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ples of p 2. The range of the tangent function is all real numbers. Because the tangent function has a period of domain, p, for any number t in its tan 1 t Β± p 2 tan t. Example 3 Finding All t-values State all values of t for which tan t is 1. Solution p, The tangent function repeats its pattern of y-values at interval... |
p. 4 3 Solution The graph of g is the graph of t f 1 2 sin t 1 2 compressed vertically by a factor of, as shown in Figure 7.1-6. reflected across the x-axis and y 1 β1 1 g(t) = β sin t 2 t 2Ο Ο f(t) = sin t β2Ο βΟ Figure 7.1-6 β Example 6 Vertical Shift tan t 5 on the interval 3p, 3p. 4 3 Graph h t 2 1 Solution The gra... |
f whose graph is symthe graph, metric with respect to the origin is called an odd function. x, y 2 1 Odd Function A function f is odd if f(x) f(x) for every x in the domain of f. The graph of an odd function is symmetric with respect to the origin. f For example, t 1 t 1 sin tan 2 2 1 sin t t 2 sin t tan t and g t 2 1... |
sin t increasing? 16. For what values of t on the interval cos t decreasing? g t 2 1 17. For what values of t on the interval tan t greater than 1? 18. For what values of t on the interval tan t less than 0? 19. For what values of t on the interval tan t increasing? h t 2 1 3p, p is 4 2p, 2p 2p, 2p is is 4 4 p, 2p is ... |
2Ο 60. Scientists theorize that the average temperature at a specific location fluctuates from cooler to warmer and then to cooler again over a long period of time. The graph shows a theoretical prediction of the average summer temperature for the last 150,000 years for a location in Alaska. d. e. f. 3 3 Ο Ο 3 3 2Ο 2Ο... |
secant, and cotangent graphs y sin t, y cos t, and y tan t The graphs of that were developed in Section 7.1 are closely related to the graphs of the reciprocal functions y csc t, y sec t, and y cot t that are studied in this section. Graph of the Cosecant Function The general shape of the graph of the graph of the sin... |
β3 csc t y = β3sin t t Ο 2 Ο 2Ο 3Ο 2 β2Ο β 3Ο 2 βΟ β Ο 2 Figure 7.2-1 β Graph of the Secant Function t The graph of 2 that the graph of f 1 sec t f 1 t 2 csc t is related to the cosine graph in the same way is related to the sine graph. Graphing Exploration Graph the two functions below on the same screen in a viewing... |
Therefore the domain of p, and cot t t f 1 2 cot t consists of all real numbers except integer multiples of cot t is the set of real numbers. The graph of Section 7.2 Graphs of the Cosecant, Secant, and Cotangent Functions 489 Graph of the Cotangent Function y 4 2 0 β2 β4 y = tan t f(t) = cot t t Ο 2 Ο 2Ο 3Ο 2 β2Ο β 3... |
/Odd secant sec t t f 1 2 all real numbers except odd multiples of p 2 all real numbers less 1 than or equal to greater than or equal to 1 or cosecant csc t t f 1 2 all real numbers except multiples of p all real numbers less 1 than or equal to greater than or equal to 1 or cotangent cot t t f 1 2 all real numbers exce... |
6 6 4 2 β2 β4 β6 6 4 2 β2 β4 β6 6 4 2 β2 β4 β6 0 0 0 0 t 1Ο t 1Ο t 1Ο t 1Ο Ο 2 Ο 2 Ο 2 Ο 2 18. f t 2 1 1 2 cot t 1 19. f t 2 1 1 2 cot t 1 20. t f 1 2 2 cot t 1 21. t f 1 2 2 cot t 1 In Exercises 22 β 25, match graph a or b with each function. a. b. β1Ο β 3Ο 4 β Ο 2 β Ο 4 β1Ο β 3Ο 1 β2 β3 β4 β5 5 4 3 2 1 β1 β2 β3 β4 β5... |
one cycle of the given function. 34. 36. 38 sec 2t 35. f t 2 1 cot 3t 4 5 csc t a p 2 b 37. f t 2 1 3 4 csc t 2 3 sec t p 2 1 39. Critical Thinking Show graphically that the equation sec t t but none between has infinitely many solutions, p 2 and p 2. 40. Critical Thinking A rotating beacon is positioned 5 yards from ... |
repeats the same pattern. Period Before proceeding to the discussion about functions that have different periods, it will be helpful to consider functions of the form t f 1 2 sin bt and g 1 t 2 cos bt where b is a constant. The constant b changes the period of the sine or cosine function. Its effect on the graph is to... |
