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Precalculus

6.2 Graphs of the Other Trigonometric Functions

Precalculus6.2 Graphs of the Other Trigonometric Functions
  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Review Exercises
    15. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Equations
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  14. A | Basic Functions and Identities
  15. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

Learning Objectives

In this section, you will:
  • Analyze the graph of  y=tan x.
  • Graph variations of  y=tan x.
  • Analyze the graphs of  y=sec x  and  y=csc x.
  • Graph variations of  y=sec x  and  y=csc x.
  • Analyze the graph of  y=cot x.
  • Graph variations of  y=cot x.

We know the tangent function can be used to find distances, such as the height of a building, mountain, or flagpole. But what if we want to measure repeated occurrences of distance? Imagine, for example, a police car parked next to a warehouse. The rotating light from the police car would travel across the wall of the warehouse in regular intervals. If the input is time, the output would be the distance the beam of light travels. The beam of light would repeat the distance at regular intervals. The tangent function can be used to approximate this distance. Asymptotes would be needed to illustrate the repeated cycles when the beam runs parallel to the wall because, seemingly, the beam of light could appear to extend forever. The graph of the tangent function would clearly illustrate the repeated intervals. In this section, we will explore the graphs of the tangent and other trigonometric functions.

Analyzing the Graph of y = tan x

We will begin with the graph of the tangent function, plotting points as we did for the sine and cosine functions. Recall that

tanx= sinx cosx tanx= sinx cosx

The period of the tangent function is π π because the graph repeats itself on intervals of kπ kπ where k k is a constant. If we graph the tangent function on π 2 π 2 to π 2 , π 2 , we can see the behavior of the graph on one complete cycle. If we look at any larger interval, we will see that the characteristics of the graph repeat.

We can determine whether tangent is an odd or even function by using the definition of tangent.

tan(x)= sin(x) cos(x) Definition of tangent.              = sinx cosx Sine is an odd function, cosine is even.              = sinx cosx The quotient of an odd and an even function is odd.              =tanx Definition of tangent. tan(x)= sin(x) cos(x) Definition of tangent.              = sinx cosx Sine is an odd function, cosine is even.              = sinx cosx The quotient of an odd and an even function is odd.              =tanx Definition of tangent.

Therefore, tangent is an odd function. We can further analyze the graphical behavior of the tangent function by looking at values for some of the special angles, as listed in Table 1.

x x π 2 π 2 π 3 π 3 π 4 π 4 π 6 π 6 0 π 6 π 6 π 4 π 4 π 3 π 3 π 2 π 2
tan( x ) tan( x ) undefined 3 3 –1 3 3 3 3 0 3 3 3 3 1 3 3 undefined
Table 1

These points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. If we look more closely at values when π 3 <x< π 2 , π 3 <x< π 2 , we can use a table to look for a trend. Because π 3 1.05 π 3 1.05 and π 2 1.57, π 2 1.57, we will evaluate x x at radian measures 1.05<x<1.57 1.05<x<1.57 as shown in Table 2.

x x 1.3 1.5 1.55 1.56
tan  x tan  x 3.6 14.1 48.1 92.6
Table 2

As x x approaches π 2 , π 2 , the outputs of the function get larger and larger. Because y=tanx y=tanx is an odd function, we see the corresponding table of negative values in Table 3.

x x −1.3 −1.5 −1.55 −1.56
tanx tanx −3.6 −14.1 −48.1 −92.6
Table 3

We can see that, as x x approaches π 2 , π 2 , the outputs get smaller and smaller. Remember that there are some values of x x for which cosx=0. cosx=0. For example, cos( π 2 )=0 cos( π 2 )=0 and cos( 3π 2 )=0. cos( 3π 2 )=0. At these values, the tangent function is undefined, so the graph of y=tanx y=tanx has discontinuities at x= π 2  and  3π 2 . x= π 2  and  3π 2 . At these values, the graph of the tangent has vertical asymptotes. Figure 1 represents the graph of y=tanx. y=tanx. The tangent is positive from 0 to π 2 π 2 and from π π to 3π 2 , 3π 2 , corresponding to quadrants I and III of the unit circle.

