1 0. The Unit Circle: Cosine and Sine The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on a circular path at a constant angular velocit. One of the goals of this section is describe the position of such an object. To that end, consider an angle in standard position and let P denote the point where the terminal side of intersects the Unit Circle. B associating the point P with the angle, we are assigning a position on the Unit Circle to the angle. The -coordinate of P is called the cosine of, written cos(, while the -coordinate of P is called the sine of, written sin(. The reader is encouraged to verif that these rules used to match an angle with its cosine and sine do, in fact, satisf the definition of a function. That is, for each angle, there is onl one associated value of cos( and onl one associated value of sin(. P (cos(, sin( Eample 0... Find the cosine and sine of the following angles.. = 70. =. = = 5. = 0 Solution.. To find cos (70 and sin (70, we plot the angle = 70 in standard position and find the point on the terminal side of which lies on the Unit Circle. Since 70 represents 4 of a counter-clockwise revolution, the terminal side of lies along the negative -ais. Hence, the point we seek is (0, so that cos ( ( = 0 and sin =.. The angle = represents one half of a clockwise revolution so its terminal side lies on the negative -ais. The point on the Unit Circle that lies on the negative -ais is (, 0 which means cos( = and sin( = 0. The etmolog of the name sine is quite colorful, and the interested reader is invited to research it; the co in cosine is eplained in Section 0.4.
2 78 Foundations of Trigonometr = 70 P (, 0 = P (0, Finding cos (70 and sin (70 Finding cos ( and sin (. When we sketch = 45 in standard position, we see that its terminal does not lie along an of the coordinate aes which makes our job of finding the cosine and sine values a bit more difficult. Let P (, denote the point on the terminal side of which lies on the Unit Circle. B definition, = cos (45 and = sin (45. If we drop a perpendicular line segment from P to the -ais, we obtain a right triangle whose legs have lengths and units. From Geometr, we get =. Since P (, lies on the Unit Circle, we have + =. Substituting = into this equation ields =, or = ± Since P (, lies in the first quadrant, > 0, so = cos (45 = = sin (45 =. = ±. and with = we have P (, P (, = = 45 Can ou show this?
3 0. The Unit Circle: Cosine and Sine As before, the terminal side of = does not lie on an of the coordinate aes, so we proceed using a triangle approach. Letting P (, denote the point on the terminal side of which lies on the Unit Circle, we drop a perpendicular line segment from P to the -ais to form a right triangle. After a bit of Geometr we find = so sin ( =. Since P (, lies on the Unit Circle, we substitute = into + = to get = 4, or = ±. Here, > 0 so = cos ( =. P (, P (, = 0 = = 0 5. Plotting = 0 in standard position, we find it is not a quadrantal angle and set about using a triangle approach. Once again, we get a right triangle and, after the usual computations, find = cos (0 = and = sin (0 = P (,. P (, = 0 0 = 0 Again, can ou show this?
4 70 Foundations of Trigonometr In Eample 0.., it was quite eas to find the cosine and sine of the quadrantal angles, but for non-quadrantal angles, the task was much more involved. In these latter cases, we made good use of the fact that the point P (, = (cos(, sin( lies on the Unit Circle, + =. If we substitute = cos( and = sin( into + =, we get (cos( + (sin( =. An unfortunate 4 convention, which the authors are compelled to perpetuate, is to write (cos( as cos ( and (sin( as sin (. Rewriting the identit using this convention results in the following theorem, which is without a doubt one of the most important results in Trigonometr. Theorem 0.. The Pthagorean Identit: For an angle, cos ( + sin ( =. The moniker Pthagorean brings to mind the Pthagorean Theorem, from which both the Distance Formula and the equation for a circle are ultimatel derived. 5 The word Identit reminds us that, regardless of the angle, the equation in Theorem 0. is alwas true. If one of cos( or sin( is known, Theorem 0. can be used to determine the other, up to a (± sign. If, in addition, we know where the terminal side of lies when in standard position, then we can remove the ambiguit of the (± and completel determine the missing value as the net eample illustrates. Eample 0... Using the given information about, find the indicated value.. If is a Quadrant II angle with sin( = 5, find cos(.. If < < with cos( = 5 5, find sin(.. If sin( =, find cos(. Solution.. When we substitute sin( = 5 into The Pthagorean Identit, cos ( + sin ( =, we obtain cos ( =. Solving, we find cos( = ± 4 5. Since is a Quadrant II angle, its terminal side, when plotted in standard position, lies in Quadrant II. Since the -coordinates are negative in Quadrant II, cos( is too. Hence, cos( = Substituting cos( = 5 5 into cos ( + sin ( = gives sin( = ± 5 = ± 5 5. Since we are given that < <, we know is a Quadrant III angle. Hence both its sine and cosine are negative and we conclude sin( = When we substitute sin( = into cos ( + sin ( =, we find cos( = 0. Another tool which helps immensel in determining cosines and sines of angles is the smmetr inherent in the Unit Circle. Suppose, for instance, we wish to know the cosine and sine of = 5. We plot in standard position below and, as usual, let P (, denote the point on the terminal side of which lies on the Unit Circle. Note that the terminal side of lies half revolution. In Eample 0.., we determined that cos ( = and sin ( 4 This is unfortunate from a function notation perspective. See Section See Sections. and 7. for details. radians short of one =. This means
