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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Graphs of Tangent and Cotangent

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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Graphs of Tangent and Cotangent

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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Graphs of Tangent and Cotangent

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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Graphs of Tangent and Cotangent

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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Graphs of Tangent and Cotangent

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

y = tan x

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Graphs of Tangent and Cotangent

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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Graphs of Tangent and Cotangent

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

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Graphs of Tangent and Cotangent

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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Graphs of Transformations of Tangent and Cotangent

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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Graphs of Transformations of Tangent and Cotangent

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

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Graphs of Transformations of Tangent and Cotangent

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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Example 2 – Graphing Tangent Curves

Graph the function:

(b) y = tan 2

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Example 2 – Graphing Tangent Curves

(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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More Trigonometric Graphs

Practice:

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

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5.4 – Day 2

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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Graphs of Cosecant and Secant

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Graphs of Cosecant and Secant

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

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Graphs of Cosecant and Secant

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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Graphs of Cosecant and Secant

Graph y = csc x.

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Graphs of Cosecant and Secant

One period of y = csc x

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Graphs of Cosecant and Secant

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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Graphs of Cosecant and Secant

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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Graphs of Cosecant and Secant

Graph y = sec x.

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Graphs of Cosecant and Secant

One period of y = sec x

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Graphs of Cosecant and Secant

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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Graphs of Transformations of Cosecant and Secant

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

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Graphs of Transformations of Cosecant and Secant

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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Example 4 – Graphing Cosecant Curves

Graph the function:

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Example 4 – Graphing Cosecant Curves

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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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Graphs of Cosecant and Secant

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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More Trigonometric Graphs

Practice:

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