Graphing Trigonometric Functions

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Graphing Trigonometric Functions

The previous lectures defined the trigonometric functions. This lecture discusses their graphs.

We will limit this introductory discussion to the graphs without transformation. We begin with the function, y  sin x . Note the subtle change in variable designation. Previously, x and y were variables used to calculate the output of a function sin  where  was the input. Now x

represents the input and y represents the output.

The input of the sine function is the measure of an angle in standard position. Allowing for multiple rotations and negative rotations, the measure of an angle in standard position can equal any real number, rational or irrational, negative or nonnegative. Moreover, for every angle measure, the output is a defined real number, Hence, the domain of sine equals , and we can assign x or t or any variable to represent domain values. Similarly, we can assign y or ^{f x}   ^or

any notation to represent range values.

Recall our original definition of the sine function. Consider the definition alongside the diagram.

The diagram reveals the basic inequality y  r . When y  r , the output,

_r^y

, equals a proper fraction. When y  r , then the output,

^y_r

, equals 1 or –1. The radius is always positive. Thus, the output

^y_r

is negative when y is negative. Likewise, the output

^y_r

is positive when y is

Primary Objective: Students sketch trigonometric functions in the Cartesian plane.

If  is an angle in standard position and   ^{x y} ^, is the point of

intersection of theta’s terminal side with x

²

 y

²

 r

²

, then sine maps the

measure of  to

^y_r

. We denote the sine function as sin  

_r^y

.

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positive. Hence, the range of sine equals the interval  ^ ^1,1  while the domain equals the interval

 ^{ } ^,  ^.

Recall our simplified definition of sine using the unit circle. Consider the definition alongside the diagram.

The value of sin  at the multiples of 30º and 45º reveal a repetitive pattern to the function especially when the reader reflects that all these same values will be repeated as the angle measures extend beyond 2  . For instance, sine has the same value at 13  6 as it does at  6 . This repetition creates a periodic function, one that repeats itself after a given interval called its period. We define a periodic function formally below.

A function ^{f x}   is periodic if  a number P such that ^{f x}   ^ ^{f x kP}  ^  ^for

integer values of k. The least period equals the smallest positive constant P.

If  is an angle in standard position and   ^{x y} ^, is the point of

intersection of the terminal side and the unit circle, x

²

 y

²

 1 , then we

define sine as sin   y .

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In the case of sine, we see that ^sin   ^x ^ ^sin  ^x ^ ² ^k ^  , so sine is a periodic function with a least period equal to 2  . One complete least period of the sine function is called a cycle. The interval

 ^ ^{ } ^,  , which has a length of 2  , is our fundamental cycle for y  sin x .

The wave pattern repeats continuously in both directions. This graph or any translation or dilation is a sine wave or a sinusoidal wave. The magnitude of the wave’s oscillation is its amplitude.

The other five trigonometric functions also represent periodic functions. Note that sine, cosine, secant, and cosecant all have a least period of 2  while tangent and cotangent have a least period of  . Only sine and cosine have real number amplitudes.

We leave graphing the remaining trigonometric functions as an exercise, but we first comment on a characteristic of the functions without real number amplitudes, namely, asymptotes, defined without technical language.

Amplitude equals the maximum displacement from a zero position. The

amplitude of a sine wave equals the absolute value of half the difference between the maximum and minimum y-values on the curve.

When the graph of a function mimics a relation, the mimicry is asymptotic

behavior, and the mimicked relation is an asymptote. If the asymptote is

linear, we call it a linear asymptote.

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Think about the reciprocal function y 

¹_x

. The reciprocal function maps inputs greater than 1 to proper fractions. Indeed, the larger the input, the smaller the output; hence, as the inputs

approach infinity, the outputs approach zero. Thus, as the x-values increase, the function y 

¹_x

mimics the horizontal line y  0 , making this line (the x-axis) a linear asymptote. In particular, we call y  0 a horizontal asymptote of y 

¹_x

.