A graph of y=tangent of x. Asymptotes at -pi over 2 and pi over 2.
Figure 1 Graph of the tangent function

Graphing Variations of y = tan x

As with the sine and cosine functions, the tangent function can be described by a general equation.

y=Atan(Bx) y=Atan(Bx)

We can identify horizontal and vertical stretches and compressions using values of A A and B. B.The horizontal stretch can typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph.

Because there are no maximum or minimum values of a tangent function, the term amplitude cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase stretching/compressing factor when referring to the constant A. A.

Features of the Graph of y = Atan(Bx)

  • The stretching factor is | A |. | A |.
  • The period is P= π | B | . P= π | B | .
  • The domain is all real numbers x, x, where x π 2| B | + π | B | k x π 2| B | + π | B | k such that k k is an integer.
  • The range is (−∞,). (−∞,).
  • The asymptotes occur at x= π 2| B | + π | B | k, x= π 2| B | + π | B | k, where k kis an integer.
  • y=Atan( Bx ) y=Atan( Bx ) is an odd function.

Graphing One Period of a Stretched or Compressed Tangent Function

We can use what we know about the properties of the tangent function to quickly sketch a graph of any stretched and/or compressed tangent function of the form f(x)=Atan(Bx). f(x)=Atan(Bx). We focus on a single period of the function including the origin, because the periodic property enables us to extend the graph to the rest of the function’s domain if we wish. Our limited domain is then the interval ( P 2 , P 2 ) ( P 2 , P 2 ) and the graph has vertical asymptotes at ± P 2 ± P 2 where P= π B . P= π B . On ( π 2 , π 2 ), ( π 2 , π 2 ), the graph will come up from the left asymptote at x= π 2 , x= π 2 , cross through the origin, and continue to increase as it approaches the right asymptote at x= π 2 . x= π 2 . To make the function approach the asymptotes at the correct rate, we also need to set the vertical scale by actually evaluating the function for at least one point that the graph will pass through. For example, we can use

f( P 4 )=Atan( B P 4 )=Atan( B π 4B )=A f( P 4 )=Atan( B P 4 )=Atan( B π 4B )=A

because tan( π 4 )=1. tan( π 4 )=1.

How To

Given the function f(x)=Atan(Bx), f(x)=Atan(Bx), graph one period.

  1. Identify the stretching factor, | A |. | A |.
  2. Identify B B and determine the period, P= π | B | . P= π | B | .
  3. Draw vertical asymptotes at x= P 2 x= P 2 and x= P 2 . x= P 2 .
  4. For A>0, A>0, the graph approaches the left asymptote at negative output values and the right asymptote at positive output values (reverse for A<0 A<0 ).
  5. Plot reference points at ( P 4 ,A ), ( P 4 ,A ), ( 0,0 ), ( 0,0 ), and ( P 4 ,−A ), ( P 4 ,−A ), and draw the graph through these points.

Example 1

Sketching a Compressed Tangent

Sketch a graph of one period of the function y=0.5tan( π 2 x ). y=0.5tan( π 2 x ).

Try It #1

Sketch a graph of f(x)=3tan( π 6 x ). f(x)=3tan( π 6 x ).

Graphing One Period of a Shifted Tangent Function

Now that we can graph a tangent function that is stretched or compressed, we will add a vertical and/or horizontal (or phase) shift. In this case, we add C C and D D to the general form of the tangent function.

f(x)=Atan(BxC)+D f(x)=Atan(BxC)+D

The graph of a transformed tangent function is different from the basic tangent function tanx tanx in several ways:

Features of the Graph of y = Atan(BxC)+D

  • The stretching factor is | A |. | A |.
  • The period is π | B | . π | B | .
  • The domain is x C B + π | B | k, x C B + π | B | k, where k k is an integer.
  • The range is (−∞,). (−∞,).
  • The vertical asymptotes occur at x= C B + π 2| B | k, x= C B + π 2| B | k, where k k is an odd integer.
  • There is no amplitude.
  • y=Atan(Bx-C)+D y=Atan(Bx-C)+D is an odd function because it is the quotient of odd and even functions (sine and cosine respectively).

