Lecture notes Trigonometry

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Chapter 10 Foundations of Trigonometry 10.1 Angles and their Measure This section begins our study of Trigonometry and to get started, we recall some basic de nitions from Geometry. 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 o in nitely distant from the other, as pictured below. The point from which the ray originates is called the initial point of the ray. P A ray with initial point P . When two rays share a common initial point they form an angle and the common initial point is called the vertex of the angle. Two examples of what are commonly thought of as angles are Q P An angle with vertex P . An angle with vertex Q. However, the two gures below also depict angles - albeit these are, in some sense, extreme cases. In the rst case, the two rays are directly opposite each other forming what is known as a straight angle; in the second, the rays are identical so the `angle' is indistinguishable from the ray itself. Q P A straight angle. 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.694 Foundations of Trigonometry Which amount of rotation are we attempting to quantify? What we have just discovered is that 1 we have at least two angles described by this diagram. Clearly these two angles have di erent measures because one appears to represent a larger rotation than the other, so we must label them di erently. In this book, we use lower case Greek letters such as (alpha), (beta), (gamma) and  (theta) to label angles. So, for instance, we have 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 2 of a revolution are measured proportionately. Thus half of a revolution (a straight angle) measures 1 1     (360 ) = 180 , a quarter of a revolution (a right angle) measures (360 ) = 90 and so on. 2 4    One revolution 360 180 90 Note that in the above gure, we have used the small square ` ' to denote a right angle, as is   commonplace in Geometry. Recall that if an angle measures strictly between 0 and 90 it is called   an acute angle and if it measures strictly between 90 and 180 it is called an obtuse angle. It is important to note that, theoretically, we can know the measure of any angle as long as we 1 The phrase `at least' will be justi ed in short order. 2 The choice of `360' is most often attributed to the Babylonians.10.1 Angles and their Measure 695 3 know the proportion it represents of entire revolution. For instance, the measure of an angle which 2 2   represents a rotation of of a revolution would measure (360 ) = 240 , the measure of an angle 3 3 1 1   which constitutes only of a revolution measures (360 ) = 30 and an angle which indicates 12 12  no rotation at all is measured as 0 .    240 30 0  Using our de nition 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 dicult 360  to imagine an angle with measure smaller than 1 . There are two ways to subdivide degrees. The  rst, and most familiar, is decimal degrees. For example, an angle with a measure of 30:5 would   30:5 61 represent a rotation halfway between 30 and 31 , or equivalently, = of a full rotation. This 360 720 p  4 can be taken to the limit using Calculus so that measures like 2 make sense. The second way to divide degrees is the Degree - Minute - Second (DMS) system. In this system, one degree is 5 divided equally into sixty minutes, and in turn, each minute is divided equally into sixty seconds.  0 0 00  00 In symbols, we write 1 = 60 and 1 = 60 , from which it follows that 1 = 3600 . To convert a     measure of 42:125 to the DMS system, we start by noting that 42:125 = 42 +0:125 . Converting   0  60 0 0 0 the partial amount of degrees to minutes, we nd 0:125 = 7:5 = 7 + 0:5 . Converting the  1   00 0 60 00 partial amount of minutes to seconds gives 0:5 = 30 . Putting it all together yields 0 1    42:125 = 42 + 0:125  0 = 42 + 7:5  0 0 = 42 + 7 + 0:5  0 00 = 42 + 7 + 30  0 00 = 42 7 30     0 00 0 1 1 On the other hand, to convert 117 15 45 to decimal degrees, we rst compute 15 = and 0 60 4    00 1 1 45 = . Then we nd 00 3600 80 3 This is how a protractor is graded. 4 Awesome math pun aside, this is the same idea behind de ning irrational exponents in Section 6.1. 5 Does this kind of system seem familiar?696 Foundations of Trigonometry  0 00  0 00 117 15 45 = 117 + 15 + 45   1 1  = 117 + + 4 80  9381 = 80  = 117:2625  Recall that two acute angles are called complementary angles if their measures add to 90 . Two angles, either a pair of right angles or one acute angle and one obtuse angle, are called  supplementary angles if their measures add to 180 . In the diagram below, the angles and are supplementary angles while the pair and  are complementary angles.  