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7.68 This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 7 The Unit Circle: Sine and Cosine Functions 881 sin t = y sin(−t) = −y sin t ≠ sin(−t) cos t = x cos(−t) = x cos t = cos(−t) tan(t) = y x tan(−t) = − y x tan t ≠ tan(−t) sec t = 1 x sec(−t) = 1 x csc t = 1 y csc(−t) = 1 −y cot t =... |
882 Chapter 7 The Unit Circle: Sine and Cosine Functions Fundamental Identities We can derive some useful identities from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships: tan t = sin t cos t sec t = 1 cos t csc... |
can use these fundamental identities to derive alternate forms of the Pythagorean Identity, cos2 t + sin2 t = 1. One form is obtained by dividing both sides by cos2 t. cos2 t cos2 t = + sin2 t cos2 t 1 cos2 t 1 + tan2 t = sec2 t The other form is obtained by dividing both sides by sin2 t. cos2 t sin2 t = + sin2 t sin2... |
we discussed at the beginning of the chapter, a function that repeats its values in regular intervals is known as a periodic function. The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or 2π, will result in the same outputs fo... |
(t) = 1 2, fin sec(t), csc(t), tan(t), cot(t). Solution sec t = 1 cos t = 1 1 2 = 2 csc t = 1 tan t = sin t sin cos t = − 2 3 3 = − 3 cot t = 1 tan.33 sin(t) = 2 2 and cos(t) = 2 2, fin sec(t), csc(t), tan(t), and cot(t) Evaluating Trigonometric Functions with a Calculator We have learned how to evaluate the six trigon... |
• Press the SIN key. • Enter the value of the angle inside parentheses. • Press the ENTER key. Example 7.34 Evaluating the Secant Using Technology Evaluate the cosecant of 5π 7. Solution For a scientific calculator, enter information as follows: 1/(5 × π / 7) SIN = ⎛ csc ⎝ 5π 7 ⎞ ⎠ ≈ 1.279 7.34 Evaluate the cotangent ... |
tan π 3 sec π 3 csc π 3 cot π 3 260. 261. 262. 263. tan 8π 3 sec 4π 3 csc 2π 3 cot 5π 3 264. tan 225° 265. sec 300° 266. csc 150° 267. cot 240° 268. tan 330° 269. sec 120° 270. csc 210° 271. cot 315° 272. If sin t = 3 4 cos t, sec t, csc t, tan t, and cot t., and t is in quadrant II, find the following exercises, use ... |
9 cot 4π 7 sec π 10 tan 5π 8 sec 3π 4 Chapter 7 The Unit Circle: Sine and Cosine Functions 891 291. csc π 4 292. tan 98° 293. cot 33° 294. cot 140° 295. sec 310° Extensions For the following exercises, use identities to evaluate the expression. 296. If tan(t) ≈ 2.7, and sin(t) ≈ 0.94, find cos(t). 297. If tan(t) ≈ 1.3... |
-World Applications The amount of sunlight 306. ⎛ modeled by the function h = 15cos ⎝ in a certain city can be d⎞ ⎠, where h represents the hours of sunlight, and d is the day of the year. Use the equation to find how many hours of sunlight there are on February 10, the 42nd day of the year. State the period of the fun... |
of an angle measured clockwise from the positive x-axis opposite side in a right triangle, the side most distant from a given angle period the smallest interval P of a repeating function f such that f (x + P) = f (x) positive angle description of an angle measured counterclockwise from the positive x-axis Pythagorean ... |
sc t cos t = 1 sec t tan t = 1 cot t csc t = 1 sin t sec t = 1 cos t cot t = 1 tan t tan t = cot ⎛ cos t = sin ⎝ π 2 π ⎛ sin t = cos ⎝ 2 π 2 π 2 π ⎛ sec t = csc ⎝ 2 ⎛ cot t = tan ⎝ ⎛ ⎝ − t⎞ ⎠ − t⎞ ⎠ − t⎞ ⎠ − t⎞ ⎠ − t⎞ ⎠ Cosine Sine cos t = x sin t = y Pythagorean Identity cos2 t + sin2 t = 1 Tangent function Secant fun... |
See Example 7.7. • The length of a circular arc is a fraction of the circumference of the entire circle. See Example 7.8. • The area of sector is a fraction of the area of the entire circle. See Example 7.9. • An object moving in a circular path has both linear and angular speed. • The angular speed of an object trave... |
. • When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity is also useful for determining the sines and cosines of special angles. See Example 7.20. • Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for enteri... |