2p 1 2 sin 2 4p, 1 2 a. t b The function f sin t 2 1 1 2 a t b as shown in Figure 7.3-5. β2Ο β1Ο f(t) = sin t 2 1Ο t 2Ο y 1 β1 1 cycle Figure 7.3-5 β 496 Chapter 7 Trigonometric Graphs CAUTION t A calculator may not produce an accurate graph of g cosbt for large values of b. For instance, the graph of sin 50t sinbt or... |
retches or Compressions y g(t) = 7 cos 3t Graph each function. k(t) = cos 3t a. g t 2 1 7 cos 3t b. h t 2 1 1 3 sin t 2 7 1 βΟ β 2Ο 3 β Ο 3 Ο 3 2Ο 3 t Ο Solution a. The function is the function by 7. Consequently, the graph of g is the graph of k (see Example 1a) stretched vertically by a factor of 7. multiplied g k t ... |
Solution 2 sin 4t. Then graph f on f t 2 1 The amplitude of 2 sin 4t is t f 1 2 t f 1 2p b 2 sin 4t 2 p 2p 2 4. is a 0 0 2 0 0 2, and the period of So the graph of f consists of cycles that are p 2 long and rise and fall between the heights of 2 and 2. To graph this function, be sure to notice that its graph is the re... |
is In Exercises 19β38, describe the transformations that change the graph of f into the graph of g. State the amplitude (if any) and the period of g. Section 7.3 Periodic Graphs and Amplitude 499 In Exercises 39 β 44, sketch at least one cycle of the graph of each function. 39. 41. 43 cos t 2 2 tan 3t 3.5 sin 2pt 40. ... |
t 2 49. f t 2 1 5 tan t 3 46. 48 cos 2t 3 sin t 2 50. f t 2 1 3 tan 2t In Exercises 51β56, write an equation for a sine function with the given information. 60. 61. 62. 63. 64 sin 2t; 0 t p cos 3t; 0 t p cos t 2 ; 2p t p sin t 3 ; 2p t p 3 sin 2pt; 1.5 t 1.5 65. The current generated by an AM radio transmitter 2 1 t A... |
are constants, and you will determine how these constants affect the graphs of the functions. Vertical Shifts Recall from Section 3.4 that adding a constant to the rule of a function shifts the graph vertically. Example 1 illustrates a vertical shift in combination with a reflection and a change in amplitude. Example ... |
β1 h(t) = cos( t β 2Ο 3 ) t 2Ο Ο 2Ο 3 Figure 7.4-3 β2Ο βΟ f(t) = cos t Section 7.4 Periodic Graphs and Phase Shifts 503 The cycle of cos h t 1 2 t f 2 1 t 2p 3 b a 2p 3. phase shift of cos t that begins at t 0 becomes a cycle of that begins at t 2p 3. Thus, the function h has a β Combined Transformations Now that you ... |
both f and k is A similar analysis applies to the function g a 0 t 2 0 1 and both have period bt c a cos d. 1 2 2p b. If and b 77 0, a 0 f(t) a sin(bt c) d then each of the functions and g(t) a cos(bt c) d has the following characteristics: amplitude a 00 00 phase shift c b period 2p b vertical shift d Example 4 Combi... |
line β β2Ο 4 2 β2 β4 3Ο 4 Ο 4 Figure 7.4-7 Example 6 Identifying Graphs Find a sine function and a cosine function whose graphs look like the graph shown in Figure 7.4-7. Solution 2Ο This graph appears to have an amplitude of 2 and to be centered vertib 1. so cally on the horizontal axis. The period appears to be 2 co... |
is not an identity. b. In Figure 7.4-9, the graphs of cos f t 2 1 p 2 Q t g and t 1 R 2p, 2p sin t 2 appear to coincide on the interval values for f and g, shown in Figure 7.4-10, also supports the idea Comparing a table of. 3 4 that f t 2 1 cos p 2 Q t R and g t 2 1 sin t are equivalent functions. β2Ο 4 β4 2Ο Figure ... |
7 Trigonometric Graphs Exercises 7.4 In Exercises 1β20, state the amplitude, period, phase shift, and vertical shift of the function. 1. h t 2 1 cos t 1 1 2 2. m t 2 1 7 cos t 3 2 1 3. f t 2 1 5. k 7. p t t 2 2 1 1 8. f t 2 1 9. h t 2 1 4. k t 2 1 cos 2pt a 3 b 6. g t 2 1 3 sin 2t p 2 1 5 sin 2t t p 4 2 3p t 1 sin 1 6... |
given graph. d 1 1 2 2 whose graph appears to be 31. 32. 33. 34. 35. 36. y 12 0 β12 18 0 β18 1 β1 1 β1 1 2 1 2 β 3.5 β3. 3Ο 2 Ο 2 3Ο 4 t t Ο 8 Ο 4 5Ο 16 Ο 3 Ο 37. 38. 39. 40. t 3Ο t Ο 2 4Ο 3 Ο 4 2Ο 2 β4 y 0 β2 β4 β6 β8 β10 10 8 6 4 2 0 βΟ y β2 β4 β6 β8 β10 Section 7.4 Periodic Graphs and Phase Shifts 509 47. f t 2 1 s... |
3t 44. q t 2 1 2 3 cos 3 2 t 45. h t 2 1 3 sin 2t p 2 b a 46. p t 2 1 3 cos 3t p 1 2 In Exercises 62β63, explain why there could not possibly be constants a, b, and c such that the graph of g coincides with the graph of f. a sin bt c t 1 2 62. 63 sin 2 3t 1 1 sin 2t cos 3t 2 3 cos 4t 1 1 2 510 Chapter 7 Trigonometric ... |
