Copyright © 2007 Pearson Education, Inc. Slide Graphs of the Other Trigonometric Functions Graphs of the Cosecant and Secant Functions Cosecant values are reciprocals of the corresponding sine values. –If sin x = 1, the value of csc x is 1. Similarly, if sin x = –1, then csc x = –1. –When 0 1. Similarly, if –1 < sin x < 0, then csc x < –1. –When approaches 0, the gets larger. The graph of y = csc x approaches the vertical line x = 0. –In fact, the vertical asymptotes are the lines x = n .

Copyright © 2007 Pearson Education, Inc. Slide Graphs of the Cosecant and Secant Functions A similar analysis for the secant function can be done. Plotting a few points, we have the solid lines representing the curves for the cosecant and secant functions.

Copyright © 2007 Pearson Education, Inc. Slide Graphs of the Cosecant and Secant Functions Cosecant Function –Discontinuous at values of x of the form x = n , and has vertical asymptotes at these values. –No x-intercepts. –Its period is 2  with no amplitude. –Symmetric with respect to the origin, and is an odd function. Secant Function –Discontinuous at values of x of the form (2n + 1), and has vertical asymptotes at these values. –No x-intercepts. –Its period is 2  with no amplitude. –Symmetric with respect to the y-axis, and is an even function.

Copyright © 2007 Pearson Education, Inc. Slide Sketching Traditional Graphs of the Cosecant and Secant Functions To graph y = a csc bx or y = a sec bx, with b > 0, 1.Graph the corresponding reciprocal function as a guide, using a dashed curve. 2.Sketch the vertical asymptotes. They will have equations of the form x = k, k an x-intercept of the guide function. 3.Sketch the graph of the desired function by drawing the U-shaped branches between adjacent asymptotes. To Graph Use as a Guide y = a csc bx y = a sin bx y = a sec bx y = a cos bx

Copyright © 2007 Pearson Education, Inc. Slide Graphing y = a sec bx ExampleGraph SolutionThe guide function is One period of the graph lies along the interval that satisfies the inequality Dividing this interval into four equal parts gives the key points (0,2), ( ,0), (2 ,–2), (3 ,0), and (4 ,2), which are joined with a smooth dashed curve.

Copyright © 2007 Pearson Education, Inc. Slide Graphing y = a sec bx Sketch vertical asymptotes where the guide function equals 0 and draw the U-shaped branches, approaching the asymptotes.

Copyright © 2007 Pearson Education, Inc. Slide Graphs of Tangent and Cotangent Functions Tangent –Its period is  and it has no amplitude. –Its values are 0 when sine values are 0, and undefined when cosine values are 0. –As x goes from tangent values go from –  to , and increase throughout the interval. –The x-intercepts are of the form x = n .

Copyright © 2007 Pearson Education, Inc. Slide Graphs of Tangent and Cotangent Functions Cotangent –Its period is  and it has no amplitude. –Its values are 0 when cosine values are 0, and undefined when sine values are 0. –As x goes from 0 to , cotangent values go from  to – , and decrease throughout the interval. –The x-intercepts are of the form x = (2n + 1)

Copyright © 2007 Pearson Education, Inc. Slide Sketching Traditional Graphs of the Tangent and Cotangent Functions To graph y = a tan bx or y = a cot bx, with b > 0, 1.The period is To locate two adjacent vertical asymptotes, solve the following equations for x: 2.Sketch the two vertical asymptotes found in Step 1. 3.Divide the interval formed by the vertical asymptotes into four equal parts. 4.Evaluate the function for the first-quarter point, midpoint, and third-quarter point, using x-values from Step 3. 5.Join the points with a smooth curve approaching the vertical asymptotes. For y = a tan bx:bx = and bx = For y = a cot bx:bx = 0andbx = .

Copyright © 2007 Pearson Education, Inc. Slide Graphing y = a cot bx ExampleGraph SolutionSince the function involves cotangent, we can locate two adjacent asymptotes by solving the equations: Dividing the interval into four equal parts and finding the key points, we get