Supplementary Angles Complementary Angles In practice, the distinction between the angle itself and its measure is blurred so that the sentence   ` is an angle measuring 42 ' is often abbreviated as ` = 42 .' It is now time for an example.   0 00 Example 10.1.1. Let = 111:371 and = 37 28 17 . 1. Convert to the DMS system. Round your answer to the nearest second. 2. Convert to decimal degrees. Round your answer to the nearest thousandth of a degree. 3. Sketch and . 4. Find a supplementary angle for . 5. Find a complementary angle for . Solution.    1. To convert to the DMS system, we start with 111:371 = 111 + 0:371 . Next we convert     0 00 60 60  0 0 0 0 0 00 0:371 = 22:26 . Writing 22:26 = 22 + 0:26 , we convert 0:26 = 15:6 . Hence,  0 1 1    111:371 = 111 + 0:371  0 = 111 + 22:26  0 0 = 111 + 22 + 0:26  0 00 = 111 + 22 + 15:6  0 00 = 111 22 15:6  0 00 Rounding to seconds, we obtain  111 22 16 .10.1 Angles and their Measure 697       0 1 7 00 1 17 2. To convert to decimal degrees, we convert 28 = and 17 = . Putting 0 0 60 15 3600 3600 it all together, we have  0 00  0 00 37 28 17 = 37 + 28 + 17    7 17 = 37 + + 15 3600  134897 = 3600   37:471   3. To sketch , we rst note that 90 180 . If we divide this range in half, we get     90 135 , and once more, we have 90 112:5 . This gives us a pretty good 6   estimate for , as shown below. Proceeding similarly for , we nd 0 90 , then       0 45 , 22:5 45 , and lastly, 33:75 45 . Angle Angle  4. To nd a supplementary angle for , we seek an angle  so that + = 180 . We get      = 180 = 180 111:371 = 68:629 .  5. To nd a complementary angle for , we seek an angle so that + = 90 . We get    0 00 = 90 = 90 37 28 17 . While we could reach for the calculator to obtain an 7 approximate answer, we choose instead to do a bit of sexagesimal arithmetic. We rst   0 00  0 00  0 00  0 rewrite 90 = 90 0 0 = 89 60 0 = 89 59 60 . In essence, we are `borrowing' 1 = 60 0 00 8 from the degree place, and then borrowing 1 = 60 from the minutes place. This yields,   0 00  0 00  0 00  0 00 = 90 37 28 17 = 89 59 60 37 28 17 = 52 31 43 .   Up to this point, we have discussed only angles which measure between 0 and 360 , inclusive. Ultimately, we want to use the arsenal of Algebra which we have stockpiled in Chapters 1 through 9 to not only solve geometric problems involving angles, but also to extend their applicability to other real-world phenomena. A rst step in this direction is 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 6 If this process seems hauntingly familiar, it should. Compare this method to the Bisection Method introduced in Section 3.3. 7 Like `latus rectum,' this is also a real math term. 8 This is the exact same kind of `borrowing' you used to do in Elementary School when trying to nd 300 125. Back then, you were working in a base ten system; here, it is base sixty.Terminal Side 698 Foundations of Trigonometry 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 9 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. Initial Side Initial Side   A positive angle, 45 A negative angle,45 At this point, we also extend our allowable rotations to include angles which encompass more than  one revolution. For example, to sketch an angle with measure 450 we start with an initial side,  rotate counter-clockwise one complete revolution (to take care of the ` rst' 360 ) then continue  with an additional 90 counter-clockwise rotation, as seen below.  450 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 classi ed 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 quadrantal angle. Two angles in standard 10  position are called coterminal if they share the same terminal side. In the gure below, = 120  and =240 are two coterminal Quadrant II angles drawn in standard position. Note that   = + 360 , or equivalently, = 360 . We leave it as an exercise to the reader to verify that  11 coterminal angles always di er by a multiple of 360 . More precisely, if and are coterminal  12 angles, then = + 360 k where k is an integer. 9 `widdershins' 10 Note that by being in standard position they automatically share the same initial side which is the positivex-axis. 11 It is worth noting that all of the pathologies of Analytic Trigonometry result from this innocuous fact. 12 Recall that this means k = 0;1;2;:::. Terminal Side10.1 Angles and their Measure 699 y 4 3  = 120 2 1 x 4321 1 2 3 4 1  =240 2 3 4   Two coterminal angles, = 120 and =240 , in standard position. Example 10.1.2. Graph each of the (oriented) angles below in standard position and classify them according to where their terminal side lies. Find three coterminal angles, at least one of which is positive and one of which is negative.     