5 0. The Unit Circle: Cosine and Sine 7 that the point on the terminal side of the angle, when plotted in standard position, is ( From the figure below, it is clear that the point P (, we seek can be obtained b reflecting that point about the -ais. Hence, cos ( 5 = and sin ( 5 =.,. P (, = 5 P, = 5, In the above scenario, the angle is called the reference angle for the angle 5. In general, for a non-quadrantal angle, the reference angle for (usuall denoted is the acute angle made between the terminal side of and the -ais. If is a Quadrant I or IV angle, is the angle between the terminal side of and the positive -ais; if is a Quadrant II or III angle, is the angle between the terminal side of and the negative -ais. If we let P denote the point (cos(, sin(, then P lies on the Unit Circle. Since the Unit Circle possesses smmetr with respect to the -ais, -ais and origin, regardless of where the terminal side of lies, there is a point Q smmetric with P which determines s reference angle, as seen below. P = Q P Q Reference angle for a Quadrant I angle Reference angle for a Quadrant II angle
6 7 Foundations of Trigonometr Q Q P P Reference angle for a Quadrant III angle Reference angle for a Quadrant IV angle We have just outlined the proof of the following theorem. Theorem 0.. 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. In light of Theorem 0., it pas to know the cosine and sine values for certain common angles. In the table below, we summarize the values which we consider essential and must be memorized. Cosine and Sine Values of Common Angles (degrees (radians cos( sin( Eample 0... Find the cosine and sine of the following angles.. = 5. =. = = 7 Solution.. We begin b plotting = 5 in standard position and find its terminal side overshoots the negative -ais to land in Quadrant III. Hence, we obtain s reference angle b subtracting: = 80 = 5 80 = 45. Since is a Quadrant III angle, both cos( < 0 and
7 0. The Unit Circle: Cosine and Sine 7 sin( < 0. The Reference Angle Theorem ields: cos (5 = cos (45 = sin (5 = sin (45 =. and. The terminal side of =, when plotted in standard position, lies in Quadrant IV, just sh of the positive -ais. To find s reference angle, we subtract: = = =. Since is a Quadrant IV angle, cos( > 0 and sin( < 0, so the Reference Angle Theorem gives: cos ( ( = cos = and sin ( ( = sin =. = 5 45 = Finding cos (5 and sin (5 Finding cos ( ( and sin. To plot = 5 4, we rotate clockwise an angle of 5 4 from the positive -ais. The terminal side of, therefore, lies in Quadrant II making an angle of = 5 4 = 4 radians with respect to the negative -ais. Since is a Quadrant II angle, the Reference Angle Theorem gives: cos ( 5 ( 4 = cos 4 = and sin ( 5 ( 4 = sin 4 =. 4. Since the angle = 7 measures more than =, we find the terminal side of b rotating one full revolution followed b an additional = 7 = radians. Since and are coterminal, cos ( ( 7 = cos = and sin ( ( 7 = sin =. 4 = 7 = 5 4 Finding cos ( 5 4 ( and sin 5 4 Finding cos ( ( 7 and sin 7
8 74 Foundations of Trigonometr The reader ma have noticed that when epressed in radian measure, the reference angle for a non-quadrantal angle is eas to spot. Reduced fraction multiples of with a denominator of have as a reference angle, those with a denominator of 4 have 4 as their reference angle, and those with a denominator of have as their reference angle. The Reference Angle Theorem in conjunction with the table of cosine and sine values on Page 7 can be used to generate the following figure, which the authors feel should be committed to memor. (, (, ( 5, 4 (0, (, 4 (, (, (, 0 0, (, 0 (, (, ( 7 5 4, ( (, (,, (0, Important Points on the Unit Circle For once, we have something convenient about using radian measure in contrast to the abstract theoretical nonsense about using them as a natural wa to match oriented angles with real numbers!
9 0. The Unit Circle: Cosine and Sine 75 The net eample summarizes all of the important ideas discussed thus far in the section. Eample Suppose is an acute angle with cos( = 5.. Find sin( and use this to plot in standard position.. Find the sine and cosine of the following angles: (a = + (b = (c = (d = + Solution.. Proceeding as in Eample 0.., we substitute cos( = 5 into cos ( + sin ( = and find sin( = ±. Since is an acute (and therefore Quadrant I angle, sin( is positive. Hence, sin( =. To plot in standard position, we begin our rotation on the positive -ais to the ra which contains the point (cos(, sin( = ( 5,. ( 5, Sketching. (a To find the cosine and sine of = +, we first plot in standard position. We can imagine the sum of the angles + as a sequence of two rotations: a rotation of radians followed b a rotation of radians. 7 We see that is the reference angle for, so b The Reference Angle Theorem, cos( = ± cos( = ± 5 and sin( = ± sin( = ±. Since the terminal side of falls in Quadrant III, both cos( and sin( are negative, hence, cos( = 5 and sin( =. 7 Since + = +, ma be plotted b reversing the order of rotations given here. You should do this.
