Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Homework, Page 401 Identify the graph of each function. 1. Graphs of one period of csc x and 2csc x are shown. The graph of csc x is in blue, y and the graph of 2csc x is in red. x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 1 Homework, Page 401 Describe the graph of the function in terms of a basic trigonometric function. Locate the vertical asymptotes and graph two periods of the function. 5. y y tan 2 x y tan 2 x p 2 Vertical asymptotes at 2n 1 x ,n 4 any integer x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 2 Homework, Page 401 Describe the graph of the function in terms of a basic trigonometric function. Locate the vertical asymptotes and graph two periods of the function. y 9. y 2cot 2 x y 2cot 2 x p 2 Vertical asymptotes at 2n x , n any integer 4 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 3 Homework, Page 401 Match the trigonometric function with its graph. Then give Xmin and Xmax for the viewing window in which the graph is shown. 13. y 2tan x y 2 tan x matches graph (a) which has an Xmin of and an Xmax of Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 4 Homework, Page 401 Analyze each function for Domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. 17. y cot x Domain : x : x n , where n is any integer Range : y : y or , ; continuous on domain; decreasing on domain; symmetrical about the origin; unbounded; no extrema; no horizontal asymptotes; vertical asymptotes at x n ; lim cot x and x lim cot x do not exist x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 5 Homework, Page 401 Describe the transformations required to obtain the graph of the given function from a basic trigonometric function. 21. y 3tan x To obtain the graph of y 3 tan x from the graph of y tan x, apply a vertical stretch of 3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 6 Homework, Page 401 Describe the transformations required to obtain the graph of the given function from a basic trigonometric function. 25. y 3cot 1 x 2 1 y 3cot x p 2 1 2 2 1 To obtain the graph of y 3cot x from the graph 2 of y cot x, apply a horizontal stretch of 2, a vertical stretch of 3, and reflect about the x -axis. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 7 Homework, Page 401 Solve for x in the given interval, using reference triangles in the proper quadrants. 29. sec x 2, 0 x 2 1 1 2 cos x x cos x 2 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 8 Homework, Page 401 Solve for x in the given interval, using reference triangles in the proper quadrants. 33. csc x 1, 2 x 5 2 1 1 sin x 1 x 5 2 2 sin x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 9 Homework, Page 401 Use a calculator to solve for x in the given interval. 37. cot x 0.6, 3 2 x 2 1 tan x 0.6 x 1.030 2 5.253 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 10 Homework, Page 401 41. The figure shows a unit circle and an angle t whose terminal side is in Quadrant III. y x^2 + y^2 = 1 P1 (-a, -b) t t-pi x P2 (a, b) (a) If the coordinates of P2 are (a, b), explain why the coordinates of point P1 on the circle and the terminal side of the angle t – π are (-a, -b). The line connecting P1 and P2 is a straight-line passing through the origin, so the points at which the line intersects the unit circle are reflections about the origin. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 11 Homework, Page 401 b 41. (b) Explain why tan t . a b tan t because the definition of the tangent of an a angle is opposite over adjacent and the opposite side of the triangle has measure b and the adjacent side has measure a. (c) Find tan t – π and show that tan (t) = tan (t – π). b b tan t tan t a a Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 12 Homework, Page 401 41. . (d) Explain why the period of the tangent function is π. The tangent function has a period of because the as shown in the example above, its values repeat after radians. (e) Explain why the period of the cotangent function is π. If the tangent function has a period of , its reciprocal function cotangent must also have a period of radians. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 13 Homework, Page 401 45. The Bolivar Lighthouse is located on a small island 350 ft from the shore of the mainland. 