1. = 60 2. =225 3. = 540 4.  =750 Solution.  1. To graph = 60 , we draw an angle with its initial side on the positive x-axis and rotate  60 1 counter-clockwise = of a revolution. We see that is a Quadrant I angle. To nd angles  360 6  which are coterminal, we look for angles  of the form  = + 360 k, for some integer k.       Whenk = 1, we get = 60 +360 = 420 . Substitutingk =1 gives = 60360 =300 .    Finally, if we let k = 2, we get  = 60 + 720 = 780 .   225 5 2. Since =225 is negative, we start at the positive x-axis and rotate clockwise = of  360 8 a revolution. We see that is a Quadrant II angle. To nd coterminal angles, we proceed as     before and compute  =225 + 360 k for integer values of k. We nd 135 ,585 and   495 are all coterminal with225 . y y 4 4 3 3 2 2  = 60 1 1 x x 4321 1 2 3 4 4321 1 2 3 4 1 1 2 2  =225 3 3 4 4   = 60 in standard position. =225 in standard position.700 Foundations of Trigonometry  3. Since = 540 is positive, we rotate counter-clockwise from the positive x-axis. One full 1   revolution accounts for 360 , with 180 , or of a revolution remaining. Since the terminal 2 side of lies on the negativex-axis, is a quadrantal angle. All angles coterminal with are   of the form  = 540 + 360 k, where k is an integer. Working through the arithmetic, we    nd three such angles: 180 ,180 and 900 . 4. The Greek letter  is pronounced `fee' or ` e' and since  is negative, we begin our rotation   1 clockwise from the positive x-axis. Two full revolutions account for 720 , with just 30 or 12 of a revolution to go. We nd that  is a Quadrant IV angle. To nd coterminal angles, we      compute  =750 + 360 k for a few integers k and obtain390 ,30 and 330 . y y 4 4 3 3  = 540 2 2 1 1 x x 4321 1 2 3 4 4321 1 2 3 4 1 1 2 2   =750 3 3 4 4   = 540 in standard position.  =750 in standard position. Note that since there are in nitely many integers, any given angle has in nitely many coterminal angles, and the reader is encouraged to plot the few sets of coterminal angles found in Example 10.1.2 to see this. We are now just one step away from completely marrying angles with the real numbers and the rest of Algebra. To that end, we recall this de nition from Geometry. De nition 10.1. The real number  is de ned to be the ratio of a circle's circumference to its diameter. In symbols, given a circle of circumference C and diameter d, C  = d While De nition 10.1 is quite possibly the `standard' de nition of , the authors would be remiss if we didn't mention that buried in this de nition is actually a theorem. As the reader is probably aware, the number is a mathematical constant - that is, it doesn't matter which circle is selected, the ratio of its circumference to its diameter will have the same value as any other circle. While this is indeed true, it is far from obvious and leads to a counterintuitive scenario which is explored in the Exercises. Since the diameter of a circle is twice its radius, we can quickly rearrange the C equation in De nition 10.1 to get a formula more useful for our purposes, namely: 2 = r10.1 Angles and their Measure 701 This tells us that for any circle, the ratio of its circumference to its radius is also always constant; in this case the constant is 2. Suppose now we take a portion of the circle, so instead of comparing the entire circumference C to the radius, we compare some arc measuring s units in length to the radius, as depicted below. 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 s the arc. Using proportionality arguments, it stands to reason that the ratio should also be a r constant among all circles, and it is this ratio which de nes the radian measure of an angle. s  r r s The radian measure of  is . r To get a better feel for radian measure, we note that an angle with radian measure 1 means the corresponding arc lengths equals the radius of the circler, hences =r. When the radian measure is 2, we haves = 2r; when the radian measure is 3, s = 3r, and so forth. Thus the radian measure of an angle  tells us how many `radius lengths' we need to sweep out along the circle to subtend the angle . r r r r r r r r r has radian measure 1 has radian measure 4 Since one revolution sweeps out the entire circumference 2r, one revolution has radian measure 2r = 2. From this we can nd the radian measure of other central angles using proportions, r702 Foundations of Trigonometry 1 just like we