How To

Given the function y=Atan(BxC)+D, y=Atan(BxC)+D, sketch the graph of one period.

  1. Express the function given in the form y=Atan( BxC )+D. y=Atan( BxC )+D.
  2. Identify the stretching/compressing factor, | A |. | A |.
  3. Identify B B and determine the period, P= π | B | . P= π | B | .
  4. Identify C C and determine the phase shift, C B . C B .
  5. Draw the graph of y=Atan(Bx) y=Atan(Bx) shifted to the right by C B C B and up by D. D.
  6. Sketch the vertical asymptotes, which occur at  x= C B + π 2| B | k,  x= C B + π 2| B | k, where  k   k  is an odd integer.
  7. Plot any three reference points and draw the graph through these points.

Example 2

Graphing One Period of a Shifted Tangent Function

Graph one period of the function y=−2tan(πx+π)−1. y=−2tan(πx+π)−1.

Analysis

Note that this is a decreasing function because A<0. A<0.

Try It #2

How would the graph in Example 2 look different if we made A=2 A=2 instead of −2? −2?

How To

Given the graph of a tangent function, identify horizontal and vertical stretches.

  1. Find the period P P from the spacing between successive vertical asymptotes or x-intercepts.
  2. Write f(x)=Atan( π P x ). f(x)=Atan( π P x ).
  3. Determine a convenient point (x,f(x)) (x,f(x)) on the given graph and use it to determine A. A.

Example 3

Identifying the Graph of a Stretched Tangent

Find a formula for the function graphed in Figure 4.

A graph of two periods of a modified tangent function, with asymptotes at x=-4 and x=4.
Figure 4 A stretched tangent function

Try It #3

Find a formula for the function in Figure 5.

A graph of four periods of a modified tangent function, Vertical asymptotes at -3pi/4, -pi/4, pi/4, and 3pi/4.
Figure 5

Analyzing the Graphs of y = sec x and y = cscx

The secant was defined by the reciprocal identity secx= 1 cosx . secx= 1 cosx . Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at π 2 , π 2 , 3π 2 , 3π 2 , etc. Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value.

We can graph y=secx y=secx by observing the graph of the cosine function because these two functions are reciprocals of one another. See Figure 6. The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the secant function increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.

The secant graph has vertical asymptotes at each value of x x where the cosine graph crosses the x-axis; we show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant.

Note that, because cosine is an even function, secant is also an even function. That is, sec( x )=secx. sec( x )=secx.

A graph of cosine of x and secant of x. Asymptotes for secant of x shown at -3pi/2, -pi/2, pi/2, and 3pi/2.
Figure 6 Graph of the secant function, f(x)=secx= 1 cosx f(x)=secx= 1 cosx

As we did for the tangent function, we will again refer to the constant | A | | A | as the stretching factor, not the amplitude.

Features of the Graph of y = Asec(Bx)

  • The stretching factor is | A |. | A |.
  • The period is 2π | B | . 2π | B | .
  • The domain is x π 2| B | k, x π 2| B | k, where k k is an odd integer.
  • The range is (,| A |][| A |,). (,| A |][| A |,).
  • The vertical asymptotes occur at x= π 2| B | k, x= π 2| B | k, where k k is an odd integer.
  • There is no amplitude.
  • y=Asec( Bx ) y=Asec( Bx ) is an even function because cosine is an even function.

Similar to the secant, the cosecant is defined by the reciprocal identity cscx= 1 sinx . cscx= 1 sinx . Notice that the function is undefined when the sine is 0, leading to a vertical asymptote in the graph at 0, 0, π, π, etc. Since the sine is never more than 1 in absolute value, the cosecant, being the reciprocal, will never be less than 1 in absolute value.

We can graph y=cscx y=cscx by observing the graph of the sine function because these two functions are reciprocals of one another. See Figure 7. The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the cosecant function increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.

The cosecant graph has vertical asymptotes at each value of x x where the sine graph crosses the x-axis; we show these in the graph below with dashed vertical lines.

Note that, since sine is an odd function, the cosecant function is also an odd function. That is, csc( x )=−cscx. csc( x )=−cscx.