More relevant to the trigonometric functions, the reciprocal function also has a vertical

asymptote. The reciprocals of proper fraction are numbers greater than one. The reciprocal of

¹₂

is 2. The reciprocal of

₁₀₀¹

is 100. The closer a positive number input is to zero, the further away the output is from zero. Thus, as positive x-values decrease to zero, the function y 

¹_x

mimics the vertical line x  0 , making this line (the y-axis) a linear asymptote. In particular, we call

0 x  a vertical asymptote of y 

¹_x

.

The graphs of tangent, cotangent, secant, and cosecant all have domain restrictions (which is

obvious from their definitions), and they all exhibit asymptotic behavior near these domain

restrictions. For example, cosecant, the reciprocal of sine, is undefined at x   since sine

equals zero at x   . Moreover, sine decreases to zero as x approaches  from the right, so its

reciprocal, cosecant, must grow larger and larger as x approaches  from the right, making the

vertical line x   an asymptote for the graph of cosecant.

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Example Exercise 1

Graphing Trigonometric Functions

Recall the Quotient Identity: tangent equals the ratio of sine to cosine. Generate a table of sine and cosine values.

Extending the table to

³₂^

using the convenient inputs on the unit circle diagram repeats the same pattern of outputs, so tangent has a period equal to  . We are using  

^{ }2

,

2

 as the fundamental cycle. Plotting the ordered pairs from the table and suspecting vertical asymptotes where tangent is undefined, we generate the graph.

Graph y  tan x .

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#1) #3)

#5) #7)

#9) cos θ  

¹²₁₃

and tan θ  

₁₂⁵

Graphing Trigonometric Functions

Graphing Trigonometric Functions

The previous lectures defined the trigonometric functions. This lecture discusses their graphs.

We will limit this introductory discussion to the graphs without transformation. We begin with the function, y  sin x . Note the subtle change in variable designation. Previously, x and y were variables used to calculate the output of a function sin  where  was the input. Now x

represents the input and y represents the output.

any notation to represent range values.

Recall our original definition of the sine function. Consider the definition alongside the diagram.

The diagram reveals the basic inequality y  r . When y  r , the output,

, equals a proper fraction. When y  r , then the output,

, equals 1 or –1. The radius is always positive. Thus, the output

is negative when y is negative. Likewise, the output

is positive when y is

Primary Objective: Students sketch trigonometric functions in the Cartesian plane.

If  is an angle in standard position and   x y , is the point of

intersection of theta’s terminal side with x

 y

 r

, then sine maps the

measure of  to

. We denote the sine function as sin  

.

positive. Hence, the range of sine equals the interval   1,1  while the domain equals the interval

   ,  .

Recall our simplified definition of sine using the unit circle. Consider the definition alongside the diagram.

A function f x   is periodic if  a number P such that f x    f x kP    for

integer values of k. The least period equals the smallest positive constant P.

If  is an angle in standard position and   x y , is the point of

intersection of the terminal side and the unit circle, x

 y

 1 , then we

define sine as sin   y .

In the case of sine, we see that sin   x  sin  x  2 k   , so sine is a periodic function with a least period equal to 2  . One complete least period of the sine function is called a cycle. The interval

    ,  , which has a length of 2  , is our fundamental cycle for y  sin x .

The wave pattern repeats continuously in both directions. This graph or any translation or dilation is a sine wave or a sinusoidal wave. The magnitude of the wave’s oscillation is its amplitude.

The other five trigonometric functions also represent periodic functions. Note that sine, cosine, secant, and cosecant all have a least period of 2  while tangent and cotangent have a least period of  . Only sine and cosine have real number amplitudes.

We leave graphing the remaining trigonometric functions as an exercise, but we first comment on a characteristic of the functions without real number amplitudes, namely, asymptotes, defined without technical language.

Amplitude equals the maximum displacement from a zero position. The

amplitude of a sine wave equals the absolute value of half the difference between the maximum and minimum y-values on the curve.