did with degrees. For instance, half of a revolution has radian measure (2) = , a 2 1  quarter revolution has radian measure (2) = , and so forth. Note that, by de nition, the radian 4 2 measure of an angle is a length divided by another length so that these measurements are actually dimensionless and are considered `pure' numbers. For this reason, we do not use any symbols to denote radian measure, but we use the word `radians' to denote these dimensionless units as needed. For instance, we say one revolution measures `2 radians,' half of a revolution measures ` radians,' and so forth. As with degree measure, the distinction between the angle itself and its measure is often blurred   13 in practice, so when we write ` = ', we mean  is an angle which measures radians. We 2 2 extend radian measure to oriented angles, just as we did with degrees beforehand, so that a positive 14 measure indicates counter-clockwise rotation and a negative measure indicates clockwise rotation. Much like before, two positive angles and are supplementary if + = and complementary  if + = . Finally, we leave it to the reader to show that when using radian measure, two angles 2 and are coterminal if and only if = + 2k for some integer k. Example 10.1.3. Graph each of the (oriented) angles below in standard position and classify them according to where their terminal side lies. Find three coterminal angles, at least one of which is positive and one of which is negative.  4 9 5 1. = 2. = 3. = 4.  = 6 3 4 2 Solution.  1. The angle = is positive, so we draw an angle with its initial side on the positivex-axis and 6 (=6) 1 rotate counter-clockwise = of a revolution. Thus is a Quadrant I angle. Coterminal 2 12 angles  are of the form  = + 2k, for some integer k. To make the arithmetic a bit 12  12 13 easier, we note that 2 = , thus when k = 1, we get  = + = . Substituting 6 6 6 6  12 11  24 25 k =1 gives  = = and when we let k = 2, we get  = + = . 6 6 6 6 6 6 (4=3) 4 2 2. Since = is negative, we start at the positive x-axis and rotate clockwise = of 3 2 3 a revolution. We nd to be a Quadrant II angle. To nd coterminal angles, we proceed as 6 4 6 2 before using 2 = , and compute  = + k for integer values of k. We obtain , 3 3 3 3 10 8 and as coterminal angles. 3 3 13 The authors are well aware that we are now identifying radians with real numbers. We will justify this shortly. 14 This, in turn, endows the subtended arcs with an orientation as well. We address this in short order.10.1 Angles and their Measure 703 y y 4 4 3 3 2 2 1 1  = 6 x x 4321 1 2 3 4 4321 1 2 3 4 1 1 2 2 4 = 3 3 3 4 4  4 = in standard position. = in standard position. 6 3 9 3. Since = is positive, we rotate counter-clockwise from the positive x-axis. One full 4 8  1 revolution accounts for 2 = of the radian measure with or of a revolution remaining. 4 4 8 9 8 We have as a Quadrant I angle. All angles coterminal with are of the form = + k, 4 4  7 17 where k is an integer. Working through the arithmetic, we nd: , and . 4 4 4 5 4 4. To graph  = , we begin our rotation clockwise from the positive x-axis. As 2 = , 2 2  1 after one full revolution clockwise, we have or of a revolution remaining. Since the 2 4 terminal side of  lies on the negative y-axis,  is a quadrantal angle. To nd coterminal 5 4  3 7 angles, we compute  = + k for a few integers k and obtain , and . 2 2 2 2 2 y y 4 4 3 3 5  = 2 2 2 1 1 x x 4321 1 2 3 4 4321 1 2 3 4 1 1 2 2 9 = 4 3 3 4 4 9 5 = in standard position.  = in standard position. 4 2 It is worth mentioning that we could have plotted the angles in Example 10.1.3 by rst converting them to degree measure and following the procedure set forth in Example 10.1.2. While converting back and forth from degrees and radians is certainly a good skill to have, it is best that you learn to `think in radians' as well as you can `think in degrees'. The authors would, however, be704 Foundations of Trigonometry derelict in our duties if we ignored the basic conversion between these systems altogether. Since  one revolution counter-clockwise measures 360 and the same angle measures 2 radians, we can 2 radians  radians use the proportion , or its reduced equivalent, , as the conversion factor between   360 180     radians  the two systems. For example, to convert 60 to radians we nd 60 = radians, or  180 3   180 simply . To convert from radian measure back to degrees, we multiply by the ratio . For 3  radian    5 5 180  15 example, radians is equal to radians =150 . Of particular interest is the 6 6  radians  180  fact that an angle which measures 1 in radian measure is equal to  57:2958 .  