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# Class Note for MATH 1330 at UH

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Date Created: 02/06/15

Math 1330 Section 53 Graphs of the Tangent Cotangent Secant and Cosecant Functions In this section you will learn to graph the rest of the trigonometric functions We can use some information from the graphs of gx sinx arid hx cosx to help gather information about the graph of f x tarix From the identity tarix smx cosx will have vertical asymptotes whenever cosx 0 and the graph of the tangent function will have an x intercept whenever sinx 0 372 572 cosx 0when x i3 i7i7 so this is where the graph of fxwill have vertical asymptotes arid sinx 0 when x 0 r 7 r 272 so this gives the location of the zeros of the mction we can conclude that the graph of the tangent function We know that tan j l and tan j 71 so this gives two additional points on the graph of the function f x tarix on the interval The tangent function is periodic with period 72 so these values will repeat themselves at multiples of 72 So tan k7 l and tarr 7i47z k7 1 where k is an integer Using all of this information we can generate a graph 2n 72m in O en you will need to graph the function overjust one period In this case you ll use the interval Here s the graph of fx tanx over this interval with pertinent points marked Likewise we can use some information from the graphs of gx sinx and hx cosx to help gather information about the graph of fx cotx From the identity cotx 506 s1nx function will have vertical asymptotes whenever sinx 0 and the graph of the cotangent inction will have an x intercept whenever cosx 0 we can conclude that the graph ofthe cotangent sinx 0when x 01 1 27 so this is where the graph of fx will have vertical asymptotes and cosx 0when x i so this gives the location ofthe zeros of the function We know that cot 1 and cot 71 so this gives two additional points on the graph ofthe function fx cotx on the interval 0 7 The cotangent function is periodic with period 7 so these values will repeat themselves at multiples of 7 So cot kn 1 and cotr kn 71 where k is an integer Using all of this information we can generate a graph Often you will need to graph the function over just one period In this case you ll use the interval 0 Here s the graph of f x cotx over this interval l l l l l l l l l l T S J l l l l l l l l You can take the graph of either of these basic functions and draw the graph of a more complicated function by making adjustments to the key elements of the basic function The key elemenm will be the locations of the asymptotes x intercepts and the translations of the points at g I and either 71 or 3771171 To graphfx AtanBx7 C D you will start by locating the asymptotes To do so set Ex 7 C and Ex 7 C Next find the x coordinate of the point halfway between the asymptotes Evaluate the function at this value to find the location of the translated x intercept Next find the x coordinates of the points halfway between the asymptotes and and the translated zero Evaluate the function at these values to find two more points on the graph of the function To graph gx A cotBx C D you will start by locating the asymptotes To do so set Bx C 0 and Ex C 7 Next find the x coordinate of the point halfway between the asymptotes Evaluate the function at this value to find the location of the translated x intercept Next find the x coordinates of the points halfway between the asymptotes and and the translated zero Evaluate the function at these values to find two more points on the graph of the function For both functions the period will be You will find vertical shifts and phase shifts as you did for translations of sine and cosine functions You ll also be able to take advantage of what you know about the graph of f x sinx to help you graph gx cscx Using the identity cscx you can conclude l sinx that the graph of g will have a vertical asymptote whenever sinx 0 This means that the graph ofg will have vertical asymptotes at x 0 in i27r The easiest way to draw a graph of gx cscx is to draw the graph of f x sinx sketch asymptotes at each of the zeros of f x sinx then sketch in the cosecant graph Here s the graph of fx sinx on the interval 2 4 Next we ll include the asymptotes for the cosecant graph at each point where sinx 0 Typically you lljust graph over one period 021 You ll also be able to take advantage ofwhat you know about the graph of fx cosx to help you graph gx secx Using the identity secx you cos x can conclude that the graph ofg will have a Vertical asymptote whenever cosx 0 This means that the graph ofg will have vertical asymptotes at x r g r The easiest way to draw a graph of gx secx is to draw the graph of fx cosx sketch asymptotes at each of the zeros of f x cosx then sketch in the secant graph Here s the graph of fx cosx on the interval 7 57 r F 7 7 r r 7 v J J Next we ll include the asymptotes for the secant graph 4 quotgt 7 V Now we ll include the graph of the secant function Typically you ll just graph over one period 0 27139 To graph a more complicated secant or cosecant function it s easiest to draw the graph of the underlying cosine or sine function You can then draw in asymptotes at the points that are the translations of the zeros of the underlying functions and use the framework to sketch the secant or cosecant function Example 1 Sketch fx Example 2 Sketch fx 3 c0t2x Example 3 Sketch fx 2 c0t 3x 2 Example 4 Sketch fx itan2mc 3 Example 5 Sketch fx 3sec gj ExampleG Sketch fx73csc2x1 Example 7 Sketch f4sec3

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