More Trigonometric Graphs

5.4 – Day 1

More Trigonometric Graphs

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Objectives

► Graphs of Tangent, Cotangent, Secant, and Cosecant

► Graphs of Transformation of Tangent and Cotangent

► Graphs of Transformations of Cosecant and Secant

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More Trigonometric GraphsIn this section, we graph the tangent, cotangent, secant, and cosecant functions and transformations of these functions.

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Graphs of Tangent and Cotangent

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We begin by stating the periodic properties of these functions. Sine and cosine have period 2.

Since cosecant and secant are the reciprocals of sine and cosine, respectively, they also have period 2. Tangent and cotangent, however, have period .

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Let’s use the values from the unit circle and the function y = tan x = sin x ÷ cos x to make a table of values and a graph of the tangent function.

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The graph of y = tan x approaches the vertical lines x = /2 and x = – /2. So, these lines are vertical asymptotes.

* Remember, the vertical asymptotes occur at the values that make the denominator (cos x) equal to 0.

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With the information we have so far, we can sketch the “standard” graph of y = tan x from – /2 < x < /2.

One period of y = tan x

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The complete graph of tangent is now obtained using the fact that tangent is periodic with period .

y = tan x

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Let’s use the values from the unit circle and the function y = cot x = cos x ÷ sin x to make a table of values and a graph of the cotangent function.

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The function y = cot x is graphed on the interval (0, ).

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Since cot x is undefined for x = n with n an integer, its complete graph has vertical asymptotes at these values.

One period y = cot x

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Graphs of Transformations of Tangent and Cotangent

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We now consider graphs of transformations of the tangent and cotangent functions.

Since the tangent and cotangent functions have period , the functions

y = a tan k(x-b) and y = a cot k(x-b) (k > 0)

complete one period as kx varies from 0 to , that is, for 0 kx . Solving this inequality, we get 0 x /k. So, they each have period /k.

Remember, the value of “a” stretches or compresses the graph vertically.

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Thus, one complete period of the graphs of these functions occurs on any interval of length /k.

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To sketch a complete period of these graphs, it’s convenient to select an interval between vertical asymptotes:

To find consecutive vertical asymptotes for the graph of y = a tan k(x – b), solve the equations

k(x – b) = -π/2 and k(x – b) = π/2

To find consecutive vertical asymptotes for the graph of y = a cot k(x – b), solve the equations

k(x – b) = 0 and k(x – b) = π

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Example 2 – Graphing Tangent Curves

Graph the function:

(a) y = tan 2x

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Graph the function:

(b) y = tan 2

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(a) y = tan 2x (b) y = tan 2

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Example 3 – A Shifted Cotangent Curve

Graph y = 2 cot

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Example 3 – A Shifted Cotangent Curve

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Graphs of Cosecant and Secant

It is apparent that the graphs of y = tan x and y = cot x are symmetric about the origin.

This is because tangent and cotangent are odd functions.

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Practice:

p. 405-406#1, 3, 5, 6, 11, 23, 35, 43, 53, 57

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5.4 – Day 2


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Objectives

► Graphs of Tangent, Cotangent, Secant, and Cosecant

► Graphs of Transformation of Tangent and Cotangent

► Graphs of Transformations of Cosecant and Secant

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Recall that since cosecant and secant are the reciprocals of sine and cosine, respectively, they also have period 2.

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To graph the cosecant and secant functions, we use the reciprocal identities:

and

To graph y = csc x, we take the reciprocals of the y-coordinates of the points of the graph of y = sin x. So, let’s do that!

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Graph y = csc x.

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One period of y = csc x

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The complete graph is obtained from the fact that the function cosecant is periodic with period 2.

Note that the graph has vertical asymptotes at the points where sin x = 0, that is, at x = n, for n an integer.

y = csc x

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Similarly, to graph y = sec x, we take the reciprocals of the y-coordinates of the points of the graph of y = cos x.

So, let’s do that!

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Graph y = sec x.

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One period of y = sec x

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The complete graph of y = sec x is sketched in a similar manner. Observe that the domain of sec x is the set of all real numbers other than x = ( /2) + n, for n an integer, so the graph has vertical asymptotes at those points.

y = sec x

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Graphs of Transformations of Cosecant and Secant

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An appropriate interval on which to graph one complete period is [0, 2 /k].

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To find consecutive vertical asymptotes for the graph of y = a csc k(x – b), solve the equations

k(x – b)= 0 and k(x – b)= π

To find consecutive vertical asymptotes for the graph of y = a sec k(x – b), solve the equations

k(x – b)= -π/2 and k(x – b)= π/2

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Example 4 – Graphing Cosecant Curves

Graph the function:

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Graph the function:

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Example 5 – Graphing a Secant Curve

Graph y = 3 sec

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Example 5 – Graphing a Secant Curve

y = 3 sec

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It is apparent that the graph of y = csc x is symmetric about the origin, whereas that of y = sec x is symmetric about the y-axis.

This is because cosecant is an odd function, whereas secant is an even function.

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Practice:

p. 405#2, 4, 7, 8, 15, 17, 21, 25, 31, 33, 45, 51

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