18434676-Gr12-Advance-Function-Ch52

18434676-Gr12-Advance-Function-Ch52 - 5.2 Graphs of...

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Graphs of Reciprocal Trigonometric Functions 5.2 The reciprocal trigonometric functions are related to the primary trigonometric functions. They appear in various engineering applications. For example, the cosecant function is used in modelling the radiation pattern of a radar antenna array to ensure that radar energy is not wasted in airspace, where there is little chance of fi nding an aircraft. Investigate How can you use graphs of the primary trigonometric functions to deduce the shape of the graphs of the reciprocal trigonometric functions? 1. a) To investigate the graph of y ± csc x , begin by sketching a graph of y ± sin x on the interval x [0, 2 π ]. b) Since csc x ± 1 _ sin x , the two graphs will intersect whenever sin x ± ² 1. Mark all such points on your graph. c) For what values of x will the value of csc x be undefined? Determine where these values are on the graph, and draw vertical dotted lines to represent the asymptotes at these values. 2. a) Consider the value of y ± sin x at x ± π _ 2 , and move toward x ± 0. What happens to the value of sin x ? What happens to the value of csc x ? Sketch the expected shape of the graph of y ± csc x as the value of x moves from π _ 2 to 0. b) Consider the value of y ± sin x at x ± π _ 2 , and move toward x ± π . What happens to the value of sin x ? What happens to the value of csc x ? Sketch the expected shape of the graph of y ± csc x as the value of x moves from π _ 2 to π . c) Consider the value of y ± sin x at x ± 3 π _ 2 , and move toward x ± π . What happens to the value of sin x ? What happens to the value of csc x ? Sketch the expected shape of the graph of y ± csc x as the value of x moves from 3 π _ 2 to π . Tools • scientif c calculator • grid paper Optional • graphing calculator OR • graphing soFtware 5.2 Graphs of Reciprocal Trigonometric Functions • MHR 261
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d) Consider the value of y ± sin x at x ± 3 π _ 2 , and move toward x ± 2 π . What happens to the value of sin x ? What happens to the value of csc x ? Sketch the expected shape of the graph of y ± csc x as the value of x moves from 3 π _ 2 to 2 π . 3. a) Inspect the shape of the graph of y ± csc x . Determine the domain, range, and period of the graph. b) Describe the values of x where the asymptotes occur. Why do these asymptotes occur? c) Consider the tangent lines at x ± π _ 4 for the graphs of y ± sin x and y ± csc x . How does the sign of the slope of the tangent on the sine graph at this point compare to the sign of the slope of the tangent on the cosecant graph? Consider other values of x . Is this relationship always true? d) Consider the slope of the tangent to the sine graph as x varies from π _ 4 to π _ 2 . Does the slope increase or decrease? What happens to the slope of the tangent to the cosecant graph over this same interval? Consider other intervals of x . Is this relationship always true?
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This note was uploaded on 05/15/2011 for the course ECON 103 taught by Professor Miyzaki during the Spring '11 term at Brown College.

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18434676-Gr12-Advance-Function-Ch52 - 5.2 Graphs of...

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