. See Example 7.28. • The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine. • Identities can be used to evaluate trigonometric functions. See Example 7.29 and Example 7.30. • Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce ne... |
coterminal with the given angle. 319. − 20π 11 320. 14π 5 For the following exercises, draw the angle provided in standard position on the Cartesian plane. 321. −210° 322. 75° 323. 5π 4 324. − π 3 325. Find the linear speed of a point on the equator of the earth if the earth has a radius of 3,960 miles and the earth r... |
igonometric Functions For the following exercises, find the exact value of the given expression. cos π 6 349. 350. 351. 352. tan π 4 csc π 3 sec π 4 the following exercises, use reference angles to For evaluate the given expression. 353. sec 11π 3 This content is available for free at https://cnx.org/content/col11758/1... |
370. Find ABC : sin B = 3 4, c = 12. 371. Find the missing sides of the triangle. 372. The angle of elevation to the top of a building in Chicago is found to be 9 degrees from the ground at a 900 Chapter 7 The Unit Circle: Sine and Cosine Functions This content is available for free at https://cnx.org/content/col11758... |
content/m49369/latest/), we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions. Graphing Sine and Cosine Functions Recall that the sine and cosine functions relate real number values to the x- and y-coordinates of a point on the ... |
, the shape of the graph repeats after 2π, which means the functions are periodic with a period of 2π. A periodic function is a function for which a specific horizontal shift, P, results in a function equal to the original function: f (x + P) = f (x) for all values of x in the domain of f. When this occurs, we call the... |
Asin(Bx − C) + D and y = Acos(Bx − C) + D (8.1) Determining the Period of Sinusoidal Functions Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period. In the general formula, B is rel... |
it is related to the amplitude, or greatest distance from rest. A represents the vertical stretch factor, and its absolute value |A| is the amplitude. The local maxima will be a distance |A| above the vertical midline of the graph, which is the line x = D; because D = 0 in this case, the midline is the x-axis. The loc... |
D and y = Acos(Bx − C) + D or ⎛ ⎝B⎛ y = Asin ⎝x − C B ⎞ ⎞ ⎛ ⎝B⎛ ⎠ + D and y = Acos ⎠ ⎝x − C B ⎞ ⎞ ⎠ + D ⎠ The value C B for a sinusoidal function is called the phase shift, or the horizontal displacement of the basic sine or cosine function. If C > 0, the graph shifts to the right. If C < 0, the graph shifts to the le... |
C) + D. In the given equation, notice that B = 1 and C = −. So the phase shift is or π 6 units to the left. Analysis We must pay attention to the sign in the equation for the general form of a sinusoidal function. The equation ⎞ ⎞ ⎛ ⎠ − 2. shows a minus sign before C. Therefore f (x) = sin ⎝x + ⎠ ⎞ ⎛ ⎠ − 2 can be rewr... |
phase shift is C = π. B = 0 2 = 0. Finally, D = 1, so the midline is y = 1. Analysis Inspecting the graph, we can determine that the period is π, the midline is y = 1, and the amplitude is 3. See Figure 8.15. Figure 8.15 8.5 Determine the midline, amplitude, period, and phase shift of the function y = 1 2 ⎛ cos ⎝ x 3 ... |
function in Figure 8.17. Figure 8.17 912 Chapter 8 Periodic Functions Example 8.7 Identifying the Equation for a Sinusoidal Function from a Graph Determine the equation for the sinusoidal function in Figure 8.18. Figure 8.18 Solution With the highest value at 1 and the lowest value at −5, the midline will be halfway b... |
Acos(Bx − C) + D, we will let C = 0 and D = 0 and work with a simplified form of the equations in the following examples. Given the function y = Asin(Bx), sketch its graph. 1. 2. Identify the amplitude, |A|. Identify the period, P = 2π |B|. 3. Start at the origin, with the function increasing to the right if A is posi... |
. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 8 Periodic Functions 915 Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph. 1. Express the function in the general form y = Asin(Bx − C) + D or y = Acos(Bx − C) + D. 2. 3. 4. Identify the amplitude, ... |
form and use the steps outlined in Example 8.9. y = Acos(Bx − C) + D Step 1. The function is already written in general form. Step 2. Since A = − 2, the amplitude is |A| = 2. = 2π π 2, so the period is P = 2π |B| Step 3. |B| = π 2 = 2π ⋅ 2 π = 4. The period is 4. Step 4. C = − π, so we calculate the phase shift as C B... |
8.12 Finding the Vertical Component of Circular Motion A circle with radius 3 ft is mounted with its center 4 ft off the ground. The point closest to the ground is labeled P, as shown in Figure 8.24. Sketch a graph of the height above the ground of the point P as the circle is rotated; then find a function that gives ... |
(443 feet). It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. Express a rider’s height above ground as a function of time in minutes. Solution With a diameter of 135 m, the wheel has a radius of 67.5 m. The height will oscillate with amplitude 67.5 m above and below th... |
A cos(Bx + C) + D, what constants affect the range of the function and how do they affect the range? How does the range of a translated sine function relate to 4. the equation y = A sin(Bx + C) + D? How can the unit circle be used to construct the graph of 5. f (t) = sin t? Graphical For the following exercises, graph... |
y = 2 sin(3x − 21) + 4 17. y = 5 sin(5x + 20) − 2 This content is available for free at https://cnx.org/content/col11758/1.5 Figure 8.27 Determine the amplitude, period, midline, and an 24. equation involving cosine for the graph shown in Figure 8.28. Chapter 8 Periodic Functions 923 Figure 8.28 25. Determine the ampl... |
�−5π, 5π⎤ ⎦ and Real-World Applications A Ferris wheel is 25 meters in diameter and boarded 48. from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function h(t) gives a person’s heigh... |
. 8.2.3 Analyze the graphs of y=sec x and y=csc x. 8.2.4 Graph variations of y=sec x and y=csc x. 8.2.5 Analyze the graph of y=cot x. 8.2.6 Graph variations of y=cot x. We know the tangent function can be used to find distances, such as the height of a building, mountain, or flagpole. But what if we want to measure rep... |
will help us draw our graph, but we need to determine how the graph behaves where it is undefined. If we look more closely at values when π ≈ 1.57, 3, we can use a table to look for a trend. Because π 3 ≈ 1.05 and π 2 < x < π 2 we will evaluate x at radian measures 1.05 < x < 1.57 as shown in Table 8.4. 926 Chapter 8 ... |
is for the sine and cosine functions. Instead, we will use the phrase stretching/compressing factor when referring to the constant A. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 8 Periodic Functions 927 Features of the Graph of y = Atan(Bx) • The stretching factor is |A|. • The p... |
1. 2. Identify the stretching factor, |A|. Identify B and determine the period, P = 3. Draw vertical asymptotes at x = − P 2. π |B| and x = P 2. 4. For A > 0, the graph approaches the left asymptote at negative output values and the right asymptote at positive output values (reverse for A < 0 ). 5. Plot reference poin... |
in several ways: Features of the Graph of y = Atan(Bx−C)+D • The stretching factor is |A|. • The period is π |B|. • The domain is x ≠ C B + π |B| k, where k is an integer. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 8 Periodic Functions 929 • The range is (−∞, − |A|] ∪ [|A|, ∞). ... |