a sin t whose graph looks very much like the form 2 3 cos g t graph of. Graph the new function on the same screen with g. Do the graphs appear to coincide? 1 2 sin 2 t 7 bt Section 7.4.A Excursion: Other Trigonometric Graphs 511 The results of the preceding graphing exploration suggest that the graph of looks like the... |
Trigonometric Graphs Other Trigonometric Graphs In Example 1, the variable t has the same coefficient b in both the sine and cosine terms of the functionβs rule. When this is not the case, the graph will consist of waves of varying size and shape, as shown in Figure 7.4.A-2. sin 3t cos sin 3 t 5 1 2 4 cos t 2 2 1 h t ... |
t cos t. t 2 1 β35 Figure 7.4.A-5 y y = 0.5t t y = β0.5t not to scale Figure 7.4.A-6 Graph f in a viewing window with shown in Figure 7.4.A-5. 35 t 35 and 35 y 35, as Recall that sidering the cases t 6 0. ) sign when 1 cos t 1. t 0 and Multiply each term of the inequality by t con(Remember to reverse the inequality t ... |
). β y 0.5t, and Graphing Exploration Find viewing window ranges that clearly show the graph in Example 4 when t is in the following domains. 2p t 0 0 t 2p 2p t 4p Example 5 Oscillating Behavior Analyze the graph of f(t) sin p t b. a Solution Using a wide viewing window, it is clear that the t-axis is an asymptote of t... |
2 sin 4t 5 cos 4t 3 sin 1 5 sin 0.3 sin 2t 1 4 cos 2t 3 2 3t 2 1 2t 4 1 2 2 1 2 cos 1 0.4 cos 2 3t 1 2 2t 3 1 2 In Exercises 7β16, find a viewing window that shows a complete graph of the function. 7. g(t) (5 sin 2t)(cos 5t) 8. h(t) esin t 9. f(t) t 2 cos 2t 10. g t 2 1 sin t 3 2 2 cos t 4 2 11. h t 2 1 sin 300t cos 5... |
the tangent function..... 480 Even function........................... 482 Odd function............................ 483 Graph of the cosecant function.............. 486 Domain and range of the cosecant function.... 486 Graph of the secant function................ 488 Domain and range of the secant function...... 487 D... |
about the graph of f It has no sharp corners. a. b. It crosses the horizontal axis more than once. c. It rises higher and higher as t gets larger. d. It is periodic. e. It has no vertical asymptotes. sin t? t 2 1 In Exercises 2β4, graph each function on the given interval. 2. f t 2 1 sin t 4. h t 2 1 tan t 7p 2, 7p T ... |
ii only c. d. all of them e. none of them i and iii only 17. Which of the following functions has the graph shown at left? a. b. c. d. e tan t t p tan a 2 b 1 tan t 3 tan t tan t 18. Which of the following is true about sec t? a. b. 0 0 sec 2 1 sec t 1 sin t Its graph has no asymptotes. c. d. It is a periodic function... |
Ο 5 8Ο 5 2Ο 35. State the rule of a periodic function whose graph from t 0 to t 2p closely resembles the graph at left. y 2 1 β1 β2 In Exercises 36β38, sketch the graph of at least one cycle of each function. 36. f t 2 1 1 2 cos 1 2t p 3 2 37. g t 2 1 sin 1 3 a t p b 38. g t 2 1 4 cos 2t 3 b a 5 In Exercises 39β42, det... |
in graphing the series, the series approximates the function more closely The set of all values of x for which the series converges to when x 6 1 and when x 7 1. y 1 x x2 x3 x4 on the same screen, as shown in 6 1, 6 1. x x 0 0 the function is called the interval of convergence. The function 1 6 x 6 1. 1 1 x x 1, the i... |
-2d β Find an infinite geometric series that represents the given function, and state the interval of convergence. Find a function that is approximated by the following series. State the interval of convergence. 1. y 2 1 3x 3. y 2 1 x 2. y 3 1 2x 4. y 3 1 2x 5. 1 x2 2! x4 4! x6 6! p x 1 2 6. 1 x 1 2 1 7. 1 x x2 2! 1 x3... |
that use inverse functions, basic identities, and algebra to solve trigonometric equations are considered in Section 8.3. Skills from the Sections 8.1 through 8.3 are applied to problem-solving and real-world applications in Section 8.4. NOTE In Chapter 7, the variable t was used for trigonometric functions to avoid c... |