When the graph of a function mimics a relation, the mimicry is asymptotic

behavior, and the mimicked relation is an asymptote. If the asymptote is

linear, we call it a linear asymptote.

Think about the reciprocal function y 

. The reciprocal function maps inputs greater than 1 to proper fractions. Indeed, the larger the input, the smaller the output; hence, as the inputs

approach infinity, the outputs approach zero. Thus, as the x-values increase, the function y 

mimics the horizontal line y  0 , making this line (the x-axis) a linear asymptote. In particular, we call y  0 a horizontal asymptote of y 

.

More relevant to the trigonometric functions, the reciprocal function also has a vertical

asymptote. The reciprocals of proper fraction are numbers greater than one. The reciprocal of

is 2. The reciprocal of

is 100. The closer a positive number input is to zero, the further away the output is from zero. Thus, as positive x-values decrease to zero, the function y 

mimics the vertical line x  0 , making this line (the y-axis) a linear asymptote. In particular, we call

0

x  a vertical asymptote of y 

.

The graphs of tangent, cotangent, secant, and cosecant all have domain restrictions (which is

obvious from their definitions), and they all exhibit asymptotic behavior near these domain

restrictions. For example, cosecant, the reciprocal of sine, is undefined at x   since sine

equals zero at x   . Moreover, sine decreases to zero as x approaches  from the right, so its

reciprocal, cosecant, must grow larger and larger as x approaches  from the right, making the

vertical line x   an asymptote for the graph of cosecant.

Example Exercise 1

Graphing Trigonometric Functions

Recall the Quotient Identity: tangent equals the ratio of sine to cosine. Generate a table of sine and cosine values.

Extending the table to

using the convenient inputs on the unit circle diagram repeats the same pattern of outputs, so tangent has a period equal to  . We are using  

,

 as the fundamental cycle. Plotting the ordered pairs from the table and suspecting vertical asymptotes where tangent is undefined, we generate the graph.

Graph y  tan x .

#1) #3)

#5) #7)

#9) cos θ  

and tan θ  

Application Exercise

Practice Problems Graph the following functions.

#1 f x    sin x over     ,  #2 g x    csc x over     x 

#3 y  cos x over  0, 2   #4 y  sec x over 0   x 2 

#5 c x    cot x over   0, π #6 T x    tan x over   0, π

#7 y  cos x over     ,  #8 G x    sec x over     x 

Determine the values of cos θ and tan θ for the given θ .

#9 The terminal side of θ is in quadrant II, and sin θ 

If  is an angle in standard position and   ^{x y} ^, is the point of

positive. Hence, the range of sine equals the interval  ^ ^1,1  while the domain equals the interval

 ^{ } ^,  ^.

A function ^{f x}   is periodic if  a number P such that ^{f x}   ^ ^{f x kP}  ^  ^for

If  is an angle in standard position and   ^{x y} ^, is the point of

In the case of sine, we see that ^sin   ^x ^ ^sin  ^x ^ ² ^k ^  , so sine is a periodic function with a least period equal to 2  . One complete least period of the sine function is called a cycle. The interval

 ^ ^{ } ^,  , which has a length of 2  , is our fundamental cycle for y  sin x .

#1 ^{f x}   ^ ^sin ^x ^over  ^ ^{ } ^,  ^#2 ^{g x}   ^ ^csc ^x ^over ^{  } ^ ^x ^

#3 y  cos x over  ^{0, 2} ^  ^#4 ^y ^ ^sec ^x ^over ⁰ ^{ } ^x ² ^

#5 ^{c x}   ^ ^cot ^x ^over   ^0, ^π ^#6 ^{T x}   ^ ^tan ^x ^over   ^0, ^π

#7 y  cos x over  ^ ^{ } ^,  ^#8 ^{G x}   ^ ^sec ^x ^over ^{  } ^ ^x ^