10 7 Foundations of Trigonometr Visualizing = + has reference angle (b Rewriting = as = + (, we can plot b visualizing one complete revolution counter-clockwise followed b a clockwise revolution, or backing up, of radians. We see that is s reference angle, and since is a Quadrant IV angle, the Reference Angle Theorem gives: cos( = 5 and sin( =. Visualizing = has reference angle (c Taking a cue from the previous problem, we rewrite = as = + (. The angle represents one and a half revolutions counter-clockwise, so that when we back up radians, we end up in Quadrant II. Using the Reference Angle Theorem, we get cos( = 5 and sin( =.
11 0. The Unit Circle: Cosine and Sine 77 Visualizing has reference angle (d To plot = +, we first rotate radians and follow up with radians. The reference angle here is not, so The Reference Angle Theorem is not immediatel applicable. (It s important that ou see wh this is the case. Take a moment to think about this before reading on. Let Q(, be the point on the terminal side of which lies on the Unit Circle so that = cos( and = sin(. Once we graph in standard position, we use the fact that equal angles subtend equal chords to show that the dotted lines in the figure below are equal. Hence, = cos( = 5. Similarl, we find = sin( =. P ( 5, Q (, Visualizing = + Using smmetr to determine Q(,
12 78 Foundations of Trigonometr Our net eample asks us to solve some ver basic trigonometric equations. 8 Eample Find all of the angles which satisf the given equation.. cos( =. sin( =. cos( = 0. Solution. Since there is no contet in the problem to indicate whether to use degrees or radians, we will default to using radian measure in our answers to each of these problems. This choice will be justified later in the tet when we stud what is known as Analtic Trigonometr. In those sections to come, radian measure will be the onl appropriate angle measure so it is worth the time to become fluent in radians now.. If cos( =, then the terminal side of, when plotted in standard position, intersects the Unit Circle at =. This means is a Quadrant I or IV angle with reference angle. One solution in Quadrant I is =, and since all other Quadrant I solutions must be coterminal with, we find = +k for integers k.9 Proceeding similarl for the Quadrant IV case, we find the solution to cos( = here is 5, so our answer in this Quadrant is = 5 + k for integers k.. If sin( =, then when is plotted in standard position, its terminal side intersects the Unit Circle at =. From this, we determine is a Quadrant III or Quadrant IV angle with reference angle. 8 We will more formall stud of trigonometric equations in Section 0.7. Enjo these relativel straightforward eercises while the last! 9 Recall in Section 0., two angles in radian measure are coterminal if and onl if the differ b an integer multiple of. Hence to describe all angles coterminal with a given angle, we add k for integers k = 0, ±, ±,....
13 0. The Unit Circle: Cosine and Sine 79 In Quadrant III, one solution is 7, so we capture all Quadrant III solutions b adding integer multiples of : = 7 + k. In Quadrant IV, one solution is so all the solutions here are of the form = + k for integers k.. The angles with cos( = 0 are quadrantal angles whose terminal sides, when plotted in standard position, lie along the -ais. While, technicall speaking, isn t a reference angle we can nonetheless use it to find our answers. If we follow the procedure set forth in the previous eamples, we find = + k and = + k for integers, k. While this solution is correct, it can be shortened to = + k for integers k. (Can ou see wh this works from the diagram? One of the ke items to take from Eample 0..5 is that, in general, solutions to trigonometric equations consist of infinitel man answers. To get a feel for these answers, the reader is encouraged to follow our mantra from Chapter 9 - that is, When in doubt, write it out! This is especiall important when checking answers to the eercises. For eample, another Quadrant IV solution to sin( = is =. Hence, the famil of Quadrant IV answers to number above could just have easil been written = + k for integers k. While on the surface, this famil ma look
14 70 Foundations of Trigonometr different than the stated solution of = + k for integers k, we leave it to the reader to show the represent the same list of angles. 0.. Beond the Unit Circle We began the section with a quest to describe the position of a particle eperiencing circular motion. In defining the cosine and sine functions, we assigned to each angle a position on the Unit Circle. In this subsection, we broaden our scope to include circles of radius r centered at the origin. Consider for the moment the acute angle drawn below in standard position. Let Q(, be the point on the terminal side of which lies on the circle + = r, and let P (, be the point on the terminal side of which lies on the Unit Circle. Now consider dropping perpendiculars from P and Q to create two right triangles, OP A and OQB. These triangles are similar, 0 thus it follows that = r = r, so = r and, similarl, we find = r. Since, b definition, = cos( and = sin(, we get the coordinates of Q to be = r cos( and = r sin(. B reflecting these points through the -ais, -ais and origin, we obtain the result for all non-quadrantal angles, and we leave it to the reader to verif these formulas hold for the quadrantal angles. r Q (, P (, r P (, Q(, = (r cos(, r sin( O A(, 0 B(, 0 Not onl can we describe the coordinates of Q in terms of cos( and sin( but since the radius of the circle is r = +, we can also epress cos( and sin( in terms of the coordinates of Q. These results are summarized in the following theorem. Theorem 0.. If Q(, is the point on the terminal side of an angle, plotted in standard position, which lies on the circle + = r then = r cos( and = r sin(. Moreover, cos( = r = and sin( = + r = + 0 Do ou remember wh?
