Essential Question What are the characteristics of the graph of the tangent function?

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1 8.5 Graphing Other Trigonometric Functions Essential Question What are the characteristics of the graph of the tangent function? Graphing the Tangent Function Work with a partner. a. Complete the table for = tan, where is an angle measure in radians. 0 = tan = tan b. The graph of = tan has vertical asmptotes at -values where tan is undefined. Plot the points (, ) from part (a). Then use the asmptotes to sketch the graph of = tan. MAKING SENSE OF PROBLEMS To be proficient in math, ou need to consider analogous problems and tr special cases of the original problem in order to gain insight into its solution. c. For the graph of = tan, identif the asmptotes, the -intercepts, and the intervals for which the function is increasing or decreasing over. Is the tangent function even, odd, or neither? Communicate Your Answer. What are the characteristics of the graph of the tangent function?. Describe the asmptotes of the graph of = cot on the interval < <. Section 8.5 Graphing Other Trigonometric Functions 5

2 8.5 Lesson What You Will Learn Core Vocabular Previous asmptote period amplitude -intercept transformations local maimum local minimum Eplore characteristics of tangent and cotangent functions. Graph tangent and cotangent functions. Graph secant and cosecant functions. Eploring Tangent and Cotangent Functions The graphs of tangent and cotangent functions are related to the graphs of the parent functions = tan and = cot, which are graphed below. approaches approaches.5.57 = tan Undef Undef. tan approaches tan approaches Because tan = sin cos, tan is undefined for -values at which cos = 0, such as = ± ±.57. The table indicates that the graph has asmptotes at these values. The table represents one ccle of the graph, so the period of the graph is. You can use a similar approach to graph = cot. Because cot = cos, cot is undefined for sin -values at which sin = 0, which are multiples of. The graph has asmptotes at these values. The period of the graph is also. = tan period: period: = cot STUDY TIP Odd multiples of are values such as these: ± = ± ± = ± ±5 = ± 5 Core Concept Characteristics of = tan and = cot The functions = tan and = cot have the following characteristics. The domain of = tan is all real numbers ecept odd multiples of. At these -values, the graph has vertical asmptotes. The domain of = cot is all real numbers ecept multiples of. At these -values, the graph has vertical asmptotes. The range of each function is all real numbers. So, the functions do not have maimum or minimum values, and the graphs do not have an amplitude. The period of each graph is. The -intercepts for = tan occur when = 0, ±, ±, ±,.... The -intercepts for = cot occur when = ±, ±, ± 5, ± 7,.... Chapter 8 Trigonometric Ratios and Functions

3 Graphing Tangent and Cotangent Functions The graphs of = a tan b and = a cot b represent transformations of their parent functions. The value of a indicates a vertical stretch (a > ) or a vertical shrink (0 < a < ). The value of b indicates a horizontal stretch (0 < b < ) or a horizontal shrink (b > ) and changes the period of the graph. Core Concept Period and Vertical Asmptotes of = a tan b and = a cot b The period and vertical asmptotes of the graphs of = a tan b and = a cot b, where a and b are nonzero real numbers, are as follows. The period of the graph of each function is b. The vertical asmptotes for = a tan b occur at odd multiples of The vertical asmptotes for = a cot b occur at multiples of b. b. Each graph below shows five ke -values that ou can use to sketch the graphs of = a tan b and = a cot b for a > 0 and b > 0. These are the -intercept, the -values where the asmptotes occur, and the -values halfwa between the -intercept and the asmptotes. At each halfwa point, the value of the function is either a or a. a a b b b b b b = a tan b = a cot b Graphing a Tangent Function Graph one period of g() = tan. Describe the graph of g as a transformation of the graph of f () = tan. SOLUTION The function is of the form g() = a tan b where a = and b =. So, the period is b =. Intercept: (0, 0) Asmptotes: = b = (), or = ; = b = (), or = Halfwa points: ( b, a ) = ( (), ) = (, ) ; ( b, a ) = ( (), ) = (, ) The graph of g is a vertical stretch b a factor of and a horizontal shrink b a factor of of the graph of f. Section 8.5 Graphing Other Trigonometric Functions 7