The graph of cosecant, which is shown in Figure 7, is similar to the graph of secant.

A graph of cosecant of x and sin of x. Five vertical asymptotes shown at multiples of pi.
Figure 7 The graph of the cosecant function, f(x)=cscx= 1 sinx f(x)=cscx= 1 sinx

Features of the Graph of y = Acsc(Bx)

  • The stretching factor is | A |. | A |.
  • The period is 2π | B | . 2π | B | .
  • The domain is x π | B | k, x π | B | k, where k k is an integer.
  • The range is ( ,| A | ][ | A |, ). ( ,| A | ][ | A |, ).
  • The asymptotes occur at x= π | B | k, x= π | B | k, where k k is an integer.
  • y=Acsc( Bx ) y=Acsc( Bx ) is an odd function because sine is an odd function.

Graphing Variations of y = sec x and y= csc x

For shifted, compressed, and/or stretched versions of the secant and cosecant functions, we can follow similar methods to those we used for tangent and cotangent. That is, we locate the vertical asymptotes and also evaluate the functions for a few points (specifically the local extrema). If we want to graph only a single period, we can choose the interval for the period in more than one way. The procedure for secant is very similar, because the cofunction identity means that the secant graph is the same as the cosecant graph shifted half a period to the left. Vertical and phase shifts may be applied to the cosecant function in the same way as for the secant and other functions.The equations become the following.

y=Asec( BxC )+D y=Asec( BxC )+D
y=Acsc( BxC )+D y=Acsc( BxC )+D

Features of the Graph of y = Asec(BxC)+D

  • The stretching factor is | A |. | A |.
  • The period is 2π | B | . 2π | B | .
  • The domain is x C B + π 2| B | k, x C B + π 2| B | k, where k k is an odd integer.
  • The range is (,| A |+D][| A |+D,). (,| A |+D][| A |+D,).
  • The vertical asymptotes occur at x= C B + π 2| B | k, x= C B + π 2| B | k, where k k is an odd integer.
  • There is no amplitude.
  • y=Asec( Bx-C )+D y=Asec( Bx-C )+D is an even function because cosine is an even function.

Features of the Graph of y = Acsc(BxC)+D

  • The stretching factor is | A |. | A |.
  • The period is 2π | B | . 2π | B | .
  • The domain is x C B + π | B | k, x C B + π | B | k, where k k is an integer.
  • The range is (,| A |+D][| A |+D,). (,| A |+D][| A |+D,).
  • The vertical asymptotes occur at x= C B + π |B| k, x= C B + π |B| k, where k k is an integer.
  • There is no amplitude.
  • y=Acsc( Bx-C )+D y=Acsc( Bx-C )+D is an odd function because sine is an odd function.

How To

Given a function of the form y=Asec( Bx ), y=Asec( Bx ), graph one period.

  1. Express the function given in the form y=Asec( Bx ). y=Asec( Bx ).
  2. Identify the stretching/compressing factor, | A |. | A |.
  3. Identify B B and determine the period, P= 2π | B | . P= 2π | B | .
  4. Sketch the graph of y=Acos( Bx ). y=Acos( Bx ).
  5. Use the reciprocal relationship between y=cosx y=cosx and y=secx y=secx to draw the graph of y=Asec( Bx ). y=Asec( Bx ).
  6. Sketch the asymptotes.
  7. Plot any two reference points and draw the graph through these points.

Example 4

Graphing a Variation of the Secant Function

Graph one period of f(x)=2.5sec(0.4x). f(x)=2.5sec(0.4x).

Try It #4

Graph one period of f(x)=2.5sec(0.4x). f(x)=2.5sec(0.4x).

Q&A

Do the vertical shift and stretch/compression affect the secant’s range?

Yes. The range of f( x )=Asec( BxC )+D f( x )=Asec( BxC )+D is ( ,| A |+D ][ | A |+D, ). ( ,| A |+D ][ | A |+D, ).

How To

Given a function of the form f( x )=Asec( BxC )+D, f( x )=Asec( BxC )+D, graph one period.