We summarize these conversions below. Equation 10.1. Degree - Radian Conversion:  radians To convert degree measure to radian measure, multiply by  180  180 To convert radian measure to degree measure, multiply by  radians In light of Example 10.1.3 and Equation 10.1, the reader may well wonder what the allure of radian measure is. The numbers involved are, admittedly, much more complicated than degree measure. The answer lies in how easily angles in radian measure can be identi ed with real numbers. Consider 2 2 the Unit Circle,x +y = 1, as drawn below, the angle in standard position and the corresponding arc measuring s units in length. By de nition, and the fact that the Unit Circle has radius 1, the s s radian measure of  is = = s so that, once again blurring the distinction between an angle r 1 and its measure, we have  = s. 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). Viewing the vertical linex = 1 as another real number line demarcated like the y-axis, given a real number t 0, we `wrap' the (vertical) interval 0;t around the Unit Circle in a counter-clockwise fashion. The resulting arc has a length of t units and therefore the corresponding angle has radian measure equal to t. If t 0, we wrap the interval t; 0 clockwise around the Unit Circle. Since we have de ned clockwise rotation as having negative radian measure, the angle determined by this arc has radian measure equal to t. If t = 0, we are at the point (1; 0) on the x-axis which corresponds to an angle with radian measure 0. In this way, we identify each real number t with the corresponding angle with radian measure t. 15 Note that the negative sign indicates clockwise rotation in both systems, and so it is carried along accordingly.10.1 Angles and their Measure 705 y y y 1 1 1 s t  t x x x 1 1 1 t t On the Unit Circle,  =s. Identifying t 0 with an angle. Identifying t 0 with an angle. Example 10.1.4. Sketch the oriented arc on the Unit Circle corresponding to each of the following real numbers. 3 2. t =2 3. t =2 4. t = 117 1. t = 4 Solution. 3 3 1. The arc associated with t = is the arc on the Unit Circle which subtends the angle in 4 4 3 3 radian measure. Since is of a revolution, we have an arc which begins at the point (1; 0) 4 8 proceeds counter-clockwise up to midway through Quadrant II. 2. Since one revolution is 2 radians, and t =2 is negative, we graph the arc which begins at (1; 0) and proceeds clockwise for one full revolution. y y 1 1 x x 1 1 3 t = t =2 4 3. Liket =2,t =2 is negative, so we begin our arc at (1; 0) and proceed clockwise around  the unit circle. Since  3:14 and  1:57, we nd that rotating 2 radians clockwise from 2 the point (1; 0) lands us in Quadrant III. To more accurately place the endpoint, we proceed 5 as we did in Example 10.1.1, successively halving the angle measure until we nd  1:96 8 which tells us our arc extends just a bit beyond the quarter mark into Quadrant III.706 Foundations of Trigonometry 4. Since 117 is positive, the arc corresponding to t = 117 begins at (1; 0) and proceeds counter- clockwise. As 117 is much greater than 2, we wrap around the Unit Circle several times 117 before nally reaching our endpoint. We approximate as 18:62 which tells us we complete 2 5 18 revolutions counter-clockwise with 0:62, or just shy of of a revolution to spare. In other 8 words, the terminal side of the angle which measures 117 radians in standard position is just short of being midway through Quadrant III. y y 1 1 x x 1 1 t =2 t = 117 10.1.1 Applications of Radian Measure: Circular Motion Now that we have paired angles with real numbers via radian measure, a whole world of applications awaits us. Our rst excursion into this realm comes by way of 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. Q s  r P Heres represents a displacement so thats 0 means the object is traveling in a counter-clockwise direction and s 0 indicates movement in a clockwise direction. Note that with this convention s the formula we used to de ne radian measure, namely  = , still holds since a negative value r of s incurred from a clockwise displacement matches the negative we assign to  for a clockwise rotation. In Physics, the average velocity of the object, denotedv and read as `v-bar', is de ned 16 as the average rate of change of the position of the object with respect to time. As a result, we 16 See De nition 2.3 in Section 2.1 for a review of this concept.10.1 Angles and their Measure 707 s displacement length have v = = . The quantity v has units of and conveys two ideas: the direction time time t in which the object is moving