actual solutions. is used rather than Section 8.1 Graphical Solutions to Trigonometric Equations 525 1 f x tan x The function p 2 b in this interval. Using the intersecand there is one solution of tion finder on a graphing calculator gives the approximate solution in this interval. completes one cycle on the interval ... |
Technology Tip 3 sin2 x Enter culator as on a cal- 2 1 sine and 3 sin2 x cos x 2 cosine have period is at most the period of Both The graph of f, which is shown x f in two viewing windows in Figure 8.1-5, does not repeat its pattern over so you can conclude that f has a period any interval of less than of 2p. 2p, 2p. ... |
of y 3 sin 2x f x 1 2 tan x 3 sin 2x. p, Recall from Section which is also the period of 7.3 that y tan x. has a period of 2p 2 p. Therefore, the period of f is p 2 b p 2, a, f on the interval an interval of length p. Figure 8.1-6 shows the graph of Even without the graph, it can be easily verified that there is an x-... |
x 3 cos x 1 0 8. tan x 3 cos x 9. cos4x 3 cos3x cos x 1 10. sec x tan x 3 11. sin3x 2 sin2 x 3 cos x 2 0 12. csc2x sec x 1 following. a. The solutions of sin x 1 are 5p x p, 2 2 3p 2 x, 9p, 2 7p 2, p and 11p 2,,.... b. The solutions of sin x 1 are 11p 2 x 3p, 2 p 2 x 7p, 2 5p 2,, p and 9p 2,...., 14. Use the graph of t... |
constant) does a basic equation involving the sine and cosine function have no solutions? 34. Critical Thinking Under what conditions (on the constant) does a basic equation involving the secant and cosecant function have no solutions? 8.2 Inverse Trigonometric Functions Objectives β’ Define the domain and range of the... |
8.2-2 x 2 sin 1 x is the interval 1, 1 3 4, and its range is the g The domain of p 2 T interval p 2, S 1. Inverse Sine Function For each v with 1 v 1, sin1 v is the unique number u in the interval sine is v; that is, P 2, S P 2 T whose sin1 v u exactly when sin u v. Section 8.2 Inverse Trigonometric Functions 531 Tech... |
u) u if p 2 u p 2 sin(sin1 v) v if 1 v 1 Example 2 Composition of Inverse Functions Explain why sin 1 sin a p 6 b p 6 is true but sin 1 sin a 5p 6 b 5p 6 is not true. Solution You know that sin p 6 1 2, so by substitution 1 sin sin a p 6 b sin 1 1 2 b a p 6 because p 6 is in the interval p 2 S, p 2 T. Although sin 5p ... |
3 Evaluating Inverse Cosine Expressions Evaluate the following. a. cos 1 1 2 b. cos 1 0 c. cos 1 0.63 2 1 534 Chapter 8 Solving Trigonometric Equations CAUTION 1 cos cos x 1 does not mean 1 1 or cosx. 2 Solution a. cos 1 1 2 p 3 because p 3 whose cosine is 1 2. is the unique number in the interval 0, p 4 3 b. cos 1 0 ... |
set of all real numbers and its range 1 x tan p 2 b., a Inverse Tangent Function For each real number v, tan1 v is the unique number u in the interval P 2 a, P 2 b whose tangent is v; that is, tan1v u exactly when tan u v. The properties of the inverse tangent function are similar to the properties of the inverse sine... |
22 2 b 19. 20. 22. 24. 1 1 1 sin cos sin 7 2 Hint: the answer is not 7. cos 3.5 2 1 sin tan 1 1 2 sin C 1 2D tan 12.4 1 tan cos 1 tan C 1 1 cos C 1 4 2D 8.5 2D 21. 23. 2 25. Given that u sin 1 of cos u and tan u. a 23 2 b, find the exact value Section 8.2 Inverse Trigonometric Functions 537 45. tan sin 1v A B 46. sin ... |
, has an inverse function. and x p 2 Sketch its graph. 53. Show that the restricted cosecant function, whose domain consists of all numbers x such that p x 0, 2 has an inverse function. x p 2 and Sketch its graph. 54. Show that the restricted cotangent function, whose domain is the interval function. Sketch its graph. ... |
cos x The graphs of and just two solutions (intersection points) on the interval one full period of the cosine function. in Figure 8.3-1 show that there are which is p, p, 3 4 Y2 0.6 The definition of the inverse cosine function states that cos 10.6 Using the inverse cosine function, of by using the identity 0, p is t... |
3.9897 2kp x 2.2935 : x 2.2935 k 1 p, p and. 4 3 β 540 Chapter 8 Solving Trigonometric Equations Example 3 Solving Basic Tangent Equations 5 Solve tan x 3. Solution The definition of the inverse tangent function states that Ο Ο 5 Figure 8.3-3 tan 1 3 is the number in the interval p p a 2 b 2 1 3 1.2490 x tan p a 2 of ... |