4 Graphing a Cotangent Function Graph one period of g() = cot. Describe the graph of g as a transformation of the graph of f () = cot. SOLUTION The function is of the form g() = a cot b where a = and b =. So, the period is b = =. Intercept: ( b, 0 ) ( =, 0 = (, 0) ( ) Asmptotes: = 0; = b =, or = Halfwa points: ( ) b, a ) ( =, ) = (, ) ( ; ( ) b ), a = ( ), ( = ( ), ) The graph of g is a horizontal stretch b a factor of of the graph of f. STUDY TIP Because sec = cos, sec is undefined for -values at which cos = 0. The graph of = sec has vertical asmptotes at these -values. You can use similar reasoning to understand the vertical asmptotes of the graph of = csc. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph one period of the function. Describe the graph of g as a transformation of the graph of its parent function.. g() = tan. g() = cot. g() = cot. g() = 5 tan Graphing Secant and Cosecant Functions The graphs of secant and cosecant functions are related to the graphs of the parent functions = sec and = csc, which are shown below. = sec 5 = cos = csc = sin period: Core Concept Characteristics of = sec and = csc 8 Chapter 8 Trigonometric Ratios and Functions period: The functions = sec and = csc have the following characteristics. The domain of = sec is all real numbers ecept odd multiples of. At these -values, the graph has vertical asmptotes. The domain of = csc is all real numbers ecept multiples of. At these -values, the graph has vertical asmptotes. The range of each function is and. So, the graphs do not have an amplitude. The period of each graph is.

5 To graph = a sec b or = a csc b, first graph the function = a cos b or = a sin b, respectivel. Then use the asmptotes and several points to sketch a graph of the function. Notice that the value of b represents a horizontal stretch or shrink b a factor of b, so the period of = a sec b and = a csc b is b. Graphing a Secant Function Graph one period of g() = sec. Describe the graph of g as a transformation of the graph of f () = sec. SOLUTION Step Graph the function = cos. The period is =. Step Graph asmptotes of g. Because the asmptotes of g occur when cos = 0, graph =, =, and =. Step Plot points on g, such as (0, ) and (, ). Then use the asmptotes to sketch the curve. = sec = cos The graph of g is a vertical stretch b a factor of of the graph of f. Graphing a Cosecant Function LOOKING FOR A PATTERN In Eamples and, notice that the plotted points are on both graphs. Also, these points represent a local maimum on one graph and a local minimum on the other graph. Graph one period of g() = csc. Describe the graph of g as a transformation of the graph of f () = csc. SOLUTION Step Graph the function = sin. The period is =. Step Graph asmptotes of g. Because the asmptotes of g occur when sin = 0, graph = 0, =, and =. Step Plot points on g, such as (, ) and (, ). Then use the asmptotes to sketch the curve. = sin = csc The graph of g is a vertical shrink b a factor of and a horizontal shrink b a factor of of the graph of f. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph one period of the function. Describe the graph of g as a transformation of the graph of its parent function. 5. g() = csc. g() = sec 7. g() = csc 8. g() = sec Section 8.5 Graphing Other Trigonometric Functions 9

6 8.5 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. WRITING Eplain wh the graphs of the tangent, cotangent, secant, and cosecant functions do not have an amplitude.. COMPLETE THE SENTENCE The and functions are undefined for -values at which sin = 0.. COMPLETE THE SENTENCE The period of the function = sec is, and the period of = cot is.. WRITING Eplain how to graph a function of the form = a sec b. Monitoring Progress and Modeling with Mathematics In Eercises 5, graph one period of the function. Describe the graph of g as a transformation of the graph of its parent function. (See Eamples and.) 5. g() = tan. g() = tan 7. g() = cot 8. g() = cot 9. g() = cot 0. g() = cot. g() = tan. g() = tan. ERROR ANALYSIS Describe and correct the error in finding the period of the function = cot. Period: b =. ERROR ANALYSIS Describe and correct the error in describing the transformation of f () = tan represented b g() = tan 5. A vertical stretch b a factor of 5 and a horizontal shrink b a factor of.. USING EQUATIONS Which of the following are asmptotes of the graph of = tan? A = 8 B = C = 0 D = 5 8 In Eercises 7, graph one period of the function. Describe the graph of g as a transformation of the graph of its parent function. (See Eamples and.) 7. g() = csc 8. g() = csc 9. g() = sec 0. g() = sec. g() = sec. g() = sec. g() = csc. g() = csc ATTENDING TO PRECISION In Eercises 5 8, use the graph to write a function of the form = a tan b ANALYZING RELATIONSHIPS Use the given graph to graph each function. a. f () = sec b. f () = csc = cos = sin Chapter 8 Trigonometric Ratios and Functions