15 0. The Unit Circle: Cosine and Sine 7 Note that in the case of the Unit Circle we have r = + =, so Theorem 0. reduces to our definitions of cos( and sin(. Eample Suppose that the terminal side of an angle, when plotted in standard position, contains the point Q(4,. Find sin( and cos(.. In Eample 0..5 in Section 0., we approimated the radius of the earth at 4.8 north latitude to be 90 miles. Justif this approimation if the radius of the Earth at the Equator is approimatel 90 miles. Solution.. Using Theorem 0. with = 4 and =, we find r = (4 + ( = 0 = 5 so that cos( = r = 4 = and = r = = Assuming the Earth is a sphere, a cross-section through the poles produces a circle of radius 90 miles. Viewing the Equator as the -ais, the value we seek is the -coordinate of the point Q(, indicated in the figure below Q(4, 4.8 Q (, 90 4 The terminal side of contains Q(4, A point on the Earth at 4.8 N Using Theorem 0., we get = 90 cos (4.8. Using a calculator in degree mode, we find 90 cos ( Hence, the radius of the Earth at North Latitude 4.8 is approimatel 90 miles.
16 7 Foundations of Trigonometr Theorem 0. gives us what we need to describe the position of an object traveling in a circular path of radius r with constant angular velocit ω. Suppose that at time t, the object has swept out an angle measuring radians. If we assume that the object is at the point (r, 0 when t = 0, the angle is in standard position. B definition, ω = t which we rewrite as = ωt. According to Theorem 0., the location of the object Q(, on the circle is found using the equations = r cos( = r cos(ωt and = r sin( = r sin(ωt. Hence, at time t, the object is at the point (r cos(ωt, r sin(ωt. We have just argued the following. Equation 0.. Suppose an object is traveling in a circular path of radius r centered at the origin with constant angular velocit ω. If t = 0 corresponds to the point (r, 0, then the and coordinates of the object are functions of t and are given b = r cos(ωt and = r sin(ωt. Here, ω > 0 indicates a counter-clockwise direction and ω < 0 indicates a clockwise direction. r Q (, = (r cos(ωt, r sin(ωt = ωt r Equations for Circular Motion Eample Suppose we are in the situation of Eample Find the equations of motion of Lakeland Communit College as the earth rotates. Solution. From Eample 0..5, we take r = 90 miles and and ω = hours. Hence, the equations of motion are = r cos(ωt = 90 cos ( t and = r sin(ωt = 90 sin ( t, where and are measured in miles and t is measured in hours. In addition to circular motion, Theorem 0. is also the ke to developing what is usuall called right triangle trigonometr. As we shall see in the sections to come, man applications in trigonometr involve finding the measures of the angles in, and lengths of the sides of, right triangles. Indeed, we made good use of some properties of right triangles to find the eact values of the cosine and sine of man of the angles in Eample 0.., so the following development shouldn t be that much of a surprise. Consider the generic right triangle below with corresponding acute angle. The side with length a is called the side of the triangle adjacent to ; the side with length b is called the side of the triangle opposite ; and the remaining side of length c (the side opposite the You ma have been eposed to this in High School.