4 Therefore, the exact solution is given by u p 4 2kp and u 3p 4 2kp, where k is any integer. β Sometimes trigonometric equations can be solved by using substitution to make them into basic equations. 542 Chapter 8 Solving Trigonometric Equations Example 6 Using Substitution and Basic Equations Solve sin 2x 22 2 Solut... |
kp, p, p, Figure 8.3-4 indicates that there are three solutions in the interval is outside which is marked with vertical lines. The solution the interval, but the corresponding solution within the interval can be x 3.8713 2kp. found by letting, the solutions are x 3.8713 k 1 p, p Within in 4 3 4 3 x 3.8713 2p 2.4119, x... |
cos2 x 3 sin x 9 0 2 kp, x 5p 6 2kp, are x 0.2014 2kp, and x 3.3430 2kp, where k is any integer. The graph of in Figure 8.3-6 confirms the solution. Y1 10 cos2x 3 sin x 9 shown β 3Ο 2 Example 10 Identities and Quadratic Formula Solve sec2 x 5 tan x 2. Section 8.3 Algebraic Solutions of Trigonometric Equations 545 Solu... |
figure. v2 u2 The number v1 v2 is called the index of refraction. Angle of incidence Incident ray, speed v 1 ΞΈ 1 Refracted ray, speed v 2 ΞΈ 2 Angle of refraction 9. The index of refraction of light passing from air to, find water is 1.33. If the angle of incidence is the angle of refraction. 38Β° 546 Chapter 8 Solving ... |
2x sin x 0 53. 2 tan2x tan x 5 sec2x 54. The number of hours of daylight in Detroit on t 0 day t of a non-leap year (with 1) is given by the following function. being January 3 sin d t 2 1 2p 365 1 S t 80 2 T 12 a. On what days of the year are there exactly 11 hours of daylight? b. What day has the maximum amount of da... |
x 1 or 5p 4 x p 4 Hint: Solve the original equation by moving all terms to one side and factoring. Compare your answers with the ones above. 63. Critical Thinking Let n be a fixed positive integer. Describe all solutions of the equation sin nx 1 2. ΞΈ d a d b 56. If muzzle velocity of a rifle is 300 feet per second, at... |
usoid and the functions are called sinusoidal functions. a sin a 1 2 f t bt, is the number Recall that the amplitude of the function 0 1 units above the horand that its graph consists of waves that rise to 0 units below the horizontal axis. In other izontal axis and fall to words, the amplitude is half the distance fro... |
t 2 1 3p 4. t 3p 4, so the phase shift 3p 4 3p 4 c b c 2 c 3p 2 phase shift is c b b 2 Solve for c. Therefore, a function for the sinusoidal graph in Figure 8.4-3 is 3 sin f t 2 1 2t 3p 2 b a 1. β Simple Harmonic Motion Motion that can be described by a function of the form f t 2 1 a sin bt c 1 2 d or g t 2 1 a cos bt... |
shows the main features of the graph. The amplitude is a 1 2 1 fmax fmin2 1 2 1 8 12 1 22 2. Use the period to find b. 2p 2p 3 b b 3 so there is no phase shift and c 0. The sine wave begins at The vertical shift, d, is d fmax t 0, 10 units. fmin 2 8 1 2 12 2 10 Thus, the function giving the y-coordinate of E at time t... |
.4-7a and 8.4-7b. is 0 when t is 0. As t increases and increases h t 1 2 1 2 1 2 8, 1 2 its h(t) 8 h(t) 8 β8 t or β8 t Figure 8.4-7a Figure 8.4-7b Careful physical experiment suggests that the curve in Figure 8.4-7a, which resembles the sine graphs you have studied, is a reasonably accurate model of this process. Facts... |
Temperature Data The following table shows the average monthly temperature in Cleveland, Ohio, based on data from 1971 to 2000. Since average temperatures are not likely to vary much from year to year, the data essentially repeats the same pattern in subsequent years. So, a periodic model is appropriate. Month Tempera... |
the 12 given data points, the regression feature on a calculator 13 Figure 8.4-12 produces the following model. 23.1202 sin t f 1 2 0.5018t 2.0490 48.6927 2 1 The period of this function is approximately 2p 0.5018 12.52, which is not a very good approximation for a 12-month cycle. Because the data repeats the same pat... |
.4-13b Figure 8.4-14 Exercises 8.4 1. The original Ferris wheel, built by George Ferris for the Columbian Exposition of 1893, was much larger and slower than its modern counterparts: It had a diameter of 250 feet and contained 36 cars, each of which held 40 people; it made one revolution every 10 minutes. Suppose that ... |