(−1.25, 1), (−1,−1), and (−0.75,−3). The graph is shown in Figure 8.37. 930 Chapter 8 Periodic Functions Figure 8.37 Analysis Note that this is a decreasing function because A < 0. 8.13 How would the graph in Example 8.15 look different if we made A = 2 instead of −2? Given the graph of a tangent function, identify ho... |
is 0,, etc. Because the cosine is never more than 1 in absolute value, the secant, being, 3π 2 the reciprocal, will never be less than 1 in absolute value. We can graph y = sec x by observing the graph of the cosine function because these two functions are reciprocals of one another. See Figure 8.40. The graph of the ... |
than 1 in absolute value. We can graph y = csc x by observing the graph of the sine function because these two functions are reciprocals of one another. See Figure 8.41. The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the c... |
as the cosecant graph shifted half a period to the left. Vertical and phase shifts may be applied to the cosecant function in the same way as for the secant and other functions.The equations become the following. y = Asec(Bx − C) + D y = Acsc(Bx − C) + D (8.4) (8.5) Features of the Graph of y = Asec(Bx−C)+D • The stre... |
Secant Function Graph one period of f (x) = 2.5sec(0.4x). Solution Step 1. The given function is already written in the general form, y = Asec(Bx). Step 2. A = 2.5 so the stretching factor is 2.5. Step 3. B = 0.4 so P = 2π 0.4 = 5π. The period is 5π units. Step 4. Sketch the graph of the function g(x) = 2.5cos(0.4x). ... |
⎝ π 3 x − ⎞ ⎠ + 1. π 2 Solution ⎛ Step 1. Express the function given in the form y = 4sec ⎝ π 3 x − ⎞ ⎠ + 1. π 2 Step 2. The stretching/compressing factor is |A| = 4. Step 3. The period is Step 4. The phase shift is 2π |B| = 2π π 3 = 2π.5 Step 5. Draw the graph of y = Asec(Bx), but shift it to the right by C B = 1.5 a... |
The given function is already written in the general form, y = Acsc(Bx). Step 2. |A| = |−3| = 3, so the stretching factor is 3. Step 3. B = 4, so P = 2π 4. The period is π 2 units. π 2 = Step 4. Sketch the graph of the function g(x) = −3sin(4x). Step 5. Use the reciprocal relationship of the sine and cosecant function... |
stretching/compressing factor, |A| = 2. = 2π Step 3. The period is 2π π |B| 2 ⋅ 2 π = 4. = 2π 1 x⎞ ⎠ + 1. Step 4. The phase shift is 0 π 2 = 0. Step 5. Draw the graph of y = Acsc(Bx) but shift it up D = 1. Step 6. Sketch the vertical asymptotes, which occur at x = 0, x = 2, x = 4. The graph for this function is shown ... |
show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent, cot x has vertical asymptotes at all values of x where tan x = 0, and cot x = 0 at all values of x where tan x has its vertical asymptotes. This content is available for free at https://cnx.org/content/col11758/1.5 C... |
reference points. 6. Use the reciprocal relationship between tangent and cotangent to draw the graph of y = Acot(Bx). 7. Sketch the asymptotes. Example 8.21 Graphing Variations of the Cotangent Function Determine the stretching factor, period, and phase shift of y = 3cot(4x), and then sketch a graph. Solution Step 1. ... |
. π 2 944 Chapter 8 Periodic Functions Solution Step 1. The function is already written in the general form f (x) = Acot(Bx − C) + D. Step 2. A = 4, so the stretching factor is 4. Step 3. B =, so the period is P = = 8. = π 8 π |B| π π 8 Step 4. C = π 2, so the phase shift is C B = π 2 π 8 = 4. ⎛ Step 5. We draw f (x) =... |
interval ⎡ ⎣0, 5⎤ ⎦. c. Evaluate f (1) and discuss the function’s value at that input. Solution a. We know from the general form of y = Atan(Bt) that |A| is the stretching factor and π B is the period. Figure 8.50 We see that the stretching factor is 5. This means that the beam of light will have moved 5 ft after half... |
of the functions. 58. f (x) = 2tan(4x − 32) 59. 60. ⎛ h(x) = 2sec ⎝ ⎞ (x + 1) ⎠ π 4 ⎛ m(x) = 6csc ⎝ x + π⎞ ⎠ π 3 61. If tan x = −1.5, find tan(−x). 62. If sec x = 2, find sec(−x). 63. If csc x = −5, find csc(−x). This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 8 Periodic Functions 94... |