17 0. The Unit Circle: Cosine and Sine 7 right angle is called the hpotenuse. We now imagine drawing this triangle in Quadrant I so that the angle is in standard position with the adjacent side to ling along the positive -ais. c P (a, b c b c a According to the Pthagorean Theorem, a + b = c, so that the point P (a, b lies on a circle of radius c. Theorem 0. tells us that cos( = a c and sin( = b c, so we have determined the cosine and sine of in terms of the lengths of the sides of the right triangle. Thus we have the following theorem. Theorem 0.4. 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 hpotenuse is c, then cos( = a c and sin( = b c. Eample Find the measure of the missing angle and the lengths of the missing sides of: 0 Solution. The first and easiest task is to find the measure of the missing angle. Since the sum of angles of a triangle is 80, we know that the missing angle has measure = 0. We now proceed to find the lengths of the remaining two sides of the triangle. Let c denote the length of the hpotenuse of the triangle. B Theorem 0.4, we have cos (0 = 7 c, or c = 7 7 cos(0. Since cos (0 =, we have, after the usual fraction gmnastics, c = 4. At this point, we have two was to proceed to find the length of the side opposite the 0 angle, which we ll denote b. We know the length of the adjacent side is 7 and the length of the hpotenuse is 4, so we
18 74 Foundations of Trigonometr ( could use the Pthagorean Theorem to find the missing side and solve (7 + b = 4 for b. Alternativel, we could use Theorem 0.4, namel that sin (0 = b c. Choosing the latter, we find b = c sin (0 = 4 = 7. The triangle with all of its data is recorded below. c = 4 0 b = We close this section b noting that we can easil etend the functions cosine and sine to real numbers b identifing a real number t with the angle = t radians. Using this identification, we define cos(t = cos( and sin(t = sin(. In practice this means epressions like cos( and sin( can be found b regarding the inputs as angles in radian measure or real numbers; the choice is the reader s. If we trace the identification of real numbers t with angles in radian measure to its roots on page 704, we can spell out this correspondence more precisel. For each real number t, we associate an oriented arc t units in length with initial point (, 0 and endpoint P (cos(t, sin(t. P (cos(t, sin(t = t t = t In the same wa we studied polnomial, rational, eponential, and logarithmic functions, we will stud the trigonometric functions f(t = cos(t and g(t = sin(t. The first order of business is to find the domains and ranges of these functions. Whether we think of identifing the real number t with the angle = t radians, or think of wrapping an oriented arc around the Unit Circle to find coordinates on the Unit Circle, it should be clear that both the cosine and sine functions are defined for all real numbers t. In other words, the domain of f(t = cos(t and of g(t = sin(t is (,. Since cos(t and sin(t represent - and -coordinates, respectivel, of points on the Unit Circle, the both take on all of the values between an, inclusive. In other words, the range of f(t = cos(t and of g(t = sin(t is the interval [, ]. To summarize:
19 0. The Unit Circle: Cosine and Sine 75 Theorem 0.5. Domain and Range of the Cosine and Sine Functions: The function f(t = cos(t The function g(t = sin(t has domain (, has domain (, has range [, ] has range [, ] Suppose, as in the Eercises, we are asked to solve an equation such as sin(t =. As we have alread mentioned, the distinction between t as a real number and as an angle = t radians is often blurred. Indeed, we solve sin(t = in the eact same manner as we did in Eample 0..5 number. Our solution is onl cosmeticall different in that the variable used is t rather than : t = 7 + k or t = + k for integers, k. We will stud the cosine and sine functions in greater detail in Section 0.5. Until then, keep in mind that an properties of cosine and sine developed in the following sections which regard them as functions of angles in radian measure appl equall well if the inputs are regarded as real numbers. Well, to be pedantic, we would be technicall using reference numbers or reference arcs instead of reference angles but the idea is the same.
20 7 Foundations of Trigonometr 0.. Eercises In Eercises - 0, find the eact value of the cosine and sine of the given angle.. = 0. = 4. = 4. = 5. =. = 4 7. = 8. = 7 9. = = 4. =. = 5. = = 5. =. = 4 7. = 4 8. = 9. = 0 0. = 7 In Eercises - 0, use the results developed throughout the section to find the requested value.. If sin( = 7 5 with in Quadrant IV, what is cos(?. If cos( = 4 9 with in Quadrant I, what is sin(?. If sin( = 5 with in Quadrant II, what is cos(? 4. If cos( = with in Quadrant III, what is sin(? 5. If sin( = with in Quadrant III, what is cos(?. If cos( = 8 5 with in Quadrant IV, what is sin(? 7. If sin( = 5 5 and < <, what is cos(? 8. If cos( = 0 0 and < < 5, what is sin(? 9. If sin( = 0.4 and < <, what is cos(? 0. If cos( = 0.98 and < <, what is sin(?