h(t) in terms of the sine or cosine function under the given conditions. 5. There is an initial push upward from the equilibrium point. 6. There is an initial pull downward from the equilibrium point. Hint: What does the graph of y a sin bt look like when a 6 0? 7. The weight is pulled 6 cm above the equilibrium ) is ... |
for 1991β2002. Year Unemployed Year Unemployed 1991 1992 1993 1994 1995 1996 8.628 9.613 8.940 7.996 7.404 7.236 1997 1998 1999 2000 2001 2002 6.739 6.210 5.880 5.655 6.742 8.234 a. Sketch a scatter plot of the data, with x 0 corresponding to 1990. b. Does the data appear to be periodic? If so, find an appropriate mod... |
in the summer months causes the pendulum to increase its length by 0.01%. How much time will the clock lose in June, July, and August? Hint: These three months have a total of 92 days (7,948,800 seconds). If k is increased by 0.01%, what is f 2? 1 2 2 seconds. 8.4.A Excursion: Sound Waves Objectives β’ Find the frequen... |
frequency and corresponding musical note listed on them. A tuning fork, a microphone connected to a data collection device (such as a CBL2), and a calculator with a tuning program can be used to simulate the sounds heard through the eardrum. NOTE A tuning program can be obtained by downloading or entering the DataMate... |
.4.A-3. Find the amplitude a by finding half the difference between the maximum value and minimum value. The data graphed in Figure 8.4-3 has a maximum value of 0.043942 and a minimum value of 0.045618. a 0.043942 1 2 0.045618 2 0.04478 amplitude To find the period, find the x-value of the first maximum of the graph, f... |
. b. The graph appears to be periodic. By finding the length of one complete cycle, the period of this graph appears to be approximately 0.0077632. c. The frequency of this sound is the reciprocal of the period of the Figure 8.4.A-4 function. 1 0.0077632 128.8128607 Hz The frequency of this sound is very close to the d... |
8.2 Section 8.3 Section 8.4 Basic trigonometric equations............... 524 Intersection method...................... 524 x-intercept method....................... 525 Inverse sine function...................... 530 Inverse cosine function.................... 533 Inverse tangent function................... 535 Soluti... |
2 3. sin x sec2 x 3 4. cos2 x csc2 x tan x p 2 R Q 5 0 5. cos 2x sin x 6. 3 sin 2 x cos x 7. A weight hanging from a spring is set into motion, moving up and down. (See Figure 8.4-6 in Example 3 of Section 8.4.) Its distance in centimeters above or below the equilibrium point at time t seconds is given by d 5 sin 3t 3... |
feet per second. At what angle of elevation should it be fired in order to hit a target 3500 feet away? Hint: Use the projectile equation given for Exercises 56β59 of Section 8.3. Section 8.4 40. The following table gives the average population, in thousands, of a southern town for each month throughout the year. The ... |
function approaches, if it exists, is called a limit. In this section an informal description of a limit is illustrated with some interesting trigonometric functions, but the discussion is not intended to be complete. See Chapter 14 for a detailed discussion of limits. x If the output of a function approaches a single... |
, 2Ο cos x 1 x lim xS0 0. β There are two ways in which a limit may not exist, as shown in the next examples. The first example illustrates a function that does not have a Figure 8.C-3b limit as x approaches p 2 because function values on either side of p 2 do not approach a single real number. The second example illus... |
xS0 cos 1 xb a does not exist. β 568 Exercises Discuss the behavior of the function around the given x-value by using a table and a graph. Find the limit of each function, if it exists. 1. lim xS0 1 cos x 3. lim xSp 2 tan x x 5. lim xS0 sin 3x 3x 7. lim xS0 x tan2 x 9. lim xS0 3x sin 3x 11. lim xS0 sin 3x sin 4x 2. li... |
