radius=3,959cos(29.6911°) R=3,439. 2 radians the angular velocity of earth 24 hours 12 hours miles linear velocity r 3, 439 miles 900 12 hours hour Pre-calculus chapter 10 Beyond the Unit Circle Suppose an object is traveling in a circular path of radius r centered at the origin with constant angular velocity . If t 0 corresponds to the point (r , 0), then the x and y coordinates of the object are functions of t and are given by x r cos(t ) and y r sin(t ). Here, 0 indicates a counter-clockwise direction and 0 indicates a clockwise direction. x r cos(t ) y r sin(t ). 0 counter-clockwise 0 clockwise Pre-calculus chapter 10 Beyond the Unit Circle Find the equations of the rotation of the San Jacinto San Jacinto Community College is at 29.6911° N latitude. In this problem, now we can figure out why the radius of rotation for San Jacinto is Radius=3,959cos(29.6911°) R=3,439. 2 radians The angular velocity of earth 24 hours 12 hours r of rotation is 3439 x r cos(t ) 3439 cos( t ) 12 y r sin(t ) 3439sin( 12 t) Pre-calculus chapter 10 Beyond the Unit Circle Suppose is an acute angle residing in a right triangle. If the length of the side adjacent to is a, the length of the side opposite is b, and the length of the hypotenuse is c, then a b cos( ) and sin( ) . c c Pre-calculus chapter 10 Beyond the Unit Circle Find the measure of the missing angle and the lengths of the missing sides of: 180 30 90 60 cos(30 ) 7 7 c c cos(30 ) 3 7 c 14 3 . 3 2 b 14 3 1 7 3 b c sin(30 ) . c 3 2 3 14 3 2 7 3 2 2 Or 7 b ( ) b . 3 3 sin(30 ) Pre-calculus chapter 10 Beyond the Unit Circle We close this section by noting that we can easily extend the functions cosine and sine to real numbers by identifying a real number t with the angle t radians. Pre-calculus chapter 10