��x − ⎞ ⎠ π 2 72. f (x) = 4tan(x) 73. ⎛ f (x) = tan ⎝x + ⎞ ⎠ π 4 74. f (x) = πtan(πx − π) − π 75. f (x) = 2csc(x) 76. f (x) = − 1 4 csc(x) 77. f (x) = 4sec(3x) 78. f (x) = − 3cot(2x) 79. f (x) = 7sec(5x) 80. 81. 82. 83. f (x) = 9 10 csc(πx) ⎛ ⎝x + f (x) = 2csc ⎞ ⎠ − 1 π 4 ⎛ ⎝x − f (x) = − sec ⎞ ⎠ − 2 π 3 f (x) = 7 5 ⎛ ... |
. Graph g(x) on the interval ⎡ ⎣0, 35⎤ ⎦. b. Evaluate g(5) and interpret the information. c. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? d. Find and discuss the meaning of any vertical asymptotes. A video camera is focused on a ... |
(x). d. Calculate and interpret d⎛ ⎝− ⎞ ⎠. Round to the π 3 second decimal place. e. Calculate and interpret d⎛ ⎝ ⎞ ⎠. Round to the second π 6 decimal place. f. What is the minimum distance between the fisherman and the boat? When does this occur? Figure 8.52 A laser rangefinder is locked on a comet approaching 104. Ea... |
• Since cos(π) = − 1, then π = cos−1 (−1). ⎛ • Since tan ⎝ π 4 ⎞ ⎠ = 1, then π 4 = tan−1 (1). In previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value. For this, we need inverse functions. Recall that... |
domain of the tangent function also has the useful property that it extends from one vertical asymptote to the next instead of being divided into two parts by an asymptote. On these restricted domains, we can define the inverse trigonometric functions. • The inverse sine function y = sin−1 x means x = sin y. The inver... |
53 Figure 8.58 The tangent function and inverse tangent (or arctangent) function Relations for Inverse Sine, Cosine, and Tangent Functions For angles in the interval ⎡ ⎣− π 2, π 2 ⎤ ⎦, if sin y = x, then sin−1 x = y. For angles in the interval [0, π], if cos y = x, then cos−1 x = y. For angles in the interval ⎛ ⎝− π 2,... |
Evaluating Inverse Trigonometric Functions for Special Input Values Evaluate each of the following. a. b. c. d. ⎞ ⎠ sin−1 ⎛ ⎝ 1 2 sin−1 ⎛ ⎝− 2 2 ⎞ ⎠ cos−1 ⎛ ⎝− 3 2 ⎞ ⎠ tan−1 (1) Solution a. Evaluating sin−1 ⎛ ⎝ 1 2 ⎞ ⎠ is the same as determining the angle that would have a sine value of 1 2. In other words, what angle... |
. 8.20 Evaluate each of the following. a. b. c. d. sin−1(−1) tan−1 (−1) cos−1 (−1) cos−1 ⎛ ⎝ ⎞ ⎠ 1 2 Using a Calculator to Evaluate Inverse Trigonometric Functions To evaluate inverse trigonometric functions that do not involve the special angles discussed previously, we will need to use a calculator or other type of t... |
use of length h and the side of length p opposite to the desired angle is given, use the equation θ = sin−1 ⎛ ⎝ ⎞ ⎠. p h 3. If the two legs (the sides adjacent to the right angle) are given, then use the equation θ = tan−1 ⎛ ⎝ p a ⎞ ⎠. Example 8.27 Applying the Inverse Cosine to a Right Triangle Solve the triangle in F... |
Compositions of a trigonometric function and its inverse sin(sin−1 x) = x for − 1 ≤ x ≤ 1 cos(cos−1 x) = x for − 1 ≤ x ≤ 1 tan(tan−1 x) = x for − ∞ < x < ∞ sin−1(sin x) = x only for − π 2 ≤ x ≤ cos−1(cos x) = x only for 0 ≤ x ≤ π tan−1(tan x ) = x only for − < x < π 2 π 2 π 2 Is it correct that sin−1(sin x) = x? No. T... |
�� ⎠ 2π 3 cos−1 ⎛ ⎛ ⎝− ⎝cos ⎞ ⎞ ⎠ ⎠ π 3 Solution a. b. c. ⎡ is in ⎣− π 3 π 2, π 2 ⎤ ⎦, so sin−1 ⎛ ⎛ ⎝sin ⎝ ⎞ ⎞ ⎠ = ⎠ π 3 π 3., ⎤ ⎛ ⎦, but sin ⎝ π 2 ⎡ ⎣− is not in π 2 is in [0, π], so cos−1 ⎛ ⎛ ⎝cos ⎝ 2π 3 2π 3 d. − e. π 3 ⎛ is not in [0, π], but cos ⎝− π 3 is in [0, π], so cos−1 ⎛ ⎛ ⎝− ⎝cos ⎞ ⎞ ⎠ = ⎠ π 3 π 3. 2π 3 ⎛ ⎞... |
� ⎠. For special values of x, we can exactly evaluate the inner function and then the outer, inverse function. However, we can find a more general approach by considering the relation between the two acute angles of a right triangle where one is θ, making the other π − θ. Consider the sine and cosine of each angle of t... |