a- tions. This chapter presents many widely used trigonometric identities and specific methods for solving particular forms of trigonometric equa- tions. 9.1 Identities and Proofs Objectives β’ Identify possible identities by using graphs β’ Apply strategies to prove identities Recall that an identity is an equation that... |
2 sin2x cos x 2 cos2x sin x 1 sin x sin2x cos x cos x tan x 4 Solution Test each equation graphically to see if it might be an identity by graphing each side of the equation. 2 sin2x cos x 2 cos2x sin x a. Graph and Y2 Y1 on the same is shown darker screen, as shown in Figure 9.1-2a. The graph of Y2. than 2 sin2x Beca... |
. NOTE The definitions of the basic trigonometric ratios may help you remember the quotient and reciprocal identities. The shapes of the graphs of sine, cosine, and tangent may help you remember the periodicity and negative angle identities. Also, if you can remember the first of the Pythagorean identities, which is ba... |
learn the relationships among the trigonometric functions β’ to simplify an expression by using an equivalent form CAUTION Proving identities is not the same as solving equations. Properties that apply to equations, such as adding the same value to both sides, are not valid when verifying identities because the beginni... |
equivalent expressions. There is another technique that is often useful when dealing with fractions. Rewrite a fraction in equivalent form by multiplying its numerator and denominator by the same quantity. Example 5 Transform One Side into the Other Side Prove that sin x 1 cos x 1 cos x sin x. Solution Beginning with ... |
78 Chapter 9 Trigonometric Identities and Proof Example 7 Proving Identities that Involve Fractions Prove the first identity below, then use the first identity to prove the second identity. a. sec xΛ1 sec x cos x tan2x 2 b. sec x tan x tan x sec x cos x Solution a. Begin by transforming the left side. sec xΛ1 sec x cos... |
x tan x. tan x 1 tan x Similarly, multiply the right side of [3]. cot x 1 1 2 Λ1 1 tan x 2 cot x cot x tan x 1 tan x cot x 1 1 tan x cot x tan x. Because the left and right sides are equal to the same expression, [3] has been proven to be an identity. Therefore, conclude that cot x 1 cot x 1 1 tan x 1 tan x is also an... |
that the resulting equation appears to be an identity when you test it graphically. You need not prove the identity. a. cos x b. sec x c. sin2x d. sec2 x e. sin x cos x f. 1 sin x cos x 5. csc x tan x ____ 6. 7. sin x tan x ____ sin4x cos4 Λx sin x cos x ____ 8. tan2 x 2 1 sin x 1 sin x ____ 2 In Exercises 9β18, prove... |
. 42. 43. 44. sin x cos x 2 sin2x cos2x 1 1 1 tan x 2 2 2 sec2x sec x csc x sin x cos x 2 tan x 1 cos x sin x sin x 1 cos x 2 csc x sec x csc x 1 tan x csc x cot x 1 1 tan x csc x sec x Section 9.2 Addition and Subtraction Identities 581 54. sin x 1 cot x cos x 1 tan x cos x sin x 55. cos x 1 sin x sec x tan x 45. 1 cs... |
.2 Addition and Subtraction Identities Objectives β’ Use the addition and subtraction identities for sine, cosine, and tangent functions β’ Use the cofunction identities Many times, the input, or argument, of the sine or cosine function is the sum or difference of two angles, and you may need to simplify the expression. ... |
order to use addition or subtraction identities to find exact values, first write the argument as a sum or difference of two terms for which exact values are known, such as, and p. Section 9.2 Addition and Subtraction Identities 583 Example 1 Addition Identities Use the addition identities to find the exact values of ... |
h b β Addition and Subtraction Identities for the Tangent Function The addition and subtraction identities for sine and cosine can be used to obtain the addition and subtraction identities for the tangent function. Addition and Subtraction Identities for Tangent tan(x y) tan(x y) tan x tan y 1 tan x tan y tan x tan y ... |
2 intervals in which both sine and tangent are negative., 2p b a x y are negative numbers. Therefore x y because it is the only one of the four β Cofunction Identities Special cases of the addition and subtraction identities are the cofunction identities. 586 Chapter 9 Trigonometric Identities and Proof Cofunction Ide... |