350 ft x d (a) Express the distance d as a function of the angle x. 350 350 cos x d d 350sec x d cos x (b) If x = 1.55 rad, what is d? d 350sec x 350sec1.55 16,831.108 ft Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 14 Homework, Page 401 Find approximate solutions for the equation in the interval x 49. sec x 5cos x x 2.034, 1.107,1.107,2.034 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 15 Homework, Page 401 53. The graph of y = cot x can be obtained by a horizontal shift of (a) – tan x (b) – cot x (c) sec x (d) tan x (e) csc x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 16 Homework, Page 401 Graph both f and g in the [–π, π] by [–10, 10] viewing window. Estimate values in the interval [–π, π] for which f > g. 57. f x 5sin x and g x cot x f g on 0.439,0 0.439, Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 17 Homework, Page 401 61. Write csc x as a horizontal translation of sec x. 3 csc x sec x or csc x sec x 2 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 18 Homework, Page 401 65. A film of liquid in a thin tube has surface tension γ given by 0.5h gr sec where h is the height of liquid in the tube, ρ is the density of the liquid, g = 9.8 m/sec2 is the acceleration due to gravity, r is the radius of the tube, and φ is the angle of contact between the tube and the liquid’s surface. Whole blood has a surface tension of 0.058 N/m and a density of 1050 kg/m3. Suppose the blood rises to a height of 1.5 m in a capillary blood vessel of radius 4.7 x 10–6 m. What is the contact angle between the capillary vessel and the blood surface? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 19 Homework, Page 401 65. Cont’d 2 sec 0.5h gr h gr sec sec 2 0.058kg / sec 2 3 2 6 1.5 m 1050 kg / m 9.8 m / sec 4.7 10 m 2 0.058 1.51050 9.8 4.7 10 6 1.599 1 cos 51.290 1.599 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 20 4.6 Graphs of Composite Trigonometric Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quick Review State the domain and range of the function. 1. f ( x) -3sin 2 x 2. f ( x) | x | 2 3. f ( x) 2 cos 3 x 4. Describe the behavior of y e as x . -3 x 5. Find f g and g f , given f ( x) x 3 and g ( x) x 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 22 Quick Review Solutions State the domain and range of the function. 1. f ( x) -3sin 2 x Domain: , Range: 3,3 2. f ( x) | x | 2 Domain: , Range: 2, 3. f ( x) 2 cos 3 x Domain: , Range: 2, 2 4. Describe the behavior of y e as x . -3 x lim e x 3 x 0 5. Find f g and g f , given f ( x) x 3 and g ( x ) x 2 f g x 3; g f x 3 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 23 What you’ll learn about Combining Trigonometric and Algebraic Functions Sums and Differences of Sinusoids Damped Oscillation … and why Function composition extends our ability to model periodic phenomena like heartbeats and sound waves. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 24 Leading Questions The absolute value of a periodic function is a periodic function. Adding a linear function to a periodic function yields a new periodic function. Adding sinusoids of different periods will yield a new sinusoid. Adding sinusoids of different periods will yield a periodic function. Damped oscillation refers to a condition where the amplitude of a function varies. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 25 Example Combining the Cosine Function with x2 Graph y cos x and state whether the function 2 appears to be periodic. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 26 Example Combining the Cosine Function with x2 Graph y cos x 2 and state whether the function appears to be periodic. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 27 Absolute Values of Trigonometric Functions The absolute value of a trig function plots as a periodic function. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 28 Sums That Are Sinusoidal Functions If y a sin(b( x h )) and y a cos(b( x h )), then 1 1 1 2 2 2 y y a sin(b( x h )) a cos(b( x h )) is a 1 2 1 1 2 2 sinusoid with period 2 / | b|. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 29 Sums That Are Not Sinusoidal Functions If y a sin(b( x h )) and y f ( x) where f ( x) is not 1 1 1 2 a sin(b( x h )) or a cos(b( x h )), but another 2 2 2 2 trigonometric function, then y y is a periodic 1 2 function, but not a sinusoid. If y f ( x) is not a trigonometric function, then y y 2 1 2 is neither periodic nor sinusoidal. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 30 Example Identifying a Sinusoid Determine whether the following function is or is not a sinusoid: f ( x) 3cos x 5sin x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 31 Example Identifying a Sinusoid Determine whether the following function is or is not a sinusoid: f ( x) cos3x sin 5x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 32 Example Identifying a Non-Sinusoid Determine whether the following function is or is not a sinusoid: f ( x) 3x sin 5x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 33 Damped Oscillation The graph of y f ( x) cos bx (or y f ( x) sin bx) oscillates between the graphs of y f ( x ) and y f ( x ). When this reduces the amplitude of the wave, it is called damped oscillation. The factor f ( x) is called the damping factor. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 34 Example Working with Damped Oscillation The oscillations of a spring subject to friction are modeled by the equation y 0.43e cos1.8t . 0.55 t a Graph y and its two damping curves in the same viewing window for 0 t 12. b Approximately how long does it take for the spring to be damped so that 0.2 y 0.2? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 35 Following Questions Inverse trig function is just another name for a reciprocal function of a trig function. Arccosine (x) and cos –1 (x) are the same thing. All inverse trig functions have restricted domains. All inverse trig functions have restricted ranges. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 36 Homework Homework Assignment #31 Read Section 4.7 Page 411, Exercises: 1 – 93 (EOO) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 37 4.7 Inverse Trigonometric Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quick Review State the sign (positive or negative) of the sine, cosine, and tangent in quadrant 1. I 2. III Find the exact value. 3. cos 6 4 4. tan 3 11 5. sin 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 39 Quick Review Solutions State the sign (positive or negative) of the sine, cosine, and tangent in quadrant 1. I +,+,+ 2. III ,,+ Find the exact value. 3. cos 6 4 4. tan 3 11 5. sin 6 3/2 3 1/ 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 40 What you’ll learn about Inverse Sine Function Inverse Cosine and Tangent Functions Composing Trigonometric and Inverse Trigonometric Functions Applications of Inverse Trigonometric Functions … and why Inverse trig functions can be used to solve trigonometric equations. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 41 Inverse Sine Function f x sin x 2 x 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley f x sin x 1 1 x 1 Slide 4- 42 Inverse Sine Function (Arcsine Function) The unique angle y in the interval / 2, / 2 such that sin y x is the inverse sine (or arcsine) of x, denoted sin 1 x or arcsin x. The domain of y sin 1 x is [ 1,1] and the range is / 2, / 2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 43 Example Evaluate sin-1x Without a Calculator 1 Find the exact value without a calculator: sin 2 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 44 Example Evaluate sin-1x Without a Calculator Find the exact value without a calculator: sin sin . 10 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 45 Inverse Cosine (Arccosine Function) f x cos x 0 x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley f x cos x 1 1 x 1 Slide 4- 46 Inverse Cosine (Arccosine Function) The unique angle y in the interval 0, such that cos y x is the inverse cosine (or arccosine) of x, 1 denoted cos x or arccos x. The domain of y cos 1 x is [ 1,1] and the range is 0, . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 47 Inverse Tangent Function (Arctangent Function) f x tan x 2 x 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley f x tan x 1 x Slide 4- 48 Inverse Tangent Function (Arctangent Function) The unique angle y in the interval ( / 2, / 2) such that tan y x is the inverse tangent (or arctangent ) of x, denoted tan 1 x or arctan x. The domain of y tan 1 x is (-,) and the range is ( / 2, / 2). Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 49 End Behavior of the Tangent Function Recognizing that the graphs of inverse functions are reflected about the line y = x, we see that vertical asymptotes of y = tan x become the horizontal asymptotes of y = tan–1 x and the range of y = tan x becomes the domain of y = tan–1 x . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 50 Composing Trigonometric and Inverse Trigonometric Functions The following equations are always true whenever they are defined: sin sin 1 x x tan tan cos cos 1 x x 1 x x The following equations are only true for x values in the "restricted" domains of sin, cos, and tan: sin 1 sin x x cos 1 cos x x tan 1 tan x x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 51 Example Composing Trig Functions with Arcsine Compose each of the six basic trig functions with sin 1 x and reduce the composite function to an algebraic expression involving no trig functions. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 52 Example Applying Inverse Trig Functions A person is watching a balloon rise straight up from a place 500 ft from the launch point. a. Write θ as a function of s, the s height of the balloon. 500 ft b. Is the change in θ greater as s changes from 10 ft to 20 ft or as s changes from 200 ft to 210 ft? Explain. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 53 Example Applying Inverse Trig Functions c. In the graph of this relationship, does the x-axis represent s height and the y-axis represent θ (in degrees) or viceversa? Explain. 0,1500 by 5,80 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 54