Evaluating the Composition of an Inverse Sine with a Cosine Evaluate sin−1 ⎛ ⎛ ⎝cos ⎝ 13π 6 ⎞ ⎞ ⎠ ⎠ a. by direct evaluation. b. by the method described previously. Solution a. Here, we can directly evaluate the inside of the composition. ) = cos( cos(13π 6 π 6 π = cos( 6 + 2π) ) Now, we can evaluate the inverse functi... |
4 5, sin2 θ + cos2 θ = 1 sin2 θ + (4 5 = 1 ) 2 Use our known value for cosine. Solve for sine. sin2 θ = 1 − 16 25 sin θ = ± 9 25 = ± 3 5 Since θ = cos−1 ⎛ ⎝ 4 5 ⎞ ⎠ is in quadrant I, sin θ must be positive, so the solution is 3 5. See Figure 8.63. This content is available for free at https://cnx.org/content/col11758/... |
= 7 65 65 8.26 ⎝sin−1 ⎛ ⎛ Evaluate cos ⎝ ⎞ ⎞ ⎠. ⎠ 7 9 Example 8.32 Finding the Cosine of the Inverse Sine of an Algebraic Expression ⎝sin−1 ⎛ ⎛ Find a simplified expression for cos ⎝ ⎞ ⎞ ⎠ for − 3 ≤ x ≤ 3. ⎠ x 3 Solution We know there is an angle θ such that sin θ = x 3. sin2 θ + cos2 + cos2 θ = 1 cos2 θ = 1 − x2 9 co... |
cos−1 (−0.4) inverse functions, why is cos−1 ⎛ ⎛ ⎝− ⎝cos − π 6? ⎞ ⎞ ⎠ not equal to ⎠ π 6 108. Explain the meaning of π 6 = arcsin(0.5). 109. Most calculators do not have a key to evaluate sec−1 (2). Explain how this can be done using the cosine function or the inverse cosine function. 123. arcsin(0.23) 124. ⎛ arccos ⎝... |
� ⎠ cos−1 ⎛ ⎛ ⎝sin ⎝ ⎞ ⎞ ⎠ ⎠ π 3 tan−1 ⎛ ⎛ ⎝sin ⎝ ⎞ ⎞ ⎠ ⎠ π 3 sin−1 ⎛ ⎛ ⎝cos ⎝ −π 2 ⎞ ⎞ ⎠ ⎠ tan−1 ⎛ ⎛ ⎝sin ⎝ ⎞ ⎞ ⎠ ⎠ 4π 3 Chapter 8 Periodic Functions 135. 136. sin−1 ⎛ ⎛ ⎝sin ⎝ ⎞ ⎞ ⎠ ⎠ 5π 6 tan−1 ⎛ ⎛ ⎝sin ⎝ −5π 2 ⎞ ⎞ ⎠ ⎠ 137. ⎝sin−1 ⎛ ⎛ cos ⎝ 138. ⎝cos−1 ⎛ ⎛ sin ⎝ ⎞ ⎞ ⎠ ⎠ 4 5 ⎞ ⎞ ⎠ ⎠ 3 5 139. 140. 141. ⎝tan−1 ⎛ ⎛ sin ... |
⎞ ⎠ ⎠ Extensions the following exercises, evaluate the expression For without using a calculator. Give the exact value. 147. sin−1 ⎛ 1 ⎝ 2 cos−1 ⎛ ⎝ 2 2 ⎞ ⎠ − cos−1 ⎛ ⎝ ⎞ ⎠ − sin−1 ⎛ 3 2 ⎝ ⎞ ⎠ + sin−1 ⎛ 3 ⎝ 2 ⎞ ⎠ + cos−1 ⎛ ⎝ ⎞ ⎠ − cos−1 (1) ⎞ ⎠ − sin−1 (0) 1 2 2 2 For the sin t = following exercises, x x + 1. find the... |
truss for the roof of a house is constructed from two 162. identical right triangles. Each has a base of 12 feet and height of 4 feet. Find the measure of the acute angle adjacent to the 4-foot side. 163. The line y = 3 5 x passes through the origin in the x,y- plane. What is the measure of the angle that the line mak... |
tangent equal to a given number midline the horizontal line y = D, where D appears in the general form of a sinusoidal function periodic function a function f (x) that satisfies f (x + P) = f (x) for a specific constant P and any value of x phase shift the horizontal displacement of the basic sine or cosine function; ... |
4. • Combinations of variations of sinusoidal functions can be detected from an equation. See Example 8.5. • The equation for a sinusoidal function can be determined from a graph. See Example 8.6 and Example 8.7. • A function can be graphed by identifying its amplitude and period. See Example 8.8 and Example 8.9. • A f... |
. • Because the trigonometric functions are not one-to-one on their natural domains, inverse trigonometric functions are defined for restricted domains. • For any trigonometric function f (x), if x = f −1(y), then f (x) = y. However, f (x) = y only implies x = f −1(y) if x is in the restricted domain of f. See Example ... |
otes. 168. f (x) = − 3cos x + 3 169. f (x) = 1 4 sin x 170. ⎛ ⎝x + f (x) = 3cos ⎞ ⎠ π 6 171. ⎛ ⎝x − 2π f (x) = − 2sin 3 ⎞ ⎠ 172. ⎛ f (x) = 3sin ⎝x − ⎞ ⎠ − 4 π 4 173. f (x) = 2 ⎛ ⎛ ⎝x − 4π ⎝cos 3 ⎞ ⎞ ⎠ + 1 ⎠ 174. ⎛ ⎝3x − f (x) = 6sin ⎞ ⎠ − 1 π 6 175. f (x) = − 100sin(50x − 20) 177. ⎛ ⎝x − f (x) = 2tan ⎞ ⎠ π 6 178. f (x)... |
1) 186. What is the largest and smallest population the city may have? 187. Graph the function on the domain of [0, 40]. 188. What are the amplitude, period, and phase shift for the function? 189. Over this domain, when does the population reach 18,000? 13,000? 190. What is the predicted population in 2007? 2010? For t... |