0 6 x 6 p 2, then sin p 4 a x b? 26. If cos x 1 4 and p 2 6 x 6 p, then cos 27. If cos x 1 5 and p 6 x 6 3p 2, then sin? 28. If sin x 3 4 and 3p 2 6 x 6 2p, then cos p 4 a x b? In Exercises 13β18, rewrite the given expression in terms of cos x. sin x and 13. sin p 2 a x b 15. cos x 3p 2 b a 17. sec x p 1 2 14. cos x p... |
588 Chapter 9 Trigonometric Identities and Proof 37. If x is in the first quadrant and y is in the second quadrant, sin x 24 25 x y value of and tan sin 2 x y quadrant in which and, 1 1 lies. sin y 4 5 x y and the, 2 find the exact 50. cos 51. tan x p x p 2 2 1 1 cos x tan x 38. If x and y are in the second quadrant, ... |
sin y cos x cos y 1 1 1 1 1 sin sin sin sin cos cos 1 1 x y x y 2 2 tan x tan y tan x tan cot y cot x cot y cot x cot x tan y cot x tan y 64. cos cos 65. tan 66. cot cot y tan x cot y tan x 2 2 tan x tan y 2 cot x cot y 2 Section 9.2.A Excursion: Lines and Angles 589 9.2.A Excursion: Lines and Angles Objectives β’ Find... |
L 0 y2 x2 x1 y2 x1 x2 opposite adjacent tan p u. 2 1 [1] Use the fact that the tangent function has period and the negative angle identity for tangent to obtain tan tan p u u p tan u. 2 L tan Λu shows that slope of tan u 2 1 1 2 Combining this fact with also. 1 1 3 4 in this case β Example 1 Angle of Inclination Find ... |
using the following fact. Angle Between Two Lines If two nonvertical, nonperpendicular lines have slopes m and k, then one angle between them satisfies U tanΛ U m k 1 mk ` `. Proof Suppose L has slope k and angle of inclination and that M has slope m and angle of inclination a b. m k 1 mk By the definition of absolute... |
has slope 1 2 and M has slope 0. and M has slope 3. 11. (3, 2) and (5, 6) are on L; (0, 3) and (4, 0) are on M. 12. 1, 2 1 on M. 2 and 1 3, 3 2 are on L; 3, 3 2 1 and (6, 1) are Section 9.3 Other Identities 593 9.3 Other Identities Objectives β’ Use the following identities: double-angle power-reducing half-angle produ... |
in the third quad- sin Λx B 225 289 15 17 y x 8 x 15 17 (β8, β15) Figure 9.3-1 594 Chapter 9 Trigonometric Identities and Proof Now substitute these values in the double-angle identities to find sin 2x and cos 2x. sin Λ2x 2Λ sinΛ x cos Λx 2 15 a 17 ba 8 17 b 240 289 0.8304 cos 2x cos2x sin2x 8 a 17 b 2 2 15 a 17 b 64 ... |
1 2 sin2x cos 2x 2 cos2x 1 Section 9.3 Other Identities 595 Example 3 Use Forms of cos 2x Prove that 1 cos 2x sin 2x tan x. Solution Use the first identity in the preceding box and the double-angle identity for sine. 1 cos 2x sin 2x 1 1 2 sin2x 2 sin x cos x 1 2 2 sin2x 2 sin x cos x sin x cos x tan x β Power-Reducing... |
ine is derived from a power-reducing identity, as was the half-angle identity for sine. The half-angle identity for tangent then follows immediately since tan x 2 b a sin x 2 b a cos x 2 b a. Example 5 Half-Angle Identities Find the exact value of a. cos 5p 8 b. sin p 12 Section 9.3 Other Identities 597 use the half-an... |
on the terminal side of the angle of x The distance from triangle. 1 2, 3 2 to the origin is the hypotenuse of the Therefore 213 sin x 3 213 cos x 2 213 By the first of the half-angle identities for tangent, tan Λ x 2 1 cos x sin x 1 2 213 b a 3 213 213 2 213 3 213 213 2 3 β y x 2 x 3 13 (β2, β3) Figure 9.3-2 Section ... |
cos a Similarly, x t t 3t 2 b and y 3t. 2 sin 2t cos t 2 1 cos t cos 3t 2 cos Λa t 3t 2 b Λ cos a t 3t 2 b 2 cos 2t cos t. 2 1 Therefore, sin t sin 3t cos t cos 3t 2 sin 2t cos 2 cos 2t cos t 2 t 2 1 1 sin 2t cos 2t tan 2t. β Exercises 9.3 In Exercises 1β12, use the half-angle identities to evaluate the given expressi... |
32. sin x 0.6, for p 2 6 x 6 p 33. sin x 3 5, for 3p 2 6 x 6 2p 34. cos x 0.8, for 3p 2 6 x 6 2p 35. tan x 1 2, for p 6 x 6 3p 2 36. cot x 1, for p 6 x 6 p 2 In Exercises 37β42, assume sin x 0.6 and evaluate the given expression. Section 9.3 Other Identities 601 53. cos4x sin4x cos 2x 54. sec 2x 1 1 2 sin2x 55. cos 4x... |
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