1. Graphs of Other Trigonometric Functions

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3. We are interested in the graph of y = tan x Start with a "t" chart and let's choose values from our unit circle and find the tangent values. Tangent has a period of  so it will repeat every . x y = tan x x y would mean there is a vertical asymptote here

4. y = tan x Let's choose more values. x y = tan x x y would mean there is a vertical asymptote here Since we went from we have one complete period

5. y x period: The Unit Circle and TangentThe Unit Circle and Tangent Applet The red vertical lines are not part of the graph but are the asymptotes. Let's see what the graph would look like for y = tan x for 3 complete periods.

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8. 2.Sketch the two vertical asymptotes found in Step 1. 5.Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph to the left or right as needed. 3.Identify an x-intercept, midway between the consecutive asymptotes.

9. Step 1: Find two consecutive asymptotes.

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Example: Tangent Function 1.Find consecutive vertical asymptotes by solving for x: 5.Sketch one branch and repeat. Find the period and asymptotes and sketch the graph of y x

13. Graph Vertical asymptotes are Divide the interval from - 2  to 2  into four equal parts and plot points. y x x = - 2  x = 2 

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15. y = cot x Again the vertical lines are not part of the graph but are the asymptotes. Let's look at the tangent graph again to compare these. Notice vertical asymptotes of one are zeros of the other. y = tan x

16. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Cotangent Function Graph of the Cotangent Function 2. range: (– , +  ) 3. period:  4. vertical asymptotes: 1. domain : all real #’s Properties of y = cot x y x vertical asymptotes To graph y = cot x, use the identity. At values of x for which sin x = 0, the cotangent function is undefined and its graph has vertical asymptotes.

17. 1.Find two consecutive vertical asymptotes by finding an interval containing one period. Solve for x. 2.Sketch the two vertical asymptotes found in Step 1. 5.Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph to the left or right as needed. 3.Identify an x-intercept, midway between the consecutive asymptotes.

20. Cosecant is the reciprocal of sine One period: 2π π 2π2π 3π3π 0 −π −2π −3π Vertical asymptotes where sin θ = 0 θ csc θ sin θ

21. x y Cosecant Function Graph of the Cosecant Function 2. range: (– ,–1]  [1, +  ) 3. period:  where sine is zero. 4. vertical asymptotes: 1. domain : all real x To graph y = csc x, use the identity. Properties of y = csc x At values of x for which sin x = 0, the cosecant function is undefined and its graph has vertical asymptotes.

22. For the graph of y = f(x) = csc x, we'll take the reciprocals of the sine values. x sin x y = cosec x When we graph these rather than plot points after we see this, we'll use the sine graph as a sketching aid and then get the cosecant graph. x y 1 - 1

23. y = f(x) = csc x choose more values x sin x y = cosec x We'll use the sine graph as the sketching aid. x y 1 - 1 When the sine is 0 the cosecant will have an asymptote.

24. Using a Sine Curve to Obtain a Cosecant Curve

25. Secant is the reciprocal of cosine One period: 2π π 3π3π −2π 2π2π −π −3π 0 θ sec θ cos θ Vertical asymptotes where cos θ = 0

26. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26 y x Secant Function Graph of the Secant Function 2. range: (– ,–1]  [1, +  ) 3. period:  4. vertical asymptotes: 1. domain : all real x The graph y = sec x, use the identity. Properties of y = sec x At values of x for which cos x = 0, the secant function is undefined and its graph has vertical asymptotes.

28. Step 2: Use the x-intercepts to represent the vertical asymptotes of the secant graph. Step 3: Sketch the graph as shown in figure 4.85 on page 544.