  1. Express the function given in the form y=Asec(BxC)+D. y=Asec(BxC)+D.
  2. Identify the stretching/compressing factor, | A |. | A |.
  3. Identify B B and determine the period, 2π | B | . 2π | B | .
  4. Identify C C and determine the phase shift, C B . C B .
  5. Draw the graph of y=Asec(Bx) y=Asec(Bx) , but shift it to the right by C B C B and up by D. D.
  6. Sketch the vertical asymptotes, which occur at x= C B + π 2| B | k, x= C B + π 2| B | k, where k k is an odd integer.

Example 5

Graphing a Variation of the Secant Function

Graph one period of y=4sec( π 3 x π 2 )+1. y=4sec( π 3 x π 2 )+1.

Try It #5

Graph one period of f( x )=6sec(4x+2)8. f( x )=6sec(4x+2)8.

Q&A

The domain of cscx cscx was given to be all x x such that xkπ xkπ for any integer k. k. Would the domain of y=Acsc(BxC)+Dbex C+kπ B ? y=Acsc(BxC)+Dbex C+kπ B ?

Yes. The excluded points of the domain follow the vertical asymptotes. Their locations show the horizontal shift and compression or expansion implied by the transformation to the original function’s input.

How To

Given a function of the form y=Acsc( Bx ), y=Acsc( Bx ), graph one period.

  1. Express the function given in the form y=Acsc( Bx ). y=Acsc( Bx ).
  2. | A |. | A |.
  3. Identify B B and determine the period, P= 2π | B | . P= 2π | B | .
  4. Draw the graph of y=Asin( Bx ). y=Asin( Bx ).
  5. Use the reciprocal relationship between y=sinx y=sinx and y=cscx y=cscx to draw the graph of y=Acsc( Bx ). y=Acsc( Bx ).
  6. Sketch the asymptotes.
  7. Plot any two reference points and draw the graph through these points.

Example 6

Graphing a Variation of the Cosecant Function

Graph one period of f(x)=−3csc(4x). f(x)=−3csc(4x).

Try It #6

Graph one period of f(x)=0.5csc(2x). f(x)=0.5csc(2x).

How To

Given a function of the form f( x )=Acsc( BxC )+D, f( x )=Acsc( BxC )+D,graph one period.

  1. Express the function given in the form y=Acsc(BxC)+D. y=Acsc(BxC)+D.
  2. Identify the stretching/compressing factor, | A |. | A |.
  3. Identify B B and determine the period, 2π | B | . 2π | B | .
  4. Identify C C and determine the phase shift, C B . C B .
  5. Draw the graph of y=Acsc(Bx) y=Acsc(Bx) but shift it to the right by C B C B and up by D. D.
  6. Sketch the vertical asymptotes, which occur at x= C B + π | B | k, x= C B + π | B | k, where k k is an integer.

Example 7

Graphing a Vertically Stretched, Horizontally Compressed, and Vertically Shifted Cosecant

Sketch a graph of y=2csc( π 2 x )+1. y=2csc( π 2 x )+1.What are the domain and range of this function?

Analysis

The vertical asymptotes shown on the graph mark off one period of the function, and the local extrema in this interval are shown by dots. Notice how the graph of the transformed cosecant relates to the graph of f(x)=2sin( π 2 x )+1, f(x)=2sin( π 2 x )+1,shown as the orange dashed wave.

Try It #7

Given the graph of f(x)=2cos( π 2 x )+1 f(x)=2cos( π 2 x )+1shown in Figure 12, sketch the graph of g(x)=2sec( π 2 x )+1 g(x)=2sec( π 2 x )+1on the same axes.

A graph of two periods of a modified cosine function. Range is [-1,3], graphed from x=-4 to x=4.
Figure 12

Analyzing the Graph of y = cot x

The last trigonometric function we need to explore is cotangent. The cotangent is defined by the reciprocal identity cotx= 1 tanx . cotx= 1 tanx .Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at 0,π, 0,π,etc. Since the output of the tangent function is all real numbers, the output of the cotangent function is also all real numbers.

We can graph y=cotx y=cotxby observing the graph of the tangent function because these two functions are reciprocals of one another. See Figure 13. Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.