7 USING STRUCTURE In Eercises 9, match the equation with the correct graph. Eplain our reasoning. 9. g() = tan 0. g() = cot. g() = csc. g() = sec. g() = sec. g() = csc A. B. 0. f () = csc ; vertical stretch b a factor of and a reflection in the -ais. MULTIPLE REPRESENTATIONS Which function has a greater local maimum value? Which has a greater local minimum value? Eplain. A. f () = csc B. 8 C. D.. ANALYZING RELATIONSHIPS Order the functions from the least average rate of change to the greatest average rate of change over the interval < <. A. B. E. F. C. D. 5. WRITING Eplain wh there is more than one tangent function whose graph passes through the origin and has asmptotes at = and =.. USING EQUATIONS Graph one period of each function. Describe the transformation of the graph of its parent function. a. g() = sec + b. g() = csc c. g() = cot( ) d. g() = tan WRITING EQUATIONS In Eercises 7 0, write a rule for g that represents the indicated transformation of the graph of f. 7. f () = cot ; translation units up and units left 8. f () = tan ; translation units right, followed b a horizontal shrink b a factor of 9. f () = 5 sec ( ); translation units down, followed b a reflection in the -ais. REASONING You are standing on a bridge 0 feet above the ground. You look down at a car traveling awa from the underpass. The distance d (in feet) the car is from the base of the bridge can be modeled b d = 0 tan θ. Graph the function. Describe what happens to θ as d increases. d θ 0 ft. USING TOOLS You use a video camera to pan up the Statue of Libert. The height h (in feet) of the part of the Statue of Libert that can be seen through our video camera after t seconds can be modeled b h = 00 tan t. Graph the function using a graphing calculator. What viewing window did ou use? Eplain. Section 8.5 Graphing Other Trigonometric Functions 5

8 5. MODELING WITH MATHEMATICS You are standing 8. HOW DO YOU SEE IT? Use the graph to answer 0 feet from the base of a 0-foot building. You watch our friend go down the side of the building in a glass elevator. each question. our friend d 0 d θ ou 0 ft a. What is the period of the graph? Not drawn to scale b. What is the range of the function? a. Write an equation that gives the distance d (in feet) our friend is from the top of the building as a function of the angle of elevation θ. b. Graph the function found in part (a). Eplain how the graph relates to this situation. c. Is the function of the form f () = a csc b or f () = a sec b? Eplain. 9. ABSTRACT REASONING Rewrite a sec b in terms of cos b. Use our results to eplain the relationship between the local maimums and minimums of the cosine and secant functions.. MODELING WITH MATHEMATICS You are standing 00 feet from the base of a 00-foot cliff. Your friend is rappelling down the cliff. a. Write an equation that gives the distance d (in feet) our friend is from the top of the cliff as a function of the angle of elevation θ. 50. THOUGHT PROVOKING A trigonometric equation that is true for all values of the variable for which both sides of the equation are defined is called a trigonometric identit. Use a graphing calculator to graph the function b. Graph the function found in part (a). c. Use a graphing calculator to determine the angle of elevation when our friend has rappelled halfwa down the cliff. = tan + cot. ( ) Use our graph to write a trigonometric identit involving this function. Eplain our reasoning. 7. MAKING AN ARGUMENT Your friend states that it 5. CRITICAL THINKING Find a tangent function whose is not possible to write a cosecant function that has the same graph as = sec. Is our friend correct? Eplain our reasoning. graph intersects the graph of = + sin onl at the local minimums of the sine function. Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Write a cubic function whose graph passes through the given points. (Section.9) 5. (, 0), (, 0), (, 0), (0, ) 5. (, 0), (, 0), (, 0), (0, ) 5. (, 0), (, 0), (, 0), (, ) 55. (, 0), (, 0), (, 0), (, ) Find the amplitude and period of the graph of the function. (Section 8.) Chapter 8 Int_Math_PE_08.05.indd 5 Trigonometric Ratios and Functions /0/5 :5 PM