Describe Figure 8.65 192. At time = 0, what is the displacement of the weight? 193. At what equilibrium point equal zero? time does the displacement from the 194. What is the time required for the weight to return to its initial height of 5 inches? In other words, what is the period for the displacement function? Inve... |
� 6 226. ⎛ y = 8sin ⎝ 7π 6 x + 7π 2 ⎞ ⎠ + 6 972 Chapter 8 Periodic Functions 227. The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 68°F at midnight and the high and low temperatures during the day are 80°F and 56°F, respectively. Assuming t is... |
(x)? 238. What is the smallest possible value for f (x)? 239. Where is the function increasing on the interval [0, 2π]? For the following exercises, find and graph one period of the periodic function with the given amplitude, period, and phase shift. 240. Sine curve with amplitude 3, period π 3, and phase shift (h, k ... |
5π ⎠ 6 257. The grade of a road is 7%. This means that for every horizontal distance of 100 feet on the road, the vertical rise is 7 feet. Find the angle the road makes with the horizontal in radians. 974 Chapter 8 Periodic Functions This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 9 ... |
ometric expressions. Verifying the Fundamental Trigonometric Identities Identities enable us to simplify complicated expressions. They are the basic tools of trigonometry used in solving trigonometric equations, just as factoring, finding common denominators, and using special formulas are the basic tools of solving al... |
sin2 θ = sin2 θ + cos2 θ sin2 θ Rewrite the left side. ⎞ ⎠ Write both terms with the common denominator. = 1 sin2 θ = csc2 θ Similarly, 1 + tan2 θ = sec2 θ can be obtained by rewriting the left side of this identity in terms of sine and cosine. This gives 1 + tan2 θ = 1 + 2 ⎞ ⎠ ⎛ ⎝ sin θ cos θ 2 sin θ ⎞ = ⎠ cos θ = co... |
sin ⎝ = −1 ⎞ ⎠ π 2 This is shown in Figure 9.3. Figure 9.3 Graph of y = sin θ Recall that an even function is one in which f (−x) = f (x) for all x in the domain of f The graph of an even function is symmetric about cos( − θ) = cos θ. For example, consider corresponding inputs π 4 the y-axis. The cosine function is an ... |
ot θ. Cotangent is therefore an odd function, which means that of a negative angle as cot(−θ) = cos(−θ) sin(−θ) = cos θ cot(−θ) = − cot(θ) for all θ in the domain of the cotangent function. The cosecant function is the reciprocal of the sine function, which means that the cosecant of a negative angle will be interprete... |
(9.2) (9.3) The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle. tan(−θ) = − tan θ (9.4) 980 Chapter 9 Trigonometric Identities and Equations cot(−θ) = − cot θ sin(−θ) = − sin θ csc(−θ) = − csc θ cos(−θ) = cos θ sec(−θ) = sec θ The re... |
opportunities to use the identities and make the proper substitutions. 4. If these steps do not yield the desired result, try converting all terms to sines and cosines. Example 9.2 Verifying a Trigonometric Identity Verify tan θcos θ = sin θ. Solution We will start on the left side, as it is the more complicated side:... |
⎠ = ⎛ sin2 θ ⎝ cos2 θ ⎛ ⎜ sin2 θ cos2 θ ⎝ = sin2 θ = cos2 θ = 1 sec2 θ tan2 θ = sin2 θ cos2 θ There is more than one way to verify an identity. Here is another possibility. Again, we can start with the left side. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 9 Trigonometric Identit... |
�) − cos2(−θ) sin(−θ) − cos(−θ) = = = = = [sin(−θ)]2 − [cos(−θ)]2 sin(−θ) − cos(−θ) (−sin θ)2 − (cos θ)2 −sin θ − cos θ (sin θ)2 − (cos θ)2 −sin θ − cos θ (sin θ − cos θ)(sin θ + cos θ) −(sin θ + cos θ) (sin θ − cos θ)(sin θ + cos θ ) −(sin θ + cos θ ) = cos θ − sin θ sin(−x) = −sin x and cos(−x) = cos x Diffe ence of ... |
formulas of algebra, such as the difference of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations. For example, the equation (sin x + 1)(sin x − 1) = 0 resembles the equation (x + 1)(x − 1) = 0, which uses the factored form of the ... |