and how fast the position of the object is changing. The contribution of direction in the quantityv is either to make it positive (in the case of counter-clockwise motion) or negative (in the case of clockwise motion), so that the quantityjvj quanti es how fast the object s is moving - it is the speed of the object. Measuring  in radians we have  = thus s =r and r s r  v = = =r t t t  The quantity is called the average angular velocity of the object. It is denoted by and is t read `omega-bar'. The quantity is the average rate of change of the angle  with respect to time radians and thus has units . If is constant throughout the duration of the motion, then it can be time 17 shown that the average velocities involved, namely v and , are the same as their instantaneous counterparts, v and , respectively. In this case, v is simply called the `velocity' of the object and 18 is the instantaneous rate of change of the position of the object with respect to time. Similarly, is called the `angular velocity' and is the instantaneous rate of change of the angle with respect to time. If the path of the object were `uncurled' from a circle to form a line segment, then the velocity of the object on that line segment would be the same as the velocity on the circle. For this reason, the quantity v is often called the linear velocity of the object in order to distinguish it from the angular velocity, . Putting together the ideas of the previous paragraph, we get the following. Equation 10.2. 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. length We need to talk about units here. The units of v are , the units of r are length only, and time length radians the units of are . Thus the left hand side of the equation v =r has units , whereas time time radians lengthradians the right hand side has units length = . The supposed contradiction in units is time time resolved by remembering that radians are a dimensionless quantity and angles in radian measure lengthradians length are identi ed with real numbers so that the units reduce to the units . We are time time long overdue for an example. Example 10.1.5. 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.  19 Lakeland Community College is at 41:628 north latitude, and it can be shown that the radius of the earth at this Latitude is approximately 2960 miles. Find the linear velocity, in miles per hour, of Lakeland Community College as the world turns. Solution. To use the formula v =r, we rst need to compute the angular velocity . The earth 2 radians  makes one revolution in 24 hours, and one revolution is 2 radians, so = = , 24 hours 12 hours 17 You guessed it, using Calculus . . . 18 See the discussion on Page 161 for more details on the idea of an `instantaneous' rate of change. 19 We will discuss how we arrived at this approximation in Example 10.2.6.708 Foundations of Trigonometry where, once again, we are using the fact that radians are real numbers and are dimensionless. (For simplicity's sake, we are also assuming that we are viewing the rotation of the earth as counter- clockwise so 0.) Hence, the linear velocity is  miles v = 2960 miles  775 12 hours hour 1 revolution It is worth noting that the quantity in Example 10.1.5 is called the ordinary frequency 24 hours of the motion and is usually denoted by the variable f. The ordinary frequency is a measure of how often an object makes a complete cycle of the motion. The fact that = 2f suggests that is also a frequency. Indeed, it is called the angular frequency of the motion. On a related note, 1 the quantity T = is called the period of the motion and is the amount of time it takes for the f object to complete one cycle of the motion. In the scenario of Example 10.1.5, the period of the motion is 24 hours, or one day. The concepts of frequency and period help frame the equation v =r in a new light. That is, if is xed, points which are farther from the center of rotation need to travel faster to maintain the same angular frequency since they have farther to travel to make one revolution in one period's time. The distance of the object to the center of rotation is the radius of the circle, r, and is the `magni cation factor' which relates and v. We will have more to say about frequencies and periods in Section 11.1. While we have exhaustively discussed velocities associated with circular motion, we have yet to discuss a more natural question: if an object is moving on a circular path of radiusr with a xed angular velocity (frequency), what is the position of the object at timet? The answer to this question is the very heart of Trigonometry and is answered in the next section.10.1 Angles and their Measure 709 10.1.2 Exercises In Exercises 1 - 4, convert the angles into the DMS system. Round each of your answers to the nearest second.     