Objectives: Graphs of Other Trigonometric Functions • • • • • • Understand the graph of y = tan x. Graph variations of y = tan x. Understand the graph of y = cot x. Graph variations of y = cot x. Understand the graphs of y = csc x and y = sec x. Graph variations of y = csc x and y = sec x. Dr .Hayk Melikyan Department of Mathematics and CS [email protected] Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 The Graph of y = tan x Period: The tangent function is an odd function. tan( x) tan x The tangent function is undefined at x 2 . Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 The Tangent Curve: The Graph of y = tan x and Its Characteristics Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 The Tangent Curve: The Graph of y = tan x and Its Characteristics (continued) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Graphing Variations of y = tan x Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Graphing Variations of y = tan x (continued) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Graphing a Tangent Function Graph y = 3 tan 2x for x 3 . 4 4 A = 3, B = 2, C = 0 Step 1 Find two consecutive asymptotes. 2 Bx C 2 2 2x 2 4 x 4 An interval containing one period is , . Thus, two 4 4 consecutive asymptotes occur at x and x . 4 4 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Graphing a Tangent Function (continued) Graph y = 3 tan 2x for x 3 . 4 4 Step 2 Identify an x-intercept, midway between the consecutive asymptotes. x = 0 is midway between and . 4 4 The graph passes through (0, 0). Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Graphing a Tangent Function (continued) Graph y = 3 tan 2x for x 3 . 4 4 Step 3 Find points on the graph 1/4 and 3/4 of the way between the consecutive asymptotes. These points have y-coordinates of –A and A. 3 3tan 2x 3 3tan 2x The graph passes through 1 tan 2x 1 tan 2x , 3 and ,3 . 2x 2x 8 8 4 4 x x 8 8 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Graphing a Tangent Function (continued) Graph y = 3 tan 2x for x 3 . 4 4 Step 4 Use steps 1-3 to graph one full period of the function. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 The Cotangent Curve: The Graph of y = cot x and Its Characteristics Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 The Cotangent Curve: The Graph of y = cot x and Its Characteristics (continued) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Graphing Variations of y = cot x Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Graphing Variations of y = cot x (continued) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Graphing a Cotangent Function 1 Graph y cot x 2 2 1 A , B ,C 0 2 2 Step 1 Find two consecutive asymptotes. 0 Bx C 0 2 x 0 x2 An interval containing one period is (0, 2). Thus, two consecutive asymptotes occur at x = 0 and x = 2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Graphing a Cotangent Function (continued) 1 Graph y cot x 2 2 Step 2 Identify an x-intercept midway between the consecutive asymptotes. x = 1 is midway between x = 0 and x = 2. The graph passes through (1, 0). Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Example: Graphing a Cotangent Function (continued) 1 Graph y cot x 2 2 Step 3 Find points on the graph 1/4 and 3/4 of the way between consecutive asymptotes. These points have y-coordinates of A and –A. 1 1 3 3 cot x 1 cot x x x 2 2 2 2 4 2 2 1 1 1 1 cot x x x cot x 2 2 4 2 2 2 2 1 , 1 . 3, 1 The graph passes through and 2 2 2 2 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: Graphing a Cotangent Function (continued) 1 Graph y cot x 2 2 Step 4 Use steps 1-3 to graph one full period of the function. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 The Graphs of y = csc x and y = sec x We obtain the graphs of the cosecant and the secant curves by using the reciprocal identities 1 csc x sin x and 1 sec x . cos x We obtain the graph of y = csc x by taking reciprocals of the y-values in the graph of y = sin x. Vertical asymptotes of y = csc x occur at the x-intercepts of y = sin x. We obtain the graph of y = sec x by taking reciprocals of the y-values in the graph of y = cos x. Vertical asymptotes of y = sec x occur at the x-intercepts of y = cos x. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 The Cosecant Curve: The Graph of y = csc x and Its Characteristics Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 The Cosecant Curve: The Graph of y = csc x and Its Characteristics (continued) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 The Secant Curve: The Graph of y = sec x and Its Characteristics Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 The Secant Curve: The Graph of y = sec x and Its Characteristics (continued) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Example: Using a Sine Curve to Obtain a Cosecant Curve Use the graph of y sin x to obtain the graph of 4 y csc x . 4 The x-intercepts of the sine graph correspond to the vertical asymptotes of the cosecant graph. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 Example: Using a Sine Curve to Obtain a Cosecant Curve (continued) Use the graph of y sin x to obtain the graph of 4 y csc x y csc x . 4 4 Using the asymptotes as guides, we sketch the graph of x y sin y csc x . 4 4 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 Example: Graphing a Secant Function 3 3 Graph y = 2 sec 2x for x . 4 4 We begin by graphing the reciprocal function, y = 2 cos 2x. This equation is of the form y = A cos Bx, with A = 2 and B = 2. amplitude: A 2 2 2 2 period: B 2 We will use quarter-periods to find x-values for the five key points. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 Example: Graphing a Secant Function (continued) 3 3 Graph y = 2 sec 2x for x . 4 4 3 The x-values for the five key points are: 0, , , , and . 4 2 4 Evaluating the function y = 2 cos 2x at each of these values of x, the key points are: 3 (0,2), ,0 , , 2 , ,0 , and ,2 . 4 2 4 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 27 Example: Graphing a Secant Function (continued) 3 3 Graph y = 2 sec 2x for x . 4 4 The key points for our graph of y = 2 cos 2x are: 3 (0,2), ,0 , , 2 , ,0 , 4 2 4 and ,2 . We draw vertical asymptotes through the x-intercepts to use as guides for the graph of y = 2 sec 2x. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 28 Example: Graphing a Secant Function (continued) 3 3 Graph y = 2 sec 2x for x . 4 4 y 2sec 2 x y 2cos 2 x Copyright © 2014, 2010, 2007 Pearson Education, Inc. 29 The Six Curves of Trigonometry Copyright © 2014, 2010, 2007 Pearson Education, Inc. 30 The Six Curves of Trigonometry (continued) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 31 The Six Curves of Trigonometry (continued) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 32 The Six Curves of Trigonometry (continued) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 33 The Six Curves of Trigonometry (continued) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 34 The Six Curves of Trigonometry (continued) Copyright © 2014, 2010, 2007 Pearson Education, Inc. 35

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