we could solve each factor using the zero factor property. We could also use substitution like we did in the previous problem and let cos θ = x, rewrite the expression as 4x2 − 1, and factor (2x − 1)(2x + 1). Then replace x with cos θ and solve for the angle. 9.4 Rewrite the trigonometric expression using the differen... |
for sec t, explain why the function is undefined at certain points. All of the Pythagorean identities are related. Describe from 4. how to manipulate sin2 t + cos2 t = 1 to the other forms. equations get the to Algebraic For the following exercises, use the fundamental identities to fully simplify the expression. 5. 6... |
sin2 x cos x⎛ ⎝tan x − sec(−x)⎞ ⎠ = sin x − 1 1 + sin2 x cos2 x = 1 cos2 x + sin2 x cos2 x = 1 + 2 tan2 x (sin x + cos x)2 = 1 + 2 sin xcos x cos2 x − tan2 x = 2 − sin2 x − sec2 x Extensions For the following exercises, prove or disprove the identity. 34. 1 1 + cos x − 1 1 − cos( − x) = − 2 cot x csc x Chapter 9 Trigo... |
feet (6,168 m) above sea level. It is the highest peak in North America. (credit: Daniel A. Leifheit, Flickr) How can the height of a mountain be measured? What about the distance from Earth to the sun? Like many seemingly impossible problems, we rely on mathematical formulas to find the answers. The trigonometric ide... |
α − β. ⎝cos β, sin β⎞ ⎠; and Label two more points: A at an angle of ⎛ point B with coordinates (1, 0). Triangle POQ is a rotation of triangle AOB and thus the distance from P to Q is the same as the distance from A to B. ⎠ from the positive x-axis with coordinates ⎛ ⎝α − β⎞ ⎝α − β⎞ ⎝α − β⎞ ⎠, sin⎛ ⎝cos⎛ ⎞ ⎠ This cont... |
− 2 cos α cos β − 2 sin α sin β = 2 − 2 cos(α − β) Finally we subtract 2 from both sides and divide both sides by −2. cos α cos β + sin α sin β = cos(α − β) Thus, we have the difference formula for cosine. We can use similar methods to derive the cosine of the sum of two angles. Sum and Difference Formulas for Cosine ... |
of cos(75°). Solution As 75° = 45° + 30°, we can evaluate cos(75°) as cos(45° + 30°). This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 9 Trigonometric Identities and Equations 993 cos(α + β) = cos α cos β − sin α sin β cos(45° + 30°) = cos(45°)cos(30°) − sin(45°)sin(30°) ⎞ ⎠ − Keep in... |
sin(45° − 30°) sin(135° − 120°) Solution a. Let’s begin by writing the formula and substitute the given angles. sin(α − β) = sin α cos β − cos α sin β sin(45° − 30°) = sin(45°)cos(30°) − cos(45°)sin(30°) Next, we need to find the values of the trigonometric expressions. sin(45°) = 2 2, cos(30°) = 3 2, cos(45°) = 2 2, ... |
= 16 25 = 4 5 Using the sum formula for sine, ⎛ ⎝cos−1 1 sin 2 + sin−1 3 5 ⎞ ⎠ = sin(α + β) = sin α cos β + cos α sin 10 Using the Sum and Difference Formulas for Tangent Finding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing t... |
into the formula. tan(α + β) = ⎛ tan ⎝ π 6 + ⎞ ⎠ = π 4 tan α + tan β 1 − tan α tan β ⎛ ⎞ ⎛ ⎞ π π ⎠ + tan tan ⎝ ⎠ ⎝ tan ⎝tan ⎝ ⎝ ⎠ ⎠ 4 6 1 − ⎞ ⎞ ⎠ ⎠ Next, we determine the individual function values within the formula: This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 9 Trigonometric Id... |
ometric Identities and Equations Figure 9.9 Since cos β = − 5 13 and π < β < 3π 2, the side adjacent to β is −5, the hypotenuse is 13, and β is in the third quadrant. See Figure 9.10. Again, using the Pythagorean Theorem, we have (−5)2 + a2 = 132 25 + a2 = 169 a2 = 144 a = ±12 Since β is in the third quadrant, a = –12.... |
12 13 and cos β = − 5 13, then Then, tan β = −12 13 −5 13 = 12 5 tan(α + β) = 3 ⎞ ⎠ = 5 ⎛ 12 ⎝ 5 tan α + tan β 1 − tan α tan β 4 + 12 1 − 3 4 63 20 − 16 20 = − 63 16 = d. To find tan⎛ ⎝α − β⎞ ⎠, we have the values we need. We can substitute them in and evaluate. 1000 Chapter 9 Trigonometric Identities and Equations ta... |
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