21 0 0. The Unit Circle: Cosine and Sine 77 In Eercises - 9, find all of the angles which satisf the given equation.. sin( =. cos( =. sin( = 0 4. cos( = 5. sin( =. cos( = 7. sin( = 8. cos( = 9. cos( =.00 In Eercises 40-48, solve the equation for t. (See the comments following Theorem cos(t = 0 4. sin(t = 4. cos(t = 4. sin(t = 44. cos(t = 45. sin(t = 4. cos(t = 47. sin(t = 48. cos(t = In Eercises 49-54, use our calculator to approimate the given value to three decimal places. Make sure our calculator is in the proper angle measurement mode! 49. sin( cos(.0 5. sin( cos(07 5. sin ( 54. cos(e In Eercises 55-58, find the measurement of the missing angle and the lengths of the missing sides. (See Eample Find, b, and c. 5. Find, a, and c. c b a 45 c
22 78 Foundations of Trigonometr 57. Find, a, and b. b a Find β, a, and c. a 48 c β In Eercises 59-4, assume that is an acute angle in a right triangle and use Theorem 0.4 to find the requested side. 59. If = and the side adjacent to has length 4, how long is the hpotenuse? 0. If = 78. and the hpotenuse has length 580, how long is the side adjacent to?. If = 59 and the side opposite has length 7.4, how long is the hpotenuse?. If = 5 and the hpotenuse has length 0, how long is the side opposite?. If = 5 and the hpotenuse has length 0, how long is the side adjacent to? 4. If = 7.5 and the side opposite has length 0, how long is the side adjacent to? In Eercises 5-8, let be the angle in standard position whose terminal side contains the given point then compute cos( and sin(. 5. P ( 7, 4. Q(, 4 7. R(5, 9 8. T (, In Eercises 9-7, find the equations of motion for the given scenario. Assume that the center of the motion is the origin, the motion is counter-clockwise and that t = 0 corresponds to a position along the positive -ais. (See Equation 0. and Eample A point on the edge of the spinning o-o in Eercise 50 from Section 0.. Recall: The diameter of the o-o is.5 inches and it spins at 4500 revolutions per minute. 70. The o-o in eercise 5 from Section 0.. Recall: The radius of the circle is 8 inches and it completes one revolution in seconds. 7. A point on the edge of the hard drive in Eercise 5 from Section 0.. Recall: The diameter of the hard disk is.5 inches and it spins at 700 revolutions per minute.
23 0. The Unit Circle: Cosine and Sine A passenger on the Big Wheel in Eercise 55 from Section 0.. Recall: The diameter is 8 feet and completes revolutions in minutes, 7 seconds. 7. Consider the numbers: 0,,,, 4. Take the square root of each of these numbers, then divide each b. The resulting numbers should look hauntingl familiar. (See the values in the table on Let and β be the two acute angles of a right triangle. (Thus and β are complementar angles. Show that sin( = cos(β and sin(β = cos(. The fact that co-functions of complementar angles are equal in this case is not an accident and a more general result will be given in Section In the scenario of Equation 0., we assumed that at t = 0, the object was at the point (r, 0. If this is not the case, we can adjust the equations of motion b introducing a time dela. If t 0 > 0 is the first time the object passes through the point (r, 0, show, with the help of our classmates, the equations of motion are = r cos(ω(t t 0 and = r sin(ω(t t 0.
24 740 Foundations of Trigonometr 0.. Answers ( (. cos(0 =, sin(0 = 0. cos = 4, sin = 4 (. cos = ( ( (, sin = 4. cos = 0, sin = ( 5. cos = ( ( (, sin =. cos = 4, sin = 4 ( ( cos( =, sin( = 0 8. cos =, sin = ( ( ( cos = 4, sin = 0. cos = ( 4 4, sin = ( ( ( 5. cos = 0, sin =. cos = ( 5, sin = ( ( ( ( 7 7. cos = 4, sin = 4. cos = 4, sin = ( 5. cos ( = 0, sin ( =. cos 4 ( =, sin 4 = ( 7. cos ( = 4, sin ( = 8. cos = ( 4, sin = ( 0 9. cos = ( 0, sin = 0. cos(7 =, sin(7 = 0. If sin( = 7 5 with in Quadrant IV, then cos( = If cos( = 4 9 with in Quadrant I, then sin( = If sin( = 5 with in Quadrant II, then cos( =. 4. If cos( = with in Quadrant III, then sin( = If sin( = with in Quadrant III, then cos( = 5.. If cos( = 8 5 with in Quadrant IV, then sin( = 45 5.
25 0. The Unit Circle: Cosine and Sine If sin( = If cos( = and < <, then cos( = and < < 5 0, then sin( = If sin( = 0.4 and < <, then cos( = If cos( = 0.98 and < <, then sin( = sin( = when = + k or = 5 + k for an integer k.. cos( = when = 5 + k or = 7. sin( = 0 when = k for an integer k. + k for an integer k. 4. cos( = when = 4 + k or = 7 + k for an integer k sin( = when = + k or = + k for an integer k.. cos( = when = (k + for an integer k. 7. sin( = when = + k for an integer k. 8. cos( = when = + k or = + k for an integer k. 9. cos( =.00 never happens 40. cos(t = 0 when t = + k for an integer k. 4. sin(t = when t = k or t = cos(t = never happens. 4. sin(t = when t = 7 + k or t = + k for an integer k. + k for an integer k. 44. cos(t = when t = + k or t = 5 + k for an integer k. 45. sin(t = never happens 4. cos(t = when t = k for an integer k.
