Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 1 Homework, Page 401 Identify the graph of each function. 1.

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. 5.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 3 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. 9.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 4 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 5 Homework, Page 401 Analyze each function for Domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. 17.

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. 21.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 7 Homework, Page 401 Describe the transformations required to obtain the graph of the given function from a basic trigonometric function. 25.

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. 29.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 9 Homework, Page 401 Solve for x in the given interval, using reference triangles in the proper quadrants. 33.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 401 Use a calculator to solve for x in the given interval. 37.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page The figure shows a unit circle and an angle t whose terminal side is in Quadrant III. (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 Homework, Page (b) Explain why. (c) Find tan t – π and show that tan (t) = tan (t – π).

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page (d) Explain why the period of the tangent function is π.. (e) Explain why the period of the cotangent function is π.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page The Bolivar Lighthouse is located on a small island 350 ft from the shore of the mainland. (a) Express the distance d as a function of the angle x. (b) If x = 1.55 rad, what is d?

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 401 Find approximate solutions for the equation in the interval 49.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 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 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.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page Write csc x as a horizontal translation of sec x.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page A film of liquid in a thin tube has surface tension γ given by where h is the height of liquid in the tube, ρ is the density of the liquid, g = 9.8 m/sec 2 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 N/m and a density of 1050 kg/m 3. 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 Homework, Page Cont’d

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4.6 Graphs of Composite Trigonometric Functions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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 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. Slide 4- 25

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Combining the Cosine Function with x 2

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Combining the Cosine Function with x 2

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Absolute Values of Trigonometric Functions The absolute value of a trig function plots as a periodic function. Slide 4- 28

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Sums That Are Sinusoidal Functions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Identifying a Sinusoid

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Identifying a Sinusoid

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Identifying a Non-Sinusoid

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Damped Oscillation

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Working with Damped Oscillation

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. Slide 4- 36

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework Homework Assignment #31 Read Section 4.7 Page 411, Exercises: 1 – 93 (EOO)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4.7 Inverse Trigonometric Functions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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 Inverse Sine Function

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Inverse Sine Function (Arcsine Function)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Evaluate sin -1 x Without a Calculator

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Evaluate sin -1 x Without a Calculator

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Inverse Cosine (Arccosine Function)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Inverse Cosine (Arccosine Function)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Inverse Tangent Function (Arctangent Function)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Inverse Tangent Function (Arctangent Function)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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 Composing Trigonometric and Inverse Trigonometric Functions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Composing Trig Functions with Arcsine

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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 height of the balloon. 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 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 vice- versa? Explain.