The cotangent graph has vertical asymptotes at each value of x xwhere tanx=0; tanx=0;we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent, cotx cotxhas vertical asymptotes at all values of x xwhere tanx=0, tanx=0,and cotx=0 cotx=0at all values of x xwhere tanx tanxhas its vertical asymptotes.

A graph of cotangent of x, with vertical asymptotes at multiples of pi.
Figure 13 The cotangent function

Features of the Graph of y = Acot(Bx)

  • The stretching factor is | A |. | A |.
  • The period is P= π | B | . P= π | B | .
  • The domain is x π | B | k, x π | B | k, where k kis an integer.
  • The range is (,). (,).
  • The asymptotes occur at x= π | B | k, x= π | B | k, where k kis an integer.
  • y=Acot( Bx ) y=Acot( Bx )is an odd function.

Graphing Variations of y = cot x

We can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following.

y=Acot( BxC )+D y=Acot( BxC )+D

Features of the Graph of y = Acot(Bx−C)+D

  • The stretching factor is | A |. | A |.
  • The period is π | B | . π | B | .
  • The domain is x C B + π | B | k, x C B + π | B | k, where k k is an integer.
  • The range is (−∞,). (−∞,).
  • The vertical asymptotes occur at x= C B + π | B | k, x= C B + π | B | k, where k k is an integer.
  • There is no amplitude.
  • y=Acot(Bx) y=Acot(Bx) is an odd function because it is the quotient of even and odd functions (cosine and sine, respectively)

How To

Given a modified cotangent function of the form f( x )=Acot( Bx ), f( x )=Acot( Bx ),graph one period.

  1. Express the function in the form f( x )=Acot( Bx ). f( x )=Acot( Bx ).
  2. Identify the stretching factor, | A |. | A |.
  3. Identify the period, P= π | B | . P= π | B | .
  4. Draw the graph of y=Atan(Bx). y=Atan(Bx).
  5. Plot any two reference points.
  6. Use the reciprocal relationship between tangent and cotangent to draw the graph of y=Acot( Bx ). y=Acot( Bx ).
  7. Sketch the asymptotes.

Example 8

Graphing Variations of the Cotangent Function

Determine the stretching factor, period, and phase shift of y=3cot(4x), y=3cot(4x),and then sketch a graph.

How To

Given a modified cotangent function of the form f( x )=Acot( BxC )+D, f( x )=Acot( BxC )+D,graph one period.

  1. Express the function in the form f( x )=Acot( BxC )+D. f( x )=Acot( BxC )+D.
  2. Identify the stretching factor, | A |. | A |.
  3. Identify the period, P= π | B | . P= π | B | .
  4. Identify the phase shift, C B . C B .
  5. Draw the graph of y=Atan(Bx) y=Atan(Bx) shifted to the right by C B C B and up by D. D.
  6. Sketch the asymptotes x= C B + π | B | k, x= C B + π | B | k, where k k is an integer.
  7. Plot any three reference points and draw the graph through these points.

Example 9

Graphing a Modified Cotangent

Sketch a graph of one period of the function f( x )=4cot( π 8 x π 2 )2. f( x )=4cot( π 8 x π 2 )2.

Using the Graphs of Trigonometric Functions to Solve Real-World Problems

Many real-world scenarios represent periodic functions and may be modeled by trigonometric functions. As an example, let’s return to the scenario from the section opener. Have you ever observed the beam formed by the rotating light on a police car and wondered about the movement of the light beam itself across the wall? The periodic behavior of the distance the light shines as a function of time is obvious, but how do we determine the distance? We can use the tangent function.

Example 10

Using Trigonometric Functions to Solve Real-World Scenarios

Suppose the function y=5tan( π 4 t ) y=5tan( π 4 t )marks the distance in the movement of a light beam from the top of a police car across a wall where t tis the time in seconds and y yis the distance in feet from a point on the wall directly across from the police car.

  1. Find and interpret the stretching factor and period.
  2. Graph on the interval [ 0,5 ]. [ 0,5 ].
  3. Evaluate f( 1 ) f( 1 )and discuss the function’s value at that input.