26 74 Foundations of Trigonometr 47. sin(t = when t = + k for an integer k. 48. cos(t = when t = 4 + k or t = k for an integer k. 49. sin( cos( sin( cos( sin ( cos(e = 0, b =, c = 5. = 45, a =, c = 57. = 57, a = 8 cos(.709, b = 8 sin( β = 4, c = 59. The hpotenuse has length sin( , a = c cos( The side adjacent to has length 580 cos( The hpotenuse has length 7.4 sin( The side opposite has length 0 sin( The side adjacent to has length 0 cos( The hpotenuse has length c = c cos( = 7 4, sin( = sin( , so the side adjacent to has length. cos( = 5, sin( = cos( = 5 0 0, sin( = cos( = 5 5, sin( = r =.5 inches, ω = 9000 radians minute, =.5 cos(9000 t, =.5 sin(9000 t. Here and are measured in inches and t is measured in minutes.
27 0. The Unit Circle: Cosine and Sine 74 radians 70. r = 8 inches, ω = second, = 8 cos ( t, = 8 sin ( t. Here and are measured in inches and t is measured in seconds. 7. r =.5 inches, ω = 4400 radians minute, =.5 cos(4400 t, =.5 sin(4400 t. Here and are measured in inches and t is measured in minutes. 7. r = 4 feet, ω = 4 radians 7 second, = 4 cos ( 4 7 t, = 4 sin ( 4 7 t. Here and are measured in feet and t is measured in seconds
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
790 Foundations of Trigonometr 0.5 Graphs of the Trigonometric Functions In this section, we return to our discussion of the circular trigonometric functions as functions of real numbers and pick up where
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
Triangle Definition of sin and cos Then Consider the triangle ABC below. Let A be called. A HYP (hpotenuse) ADJ (side adjacent to the angle ) B C OPP (side opposite to the angle ) sin OPP HYP BC AB ADJ
Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions
0.6 The Inverse Trigonometric Functions 70 0.6 The Inverse Trigonometric Functions As the title indicates, in this section we concern ourselves with finding inverses of the (circular) trigonometric functions.
0606_CH0_-78.QXP //0 :6 AM Page Unit Circle Trigonometr In This Chapter. The Circular Functions. Graphs of Sine and Cosine Functions. Graphs of Other Trigonometric Functions. Special Identities.5 Inverse
LIALMC5_1768.QXP /6/4 1:4 AM Page 47 5 Trigonometric Functions Highwa transportation is critical to the econom of the United States. In 197 there were 115 billion miles traveled, and b the ear this increased
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
.6 Graphs of Functions 9.6 Graphs of Functions In Section. we defined a function as a special tpe of relation; one in which each -coordinate was matched with onl one -coordinate. We spent most of our time
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that
B_Chap0_08-05.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
Inverse Trigonometric Functions: We d like to eplore the question of inverses of the sine, tangent, and secant functions. We ll start with f ( ) sin. Recall the graph: MATH 30 Lecture 0 of 0 Well, we can
. Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the
8.7 Sstems of Non-Linear Equations and Inequalities 67 8.7 Sstems of Non-Linear Equations and Inequalities In this section, we stud sstems of non-linear equations and inequalities. Unlike the sstems of
78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
CHAPTER 5 Change of Coordinates in Two Dimensions Suppose that E is an ellipse centered at the origin. If the major and minor aes are horiontal and vertical, as in figure 5., then the equation of the ellipse
Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle - in particular
SECTION 6.3 Parametric Equations and Motion 475 What ou ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher... and wh
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
0_007.qd /7/05 :0 AM Page Section.7.7 Inverse Trigonometric Functions Inverse Trigonometric Functions What ou should learn Evaluate and graph the inverse sine function. Evaluate and graph the other inverse
Section The Circle 65 Dollars Purchase price P Book value = f(t) Salvage value S Useful life L Years t FIGURE 3 Straight-line depreciation. The Circle Definition Anone who has drawn a circle using a compass
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
Chapter 5 Trigonometric Functions 75 5. Unit Circle: Sine and Cosine Functions In this section, you will: Learning Objectives 5..1 Find function values for the sine and cosine of 0 or π 6, 45 or π 4 and
Trigonometry Notes Sarah Brewer Alabama School of Math and Science Last Updated: 25 November 2011 6 Basic Trig Functions Defined as ratios of sides of a right triangle in relation to one of the acute angles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
6.1 Angle Measure Chapter 6 Trigonometric Functions of Angles In Chapter 5, we looked at trig functions in terms of real numbers t, as determined by the coordinates of the terminal point on the unit circle.
COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two
Right Triangles and SOHCAHTOA: Finding the Length of a Side Given One Side and One Angle Preliminar Information: is an acronm to represent the following three trigonometric ratios or formulas: opposite
. EVALUATIN F TIGNMETIC FUNCTINS In this section, we obtain values of the trigonometric functions for quadrantal angles, we introduce the idea of reference angles, and we discuss the use of a calculator
Math, Trigonometr and Vectors Geometr 33º What is the angle equal to? a) α = 7 b) α = 57 c) α = 33 d) α = 90 e) α cannot be determined α Trig Definitions Here's a familiar image. To make predictive models
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures. In this brief Section we discuss the basic coordinate geometr of a circle - in particular the basic equation representing
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the
DEFINITIONS: An angle is defined as the set of points determined by two rays, or half-lines, l 1 and l having the same end point O. An angle can also be considered as two finite line segments with a common
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
The Area Bounded b a Curve 14.3 Introduction One of the important applications of integration is to find the area bounded b a curve. Often such an area can have a phsical significance like the work done
Section 0.2 Set notation and solving inequalities (5/31/07) Overview: Inequalities are almost as important as equations in calculus. Man functions domains are intervals, which are defined b inequalities.
A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which
bar5969_ch05_463-478.qd 0/5/008 06:53 PM Page 463 pinnacle 0:MHIA065:SE:CH 05: CHAPTER 5 Trigonometric Functions C TRIGONOMETRIC functions seem to have had their origins with the Greeks investigation of
Math 115 Spring 014 Written Homework 10-SOLUTIONS Due Friday, April 5 1. Use the following graph of y = g(x to answer the questions below (this is NOT the graph of a rational function: (a State the domain
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
P a g e 40 Chapter 5 The Trigonometric Functions Section 5.1 Angles Initial side Terminal side Standard position of an angle Positive angle Negative angle Coterminal Angles Acute angle Obtuse angle Complementary
. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine
1 (a) Find the curvature κ(t) of the curve r(t) = cos t, sin t, t at the point corresponding to t = Hint: You ma use the two formulas for the curvature κ(t) = T (t) r (t) = r (t) r (t) r (t) 3 Solution:
Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight
6 Polar Coordinates Section 6 Notes Page 1 In this section we will learn a new coordinate sstem In this sstem we plot a point in the form r, As shown in the picture below ou first draw angle in standard
7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work
Trigonometric Identities and Conditional Equations C TRIGONOMETRIC functions are widely used in solving real-world problems and in the development of mathematics. Whatever their use, it is often of value
88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative
2- Combining Functions 16 91. Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps
CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that
0 YEAR The Improving Mathematics Education in Schools (TIMES) Project TRIGNMETRIC FUNCTINS MEASUREMENT AND GEMETRY Module 25 A guide for teachers - Year 0 June 20 Trigonometric Functions (Measurement and
chapter Algebra Ke words substitution discriminant completing the square real and distinct imaginar rational verte parabola maimum minimum surd irrational rationalising the denominator Section. Quadratic
- Chapter 8A Chapter 8A - Angles and Circles Man applications of calculus use trigonometr, which is the stud of angles and functions of angles and their application to circles, polgons, and science. We
Chapter TRIGONOMETRIC FUNCTIONS.1 Introduction A mathematician knows how to solve a problem, he can not solve it. MILNE The word trigonometr is derived from the Greek words trigon and metron and it means
Lesson 6. Eercises, pages 7 80 A. Use technolog to determine the value of each trigonometric ratio to the nearest thousandth. a) sin b) cos ( 6 ) c) cot 7 d) csc 8 0.89 0. tan 7 sin 8 0..0. Sketch each
060_CH03_13-154.QXP 10/9/10 10:56 AM Page 13 3 Rectangular Coordinate Sstem and Graphs In This Chapter 3.1 The Rectangular Coordinate Sstem 3. Circles and Graphs 3.3 Equations of Lines 3.4 Variation Chapter
7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of
Math Learning Centre Curve Sketching A good graphing calculator can show ou the shape of a graph, but it doesn t alwas give ou all the useful information about a function, such as its critical points and
Hperbolic functions The hperbolic functions have similar names to the trigonmetric functions, but the are defined in terms of the eponential function In this unit we define the three main hperbolic functions,
5.5 Modelling Harmonic Motion Periodic behaviour happens a lot in nature. Examples of things that oscillate periodically are daytime temperature, the position of a weight on a spring, and tide level. If
Inverse Trigonometric Functions I. Four Facts About Functions and Their Inverse Functions:. A function must be one-to-one (an horizontal line intersects it at most once) in order to have an inverse function..
Section 4.1 Piecewise-Defined Functions 335 4.1 Piecewise-Defined Functions In preparation for the definition of the absolute value function, it is etremel important to have a good grasp of the concept
a p p e n d i f COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated
Functions and their Graphs Functions All of the functions you will see in this course will be real-valued functions in a single variable. A function is real-valued if the input and output are real numbers
ch.pgs1-16 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,
- Chapter A Chapter A - Rectangular Coordinate Sstem Introduction: Rectangular Coordinate Sstem Although the use of rectangular coordinates in such geometric applications as surveing and planning has been
6_5.qd // :58 PM Page CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. EXPLORATION Integrating Rational Functions Earl in Chapter, ou learned rules that allowed ou to integrate
Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important
G COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated with pairs