1. 63:75 2. 200:325 3. 317:06 4. 179:999 In Exercises 5 - 8, convert the angles into decimal degrees. Round each of your answers to three decimal places.  0  0 00  0  0 00 5. 125 50 6. 32 10 12 7. 502 35 8. 237 58 43 In Exercises 9 - 28, graph the oriented angle in standard position. Classify each angle according to where its terminal side lies and then give two coterminal angles, one of which is positive and the other negative.     9. 330 10. 135 11. 120 12. 405 5 11 5  13. 270 14. 15. 16. 6 3 4 3  7  17. 18. 19. 20. 4 3 2 4  7 5 21. 22. 23. 24. 3 2 6 3  15 13 25. 2 26. 27. 28. 4 4 6 In Exercises 29 - 36, convert the angle from degree measure into radian measure, giving the exact value in terms of .     29. 0 30. 240 31. 135 32. 270     33. 315 34. 150 35. 45 36. 225 In Exercises 37 - 44, convert the angle from radian measure into degree measure. 2 7 11 37.  38. 39. 40. 3 6 6  5   41. 42. 43. 44. 3 3 6 2710 Foundations of Trigonometry In Exercises 45 - 49, sketch the oriented arc on the Unit Circle which corresponds to the given real number. 5 45. t = 46. t = 47. t = 6 48. t =2 49. t = 12 6 50. A yo-yo which is 2.25 inches in diameter spins at a rate of 4500 revolutions per minute. How fast is the edge of the yo-yo spinning in miles per hour? Round your answer to two decimal places. 51. How many revolutions per minute would the yo-yo in exercise 50 have to complete if the edge of the yo-yo is to be spinning at a rate of 42 miles per hour? Round your answer to two decimal places. 52. In the yo-yo trick `Around the World,' the performer throws the yo-yo so it sweeps out a vertical circle whose radius is the yo-yo string. If the yo-yo string is 28 inches long and the yo-yo takes 3 seconds to complete one revolution of the circle, compute the speed of the yo-yo in miles per hour. Round your answer to two decimal places. 53. A computer hard drive contains a circular disk with diameter 2.5 inches and spins at a rate of 7200 RPM (revolutions per minute). Find the linear speed of a point on the edge of the disk in miles per hour. 54. A rock got stuck in the tread of my tire and when I was driving 70 miles per hour, the rock came loose and hit the inside of the wheel well of the car. How fast, in miles per hour, was the rock traveling when it came out of the tread? (The tire has a diameter of 23 inches.) 55. The Giant Wheel at Cedar Point is a circle with diameter 128 feet which sits on an 8 foot tall platform making its overall height is 136 feet. (Remember this from Exercise 17 in Section 20 7.2?) It completes two revolutions in 2 minutes and 7 seconds. Assuming the riders are at the edge of the circle, how fast are they traveling in miles per hour? 56. Consider the circle of radius r pictured below with central angle , measured in radians, and 1 2 subtended arc of length s. Prove that the area of the shaded sector is A = r . 2 A s (Hint: Use the proportion = .) area of the circle circumference of the circle s r  r 20 Source: Cedar Point's webpage.10.1 Angles and their Measure 711 In Exercises 57 - 62, use the result of Exercise 56 to compute the areas of the circular sectors with the given central angles and radii.  5  57.  = ; r = 12 58.  = ; r = 100 59.  = 330 ; r = 9:3 6 4   60.  =; r = 1 61.  = 240 ; r = 5 62.  = 1 ; r = 117 63. Imagine a rope tied around the Earth at the equator. Show that you need to add only 2 feet of length to the rope in order to lift it one foot above the ground around the entire equator. (You do NOT need to know the radius of the Earth to show this.) 64. With the help of your classmates, look for a proof that  is indeed a constant.712 Foundations of Trigonometry 10.1.3 Answers  0  0 00  0 00  0 00 1. 63 45 2. 200 19 30 3. 317 3 36 4. 179 59 56     5. 125:833 6. 32:17 7. 502:583 8. 237:979   9. 330 is a Quadrant IV angle 10. 135 is a Quadrant III angle     coterminal with 690 and30 coterminal with 225 and495 y y 4 4 3 3 2 2 1 1 x x 4321 1 2 3 4 4321 1 2 3 4 1 1 2 2 3 3 4 4   11. 120 is a Quadrant II angle 12. 405 is a Quadrant I angle     coterminal with 480 and240 coterminal with 45 and315 y y 4 4 3 3 2 2 1 1 x x 4321 1 2 3 4 4321 1 2 3 4 1 1 2 2 3 3 4 4 5  13. 270 lies on the positive y-axis 14. is a Quadrant II angle 6 17 7   coterminal with 90 and630 coterminal with and 6 6 y y 4 4 3 3 2 2 1 1 x x 4321 1 2 3 4 4321 1 2 3 4 1 1 2 2 3 3 4 4

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