Media

Access these online resources for additional instruction and practice with graphs of other trigonometric functions.

6.2 Section Exercises

Verbal

1.

Explain how the graph of the sine function can be used to graph y=cscx. y=cscx.

2.

How can the graph of y=cosx y=cosxbe used to construct the graph of y=secx? y=secx?

3.

Explain why the period of tanx tanxis equal to π. π.

4.

Why are there no intercepts on the graph of y=cscx? y=cscx?

5.

How does the period of y=cscx y=cscxcompare with the period of y=sinx? y=sinx?

Algebraic

For the following exercises, match each trigonometric function with one of the following graphs.

Figure 18
6.

f( x )=tanx f( x )=tanx

7.

f( x )=secx f( x )=secx

8.

f( x )=cscx f( x )=cscx

9.

f( x )=cotx f( x )=cotx

For the following exercises, find the period and horizontal shift of each of the functions.

10.

f( x )=2tan( 4x32 ) f( x )=2tan( 4x32 )

11.

h( x )=2sec( π 4 ( x+1 ) ) h( x )=2sec( π 4 ( x+1 ) )

12.

m( x )=6csc( π 3 x+π ) m( x )=6csc( π 3 x+π )

13.

If tanx=1.5, tanx=1.5,find tan( x ). tan( x ).

14.

If secx=2, secx=2,find sec( x ). sec( x ).

15.

If cscx=5, cscx=5,find csc( x ). csc( x ).

16.

If xsinx=2, xsinx=2,find ( x )sin( x ). ( x )sin( x ).

For the following exercises, rewrite each expression such that the argument x xis positive.

17.

cot( x )cos( x )+sin( x ) cot( x )cos( x )+sin( x )

18.

cos( x )+tan( x )sin( x ) cos( x )+tan( x )sin( x )

Graphical

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.

19.

f( x )=2tan( 4x32 ) f( x )=2tan( 4x32 )

20.

h( x )=2sec( π 4 ( x+1 ) ) h( x )=2sec( π 4 ( x+1 ) )

21.

m( x )=6csc( π 3 x+π ) m( x )=6csc( π 3 x+π )

22.

j( x )=tan( π 2 x ) j( x )=tan( π 2 x )

23.

p(x)=tan( x π 2 ) p(x)=tan( x π 2 )

24.

f(x)=4tan(x) f(x)=4tan(x)

25.

f(x)=tan( x+ π 4 ) f(x)=tan( x+ π 4 )

26.

f(x)=πtan( πxπ )π f(x)=πtan( πxπ )π

27.

f( x )=2csc( x ) f( x )=2csc( x )

28.

f( x )= 1 4 csc( x ) f( x )= 1 4 csc( x )

29.

f(x)=4sec( 3x ) f(x)=4sec( 3x )

30.

f(x)=3cot( 2x ) f(x)=3cot( 2x )

31.

f(x)=7sec( 5x ) f(x)=7sec( 5x )

32.

f(x)= 9 10 csc( πx ) f(x)= 9 10 csc( πx )

33.

f(x)=2csc( x+ π 4 )1 f(x)=2csc( x+ π 4 )1

34.

f(x)=sec( x π 3 )2 f(x)=sec( x π 3 )2

35.

f(x)= 7 5 csc( x π 4 ) f(x)= 7 5 csc( x π 4 )

36.

f(x)=5( cot( x+ π 2 )3 ) f(x)=5( cot( x+ π 2 )3 )

For the following exercises, find and graph two periods of the periodic function with the given stretching factor, | A |, | A |,period, and phase shift.

37.

A tangent curve, A=1, A=1,period of π 3 ; π 3 ;and phase shift ( h,k )=( π 4 ,2 ) ( h,k )=( π 4 ,2 )

38.

A tangent curve, A=−2, A=−2,period of π 4 , π 4 ,and phase shift ( h,k )=( π 4 ,−2 ) ( h,k )=( π 4 ,−2 )

For the following exercises, find an equation for the graph of each function.

39.
A graph of two periods of a modified cosecant function, with asymptotes at multiples of pi/2.
40.
A graph of a modified cotangent function. Vertical asymptotes at x=-1 and x=0 and x=1.
41.
A graph of a modified cosecant function. Vertical asymptotes at multiples of pi/4.
42.
A graph of a modified tangent function. Vertical asymptotes at -pi/8 and 3pi/8.
43.
A graph of a modified cosecant function. Vertical asymptotyes at multiples of pi.
44.
A graph of a modified secant function. Four vertical asymptotes.
45.
graph of two periods of a modified tangent function. Vertical asymptotes at x=-0.005 and x=0.005.

Technology

For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input cscx cscxas 1 sinx . 1 sinx .

46.

f(x)=| csc( x ) | f(x)=| csc( x ) |

47.

f(x)=| cot( x ) | f(x)=| cot( x ) |

48.

f(x)= 2 csc( x ) f(x)= 2 csc( x )

49.

f(x)= csc( x ) sec( x ) f(x)= csc( x ) sec( x )

50.

Graph f(x)=1+ sec 2 ( x ) tan 2 ( x ). f(x)=1+ sec 2 ( x ) tan 2 ( x ).What is the function shown in the graph?

51.

f(x)=sec( 0.001x ) f(x)=sec( 0.001x )

52.

f(x)=cot( 100πx ) f(x)=cot( 100πx )

53.

f(x)= sin 2 x+ cos 2 x f(x)= sin 2 x+ cos 2 x

Real-World Applications

54.

The function f( x )=20tan( π 10 x ) f( x )=20tan( π 10 x )marks the distance in the movement of a light beam from a police car across a wall for time x, x,in seconds, and distance f( x ), f( x ), in feet.

  1. Graph on the interval [ 0,5 ]. [ 0,5 ].
  2. Find and interpret the stretching factor, period, and asymptote.
  3. Evaluate f( 1 ) f( 1 )and f( 2.5 ) f( 2.5 )and discuss the function’s values at those inputs.
55.

Standing on the shore of a lake, a fisherman sights a boat far in the distance to his left. Let x, x,measured in radians, be the angle formed by the line of sight to the ship and a line due north from his position. Assume due north is 0 and x xis measured negative to the left and positive to the right. (See Figure 19.) The boat travels from due west to due east and, ignoring the curvature of the Earth, the distance d( x ), d( x ),in kilometers, from the fisherman to the boat is given by the function d( x )=1.5sec( x ). d( x )=1.5sec( x ).

  1. What is a reasonable domain for d( x )? d( x )?
  2. Graph d( x ) d( x )on this domain.
  3. Find and discuss the meaning of any vertical asymptotes on the graph of d( x ). d( x ).
  4. Calculate and interpret d( π 3 ). d( π 3 ).Round to the second decimal place.
  5. Calculate and interpret d( π 6 ). d( π 6 ).Round to the second decimal place.
  6. What is the minimum distance between the fisherman and the boat? When does this occur?
An illustration of a man and the distance he is away from a boat.
Figure 19
56.

A laser rangefinder is locked on a comet approaching Earth. The distance g( x ), g( x ),in kilometers, of the comet after x xdays, for x xin the interval 0 to 30 days, is given by g( x )=250,000csc( π 30 x ). g( x )=250,000csc( π 30 x ).

  1. Graph g( x ) g( x )on the interval [ 0,30 ]. [ 0,30 ].
  2. Evaluate g( 5 ) g( 5 ) and interpret the information.
  3. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond?
  4. Find and discuss the meaning of any vertical asymptotes.
57.

A video camera is focused on a rocket on a launching pad 2 miles from the camera. The angle of elevation from the ground to the rocket after x xseconds is π 120 x. π 120 x.

  1. Write a function expressing the altitude h( x ), h( x ),in miles, of the rocket above the ground after x xseconds. Ignore the curvature of the Earth.
  2. Graph h( x ) h( x )on the interval ( 0,60 ). ( 0,60 ).
  3. Evaluate and interpret the values h( 0 ) h( 0 )and h( 30 ). h( 30 ).
  4. What happens to the values of h( x ) h( x )as x x approaches 60 seconds? Interpret the meaning of this in terms